Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms
Ga\"etan Chenevier, Olivier Ta\"ibi
TL;DR
This paper introduces a new method using the Weil explicit formula to evaluate discrete series multiplicities for classical groups over integers and classifies certain automorphic representations, simplifying computations in automorphic forms theory.
Contribution
A novel approach based on the Weil explicit formula for computing automorphic form multiplicities and classifying level one algebraic cusp forms with bounded motivic weight.
Findings
New method for multiplicity evaluation in automorphic forms
Classification of level one cuspidal automorphic representations with weight ≤ 24
Simplified computation of Siegel modular cuspform spaces
Abstract
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level automorphic forms of a split classical group over , and provide numerical applications in absolute rank . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of over ( arbitrary) whose motivic weight is . In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on , and that we push further on in this paper. We use these vanishing results to obtain an arguably ``effortless'' computation of the elliptic part of the geometric side of the trace formula of , for an appropriate test function.…
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic Geometry and Number Theory · Analytic Number Theory Research
Discrete series multiplicities for classical groups over and level algebraic cusp forms
Gaëtan Chenevier
CNRS, Université Paris-Sud
and
Olivier Taïbi
CNRS, École Normale Supérieure de Lyon
Abstract.
The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level automorphic forms of a split classical group over , and provide numerical applications in absolute rank . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of over ( arbitrary) whose motivic weight is .
In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on , and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of , for an appropriate test function.
Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for : we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.
Gaëtan Chenevier and Olivier Taïbi are supported by the C.N.R.S. and by the project ANR-14-CE25.
Contents
-
2 Weil’s explicit formula for Rankin-Selberg -functions: a refined positivity criterion
-
2.1 Algebraic Harish-Chandra modules and automorphic representations.
-
2.3 The explicit formula for -functions of pairs of elements in
-
2.4.7 Another illustration: a strengthening of a vanishing result in [CvdG18]
-
3.1 Conjugacy classes and characteristic polynomials: elementary observations
-
3.2 Conjugacy classes and characteristic polynomials in the orthogonal case: spinor norms
-
3.3 Non-existence of level one regular algebraic automorphic cuspidal representations
-
5.2 Lowest-weight modules: results of Arancibia-Moeglin-Renard and of Moeglin-Renard
1. Introduction
1.1. Siegel modular forms for
We denote by and respectively the space of cuspidal Siegel modular forms for the full Siegel modular group , which are either scalar-valued of weight , or more generally vector-valued of weight in with (we refer to [vdG08] for a general introduction to Siegel modular forms). Recall that trivially vanishes when is odd, and also for (Freitag, Reznikoff, Weissauer).
The question of determining the dimension of , very classical for , has a long and rich history for . It has first been attacked for using geometric methods, in which case concrete formulas were obtained by Igusa (1962) in the scalar-valued case, and by Tsushima (1983) for the weights111More precisely, Tsushima could only prove that his formula works for , and later Petersen (2013) and Taïbi (2016) independently showed that it holds as well for and , as conjectured by Ibukiyama. . There is still no known formula for , although we have for (Ibukiyama, Skoruppa): see [CvdG18] for a discussion of these singular cases. An analogue of Igusa’s result for was proved by Tsuyumine in 1986, but only quite recently a conjectural formula was proposed by Bergström, Faber and van der Geer, in the vector valued case , based on counting genus three curves over finite fields (2011). Their formula, and more generally a formula222These are pretty huge formulas, which can’t be printed here already for , but see Theorem A loc. cit. for their general shape. for for arbitrary and was proved by the second author in [Taï17], using a method that we will recall in § 1.4. Actually, the general formulas given in [Taï17] apply to any genus and any weights with . However, they involve certain rational numbers, that we shall refer to later as masses, that are rather difficult to compute; Taïbi provided loc. cit. a number of algorithms to determine them (more precisely, certain local orbital integrals, see § 1.4) that allowed him to numerically compute those masses for .
Our first main result in this paper is a completely different and comparatively much easier method to compute the aforementioned masses. This method allows us to recover, in a uniform and rather “effortless” way, all the computations of masses done in [Taï17] for , and even to go further:
Theorem 1**.**
There is an explicit and implemented formula computing for any , and any with .
See Theorems 6 & 7 for equivalent, better formulated, statements. Theorem 1 is about Siegel modular forms of arbitrary weights such that , but with genus . A second result concerns the Siegel modular forms of arbitrary genus, but of weights (there are really finitely many relevant pairs here). It is very much in the spirit of the determination of by Chenevier-Lannes in [CL19] in the cases .
Theorem 2**.**
The dimension of for , and non scalar, is given by Table 5. The dimension of for any and any is given by Table 6.
The notations for these tables are explained in §5.3. Table 5 shows in particular that has dimension for non scalar and , and is nonzero for exactly values of . Table 6 includes the fact that vanishes whenever and , except in the three following situations:
[TABLE]
(a nonzero element in the first and last spaces has been constructed in [BFW98] and [Fre82]). We obtain for instance the following result.
Corollary 1**.**
* has dimension for , and [math] otherwise.*
An inspection of standard -functions, and general results of Böcherer, Kudla-Rallis and Weissauer, show that these four spaces are spanned by certain Siegel theta series build on Niemeier lattices (see §5.4). In a companion paper [CT19a] we come back to these constructions and study them in a much more elementary way. Combined with [CL19, Sect. 9.5], this provides an explicit construction of all the cuspidal Siegel modular eigenforms of weight and level for an arbitrary genus . In §5.4, we also prove that Eichler’s basis problem holds in weights and for arbitrary genus , completing the results of [CL19] for .
Last but not least, let us mention that in the past, several other authors have computed for a number of isolated and small pairs , sometimes with much efforts, e.g. Igusa, Witt, Erokhin, Duke-Immamoglu, 333We warn the reader that the proofs of Duke and Immamoglu in [DI98] are not valid in the case since they rely on the incorrect Corollary 3 p. 601 in [Miz91]: see [Miz19] for a recent erratum. Note also that a preliminary version of [CL19] did include a proof of the vanishing of for , and , but this statement was deleted in the published version for the same reason. Our proofs here show that all these incriminated results for were nevertheless correct, and actually do not rely anymore on the results in [Miz91]. Nebe-Venkov, and Poor-Yuen among others. We would like to stress that none of the results of this paper depend on a previous computation of the dimension of a space of Siegel modular forms, not even of ! see §1.4. Moreover, as far as we know, the dimensions given by Theorems 1 and 2 seem to recover all the previously known444More precisely, the only cases not covered by these two theorems seem to be the vanishing of for , and for the pairs with [CvdG18, Thm. 1.3]. However, this vanishing for can also be proved by arguments in the spirit of the ones employed here, as explained by the first author in an appendix of [CvdG18], and we can actually prove it as well for all with : see §2.4.7. dimensions of spaces of Siegel modular cuspforms for .
Our proof of Theorems 1 and 2 will use automorphic methods, building on a strategy developed in the recent works [CR15, CL19, Taï17]: we will review this strategy in §1.4 and §5. An important ingredient is Arthur’s endoscopic classification of the discrete automorphic spectrum of classical groups in terms of general linear groups [Art13], including the so-called multiplicity formula. A special feature of this approach is that even if we were only interested in , we would be forced to compute first the dimension of various spaces of automorphic forms for all the split classical groups over of smaller dimension. By a split classical group over we will mean here either the group scheme over , or the special orthogonal group scheme of the integral quadratic form (case even ) or (case odd) over . An important gain of this approach, however, is that in the end we do not only compute , but also the dimension of its subspace of cuspforms of any possible endoscopic type, a quantity which is arguably more interesting than the whole dimension itself: see Tables 5 & 6 for a sample of results.
1.2. Level one algebraic cusp forms on
Let be an integer and a cuspidal automorphic representation of over . We say that has level if is unramified for each prime . We say that is algebraic if the infinitesimal character of , that we may view following Harish-Chandra and Langlands as a semi-simple conjugacy class in , has its eigenvalues in , say , and with . Those eigenvalues are called the weights of , and the important integer is called the motivic weight of . The Jacquet-Shalika estimates imply for all , and in particular, (see §2.1).
The algebraic cuspidal are especially interesting to number theorists, as for such a standard conjectures (by Clozel, Langlands) predict the existence of a compatible system of pure and irreducible -adic Galois representations with same -function as , the Hodge-Tate weights of being the , and its Deligne weight being . The level assumption in this work has to be thought as a simplifying, but still interesting, one (see [CL19] for several motivations).
An important problem is thus the following. For an integer , we denote by the set of in with , and for all .
Problem 1**.**
For , determine the (finite) number of level cuspidal algebraic automorphic representations of whose weights are the , and the number of those satisfying furthermore (self-duality).
Let us say that an element in is regular if for all we have either , or , and (hence ). Despite appearences, the question of determining the for regular is very close to that discussed in § 1.1. Indeed, as was observed and used in [CR15, CL19, Taï17], the level self-dual of regular weights are the exact building blocks for Arthur’s endoscopic classification of the discrete automorphic representations of split classical groups over which are unramified at all finite places and discrete series at the Archimedean place (with a very concrete form of Arthur’s multiplicity formula, relying on [AMgR18]). As an illustration of this slogan, the following fact was observed in [CR15, Chap 9].
Key fact 1. Fix and with . Then the dimension of is an explicit function of the (finitely many) quantities with regular, and .
See [CR15, Prop. 1.11] for explicit formulas for , [Taï17, Chap. 5] for , and [CL19, Thm. 5.2] for the general recipe. This general recipe will actually be recalled in §5, in which we will also apply the recent results of [MR] to give an analoguous statement for (scalar-valued case) in the case . This last case is quite more sophisticated, in particular it also involves certain slightly irregular weights. We will come back later on the relations between the Problem above and Theorems 1 and 2.
1.3. Classification and inexistence results
Let us denote by the set of cuspidal automorphic representations of , with arbitrary, which are algebraic and of level . The second main result of this paper is a partial classification of elements in having motivic weight . The first statement of this type, proved in [CL19, Thm. F], asserts that there are exactly elements in of motivic weight : the trivial representation of , the representations of generated by the -dimensional spaces for (whose weights are ), the Gelbart-Jacquet symmetric square of (with weights ), and four other -dimensional self-dual with respective weights
[TABLE]
In this paper we significantly simplify the proof of [CL19, Thm. F]: see §2.4.6. More importantly, these simplifications allow us to prove the following theorems for motivic weights and .
Theorem 3**.**
There are exactly level cuspidal algebraic automorphic representations of over , with varying, with motivic weight , and having the weight with multiplicity :
- (i)
* representations of generated by the eigenforms in ,*
- (ii)
* representations of of weights with or ,*
- (iii)
* representations of of weights with*
[TABLE]
- (iv)
* representation of of weights .*
There are all self-dual (symplectic) and uniquely determined by their weights.
The representations in (ii) and (iii) above were first discovered in [CR15], using some conditional argument that was later made unconditional in [Taï19]. Their existence was also confirmed by the different computation of the second author in [Taï17], who also discovered the -dimensional form in (iv).
Despite our efforts, we have not been able to classify the of motivic weight such that multiplicity of the weight is . We could only prove that there is an explicit list of weights with such that (a) the weight of any such belongs to , (b) for any in there is at most one with this weight (necessarily self-dual symplectic), except for the single weight in with , for which there might also be two such which are the dual of each other: see Proposition 4.1. Nevertheless, we conjecture that all of those putative representations do not exist, except perhaps one. Indeed, we prove in §4.3 the following result, assuming a suitable form of (GRH).
Theorem 4**.**
Assume and that there exists in with motivic weight and having the weight with multiplicity . Then we have , the weights of are , , , , , as well as with multiplicity , and is the unique element of having these weights.
We now state our (weaker) classification result in motivic weight .
Theorem 5**.**
There are exactly level algebraic cuspidal self-dual automorphic representations of over , with varying, with motivic weight and a regular weight. They are self-dual (orthogonal) with respective set of weights
[TABLE]
Again, those three forms were first discovered in [CR15, Cor. 1.10 & 1.12] and confirmed in [Taï17]. Interestingly, as explained in [CR15], we expect that their Sato-Tate groups are respectively the compact groups , and . See [Che19, Thm. 6.12] for a proof that the first form, which is -dimensional, has -valued -adic Galois representations.
Proofs. Our proofs of Theorems 3 and 5 are in the same spirit of the one of [CL19, Thm. F]. As already said, all the representations appearing in the theorem were already known to exist by the works [CR15, Taï19, Taï17] (and we will give another proof of their existence in §3), so the main problem is to show that there are no others. The basic idea that we will use for doing so is to consider an hypothetical , consider an associate -function of , and show that this function cannot exist by applying to it the so-called explicit formula for suitable test functions. This is a classical method by now, inspired by the pioneering works of Stark, Odlyzko and Serre on discriminant bounds of number fields. It was developed by Mestre in [Mes86] and applied loc. cit. to the standard -function of (see also [Fer96]), then by Miller [Mil02] to the Rankin-Selberg -function of , and developed further more recently in [CL19, §9.3.4].
Two important novelties were discovered in [CL19] in order to obtain the aforementioned classification result in motivic weight . The first one, developed further in [Chea], is a finiteness result which implies that there are only finitely many level cuspidal algebraic automorphic representations of , with varying, of motivic weight . This finiteness is also valid in motivic weight assuming a suitable form of GRH. This result is effective and produces a finite but large list of possible weights for those (for instance, it leads to possible weights in motivic weight ). The hardest part is then to eliminate most of those remaining weights. The second novelty found in [CL19] was the observation that we obtain efficient constraints by applying as well the explicit formula to all the -functions where the are the known representations. See [CL19, Scholia 9.3.26 & 9.3.32] for the two useful criteria obtained there using this idea.
In this paper, we discovered a criterion that may be seen as a natural generalisation of [CL19, Scholia 9.3.26], and that happened to be (in practice, and quite surprisingly) much more efficient than the aforementioned ones. Moreover, contrary to [CL19, Scholia 9.3.32], we do not need to know any Satake parameter for the known elements of (which allow us to use test functions with aribtrary supports). Our basic idea here is to apply the explicit formula to the Rankin-Selberg -function of all linear combinations where is unknown of given weights, the with are known (in the sense that they exist and we know their weights), and the are arbitrary nonnegative real numbers. More precisely, for any test function we associate a certain symmetric bilinear form on the free vector space over , which represents the computable part of the explicit formula for the test function . Assuming a certain positivity assumption on , the quadratic form is then on the cone of generated by : see Proposition 2.2. In order to reach a contradiction we have thus to show that at least one quadratic form takes a negative value on the cone generated by . See §2.4 for the description of the minimisation algorithm that we have used for this purpose, as well as the homepage [CT19b] for related sources. One charm of this method is that although it requires some computational work to find a concrete element of that cone and a test function leading to a contradiction (and all is fair for that!), once we have found it it is quick and easy to rigorously check that we have : see §2.4.3.
1.4. The effortless computation of masses
Fix a rank split classical group over in the sense of §1.1. In other words, belongs to one of the families
[TABLE]
Assume that has discrete series, i.e. that is not isomorphic to with odd, and fix a maximal compact subgroup of . There is an analogue of Key fact with replaced by the multiplicity of any discrete series of in the space of -finite square-integrable automorphic forms over . In this paper, as in [Taï17], we will use a variant of this statement involving rather certain Euler-Poincaré characteristics. Our first aim is to state this variant (Key fact 2 below). For any dominant weight of , we denote by an irreducible representation of with highest weight and consider
[TABLE]
where denotes -cohomology. Attached to is a certain integer denoted , defined as the dimension of the standard representation of , the Langlands dual group of : concretely, we have for , and for or . The infinitesimal character of , namely “”, defines a unique regular element in with . Concretely, using the standard555The conditions on the are the following: for each , , and either (cases or ) or (case with ). notation for dominant weights of classical groups (as in [Taï17, §2]), is explicitely given by the following formulas:
[TABLE]
The promised second key fact, explained in §§4.1 and 4.2 of [Taï17] is:
Key fact 2. Fix and as above, and set . Assume we know for all regular with and , then it is equivalent to know and .
It follows from the formula above for that any regular in , with arbitrary, is of the form for a unique split classical group over and some dominant weight of . As a consequence, Key fact 2 paves the way for a computation of for all regular , by induction on .
Contrary to the case of Key fact 1, we will not reproduce here the precise claimed relation between and the quantities stated in Key fact 2, and simply refer to [Taï17, §4]. As for Key fact 1, it crucially depends on Arthur’s endoscopic classification [Art13] and on the explicit description of Archimedean Arthur packets in the regular algebraic case given by [AJ87] and [AMgR18] (they are the so-called Adams-Johnson packets, see [Taï17, §4.2.2]). The full combinatorics, when written down explicitly and case-by-case, are rather complicated but computable, and implemented since [Taï17].
The second ingredient is Arthur’s -Lefschetz trace formula developed in [Art89] applied to the relevant test function. A detailed analysis of this formula has been made in [Taï17] that we will follow below. We use this version of the trace formula as it is the one with the simplest geometric terms. Its spectral side is exactly , a quantity in principle harder to interpret (the price to pay for a simple geometric side), but this can be done precisely by Key fact 2 above.
We fix a Haar measure on , with the adèle ring of , such that the Haar measure on gives the volume . Fix a dominant weight of . We apply Arthur’s formula [Art88] to a test function of the form , where is the characteristic function of , and where is the sum, over all the discrete series of with same infinitesimal character as , of a pseudocoefficient of . According to [Art89] the resulting identity, which only depends on and , is
[TABLE]
The geometric side is a finite sum of terms indexed by Levi subgroups of of the form
[TABLE]
with a split classical groups over and . The main term, corresponding to itself and called the elliptic term, is
[TABLE]
where runs over representatives of the (finitely many) -conjugacy classes of finite order elements in whose -conjugacy class meets for each prime (see §3.1). For each such , we have denoted by its centralizer in (defined over ), choosed on a signed Haar measure with an Euler-Poincaré measure on (in the sense of [Ser71]), and we have denoted by the product over all primes of the classical orbital integrals
[TABLE]
Taïbi developped in [Taï17] a number of algorithms to enumerate the , compute their local orbital integrals (with Gross’s canonical measure), and the associated global volumes. Here we shall simply write
[TABLE]
where denotes the set of -conjugacy classes of finite order elements in (essentially, a characteristic polynomial: see §3.1) and is a certain number depending only on and called the mass of (note that is well-defined). By definition, is a concrete sum of volumes times global integral orbitals; it essentially follows from [Gro97, Theorem 9.9] and Siegel’s theorem on the rationality of the values of Artin L-functions at non-negative integers [Sie69] that we have .
The character of may be either evaluated using the (degenerate) Weyl character formula as in [CR15], or much more efficiently for small using Koike-Terrada’s formulas [KT87] as was observed in [Cheb]. Last but not least, the term in the sum defining corresponding to a proper Levi subgroup of the form (1.4.2) is expressed in terms of , as well as if , for suitable auxilliary : see [Taï17, §3.3.4] for the concrete formulas, that will not be repeated here. As a consequence, the key problem is to be able to compute the masses for .
The strategy. We are finally able to explain our strategy. Fix an integer. We may assume, by induction, that we have computed the masses for all split classical groups over such that has discrete series and , and all . By the Key fact 2 and the trace formula (1.4.1), note that we have an explicit and computable formula for for all regular with . Fix a split classical group over such that has discrete series and . Assume we have found a finite set of dominant weights of with the following two properties:
(P1) For all we have .
(P2) the matrix has rank .
Then from (P1) and the Key fact 2 we know for all . It follows that for all we know as well, by the trace formula (1.4.1), hence also since the non-elliptic geometric terms are also known by induction. By (1.4.5) and (P2), we deduce the masses for all by solving a linear system. As a consequence, for an arbitrary dominant weight of we may then compute , hence by (1.4.1), and by Key fact 2.
Amusingly, the conclusion is that we end by proving the existence of self-dual cusp forms for mostly by showing that many others do not exist ! (namely the ones with weights of the form with in .)
A simple example. Let us illustrate this method in the (admittedly too) simple case and . In this case has classes, of respective order and : say has order . The dominant weights of are of the form for a unique integer , and will be simply denoted by . For , we have , , the analysis of the spectral side gives
[TABLE]
with the Kronecker symbol, and the geometric side is
[TABLE]
Assume we know that there is no cuspidal modular form for of usual weight . This may for instance be shown by applying the explicit formula to the Hecke -function of a putative eigenform of such a weight, as observed in [Mes86, Rem. III.1]. This also follows very easily from the methods of §1.3. Using and the identity for we obtain with the linear system
[TABLE]
Luckily, the matrix on the left-hand side is invertible: we find , , and . As a consequence, we recover the classical formula for .
Remark 1.1**.**
In certain classes in , there is no element whose -conjugacy class meets for each and such that has discrete series: this forces by (1.4.3). This actually explains above.
This remark will lead us in §3.2, and following [Taï17, Remark 3.2.8], to replace by a smaller set , and to rather apply our strategy with replaced by in (P2). See also §3.1 for other more elementary reductions, using the center of or an outer automorphism of .
The crucial last ingredient for this method to work is to be able to find sufficiently many such that . We will use of course for this the method explained in §1.3 (the explicit formula for Rankin-Selberg -functions). Rather miraculously, it provides enough vanishing results up to rather a high rank: see §3.3 for a proof of the following final theorem, obtained by applying only this “effortless” strategy:
Theorem 6**.**
(“Effortless” computation) Assume with , or with . Then the masses , for all , are given in [CT19b].
As already said, these results are in accordance with all the orbital integral computations done in [Taï17]. This method is both conceptually simpler and faster: for comparison, computing all the orbital integrals for takes several weeks, whereas finding and solving the linear system to determine all the , for only takes minutes on the same computer. The cases with are new. Last but not least, if we combine the methods of this paper with Taïbi’s, we obtain the following new result in the symplectic case (see §3.3).
Theorem 7**.**
The masses , for all , are given in [CT19b].
Following Key fact 2, these two theorems allow to compute for all those and an arbitrary weight , as well as the quantity for any for : see loc. cit. for tables.
2. Weil’s explicit formula for Rankin-Selberg -functions: a refined
positivity criterion
2.1. Algebraic Harish-Chandra modules and automorphic representations.
Let be a cuspidal automorphic representation of over . As recalled in §1.2, we say that has level if is unramified for every prime . We also explained loc. cit. what it means for to be algebraic. It will be useful to have an alternative point of view on this last condition in terms of the Langlands correspondence (see e.g. [CL19, §8.2.12]). Recall that denotes the set of level algebraic cuspidal automorphic representations of (with varying).
We denote by the Weil group of : we have , where is the element of and with for all . The Langlands correspondence for is a natural bijection between the set of isomorphism classes of irreducible admissible Harish-Chandra modules for , and the set of isomorphism classes of -dimensional (complex, continuous and semi-simple) representations of [Kna94]. We say that the Harish-Chandra module is algebraic if every element in the center of acts as a homothety with factor in . In particular, factors through the (compact) quotient of by , which is an extension of by the unit circle. We denote by the trivial representation of , by its unique order character, and for we set where . Up to isomorphism, the irreducible representations of trivial on are
[TABLE]
We also have , and if and only if .
If is a cuspidal automorphic representation of over , Clozel’s purity lemma (or the Archimedean Jacquet-Shalika estimates) shows that the Harish-Chandra module is algebraic in the sense above if, and only if, is algebraic in the sense of §1.2 [CL19, Prop. 8.2.13]. Moreover, for all the multiplicity of the weight of is the same as the multiplicity of the character in the restriction of to . In other words, all the weights [math] (resp. with ) of are explained by occurences of or (resp. of ) in . It will be convenient to introduce:
- •
the Grothendieck ring of complex, continuous, finite dimensional, representations of which are trivial on its central subgroup ,
- •
for an integer, the sugroup of generated by the with and , and also by and in the case is even.
An element of will be called effective if it is the class of a representation of . The weights of such a are the in such that occurs in , counted with the multiplicity of in (a nonnegative number as is effective).
It follows from this discussion that for in , we have for some integer , and that the smallest such integer coincides with the motivic weight of introduced in §1.2. Moreover, the weights of are that of .
For later use, we now recall Langlands’ definition for the -factors and -factors of algebraic Harish-Chandra modules. The -factor of an element in is a meromorphic function on the whole complex plane, characterized by the additivity property and the following axioms [Tat79]:
[TABLE]
in which is the classical gamma function. Similarly,666What we denote by here is what Tate denotes in [Tat79, (3.4)], with choice of additive character , and with the standard Lebesgue measure on . the -factor of is the element of characterized by the additivity property and the identities and for every integer .
2.2. Regular and self-dual elements of
Let be a level algebraic cuspidal automorphic representation of over . We will say that is regular if the representation of is multiplicity free. It is thus equivalent to say that for each weight of , either has multiplicity or we have and contains both and with multiplicity . This latter case can only occur of course if both the motivic weight of and are even. Moreover, we observe that:
- •
if all the nonzero weights of have multiplicity , and if the weight [math] has multiplicity , then is regular if, and only if, we have .
- •
is regular if, and only if, the vector , where are the weights of , is regular in the sense of §1.2.
- •
if is regular and is odd, then contains .
Indeed, as a trivial central character, we must have , and we conclude by the formula for .
Assume now that is self-dual, that is, isomorphic to its contragredient (or dual) . Then is either symplectic or orthogonal in the sense of Arthur [Art13, Thm. 1.4.1]. Moreover, if is symplectic (resp. orthogonal) then preserves a nondegenerate alternating (resp. symmetric) pairing [Art13, Thm. 1.4.2]. In particular, if is self-dual, and if some weight of has multiplicity (e.g. if is regular), then is symplectic if and only if is odd [CL19, Prop. 8.3.3].
2.3. The explicit formula for -functions of pairs of elements in
Let and be level cuspidal automorphic representations of and respectively. For prime we denote by the semi-simple conjugacy class in associated with the unramified representation , following Langlands, under the Satake isomorphism. The Rankin-Selberg -function of and is the Euler product
[TABLE]
By fundamental works of Jacquet, Piatetski-Shapiro, and Shalika [JS81, JPSS83], this Euler product is absolutely convergent for , and the completed -function
[TABLE]
has a meromorphic continuation to and a functional equation of the form
[TABLE]
where is a certain nonzero complex number (it does not depend on as has level ). We set .
Assuming and are algebraic, the only case of interest here, the factor in (2.3.1) is given by the recipe recalled in §2.1, and we simply have
[TABLE]
Note that the ring structure of is determined by the relations and .
By Moeglin and Waldspurger [MW89, App.], is entire unless we have , in which case the only poles are simple and at . Moreover, is bounded in vertical strips away from its poles by Gelbart and Shahidi [GS01]. All those analytic properties are key to establishing the Weil explicit formula (for which we refer to Poitou [Poi77b, §1]) in this context. The general formalism of Mestre [Mes86, §I] applies verbatim: we refer to [CL19, Chap. 9, Sect. 3] for the details and only recall here what we need to prove our criterion.
We denote by the -vector space with basis . We fix a test function , that is an even function satisfying the axioms (i), (ii) and (iii) of [Mes86, §I.2] with the constant loc. cit. equal to [math] (see also [Poi77b, §1]). The reader will not lose anything here assuming simply that is compactly supported and of class . We denote by the Fourier transform of , with the convention . Following [CL19, Chap. 9, Sect. 3], we first define five symmetric bilinear forms on , that we denote by , , , and . The first three of them depend on the choice of . They are uniquely determined by their values on any :
(a) , the sum being over all primes and integers .
(b) , where is the linear map defined by
[TABLE]
We will also denote abusively by the real-valued symmetric bilinear form on defined by . With these abusive notations we have .
(c) is the limit of , the sum being over the zeros of with and , when the real number goes to .
(d) if , and otherwise (Kronecker symbol).
(e) if and are self-dual with , and otherwise.
The main result is that for any test function we have the equality of bilinear forms
[TABLE]
on the space : see [Mes86, §I.2] and [CL19, Prop. 9.3.9]. We finally define a last bilinear form on by the formula
[TABLE]
In our applications, it will represent the "computable" part of the explicit formula. Note that for any test function , both and are real numbers, and if is non-negative then they are both non-negative.
Definition 2.1**.**
Let be a test function. We will say that satisfies (POS) if we have for all , and for all with .
Proposition 2.2**.**
Let be a test function satisfying (POS). Then for any integer , any in and any nonnegative real numbers , we have
[TABLE]
*Proof. * By density of the rationals in , we may assume that the are rational numbers, and even that they are integers by homogeneity of the quadratic form . But in this case, the statement is [CL19, Cor. 9.3.12]. As the proof is very simple, we give a direct argument. By (2.3.5) we have . By definition (a) and the assumption , the symmetric bilinear form is positive (semi-definite) on . It is thus enough to show that has nonnegative coefficients in the natural basis of , i.e. that we have for all . But this follows from the definition of , the assumption on , and the fact that if we have then has a zero at by the functional equation (2.3.2).
2.4. Applications
In what follows we will apply Proposition 2.2 to disprove the existence of representations in such that is a given algebraic representation, using the knowledge that there are representations in with known Archimedean components.
2.4.1. The basic inequalities
Before doing so, we first recall the following basic but important consequence of the explicit formula, that we derive here as a very special case of Proposition 2.2 (see also [CL19, Cor. 9.3.12 & 9.3.14]).
Corollary 2.3**.**
Let be a test function satisfying (POS) and fix in . If there is an element in with then we have the inequality
[TABLE]
More generally, if there are distinct elements in with for all , then we have
[TABLE]
*Proof. * Consider the element of . We have by Proposition 2.2. We clearly have777 We actually have . Indeed, if are in with , then . To see this, we may assume and are self-dual, either both symplectic or both orthogonal (they have the same motivic weight by assumption), and the assertion follows then from the general property proved in [Art13, Thm. 1.5.3 (b)]. Alternatively, we can easily check for . , by the inequality . We conclude by the equalities and .
Establishing inequality (2.4.1) is the original application of the explicit formula for Rankin-Selberg -function to prove the nonexistence of certain in with given . It was used by 888In the context of Artin -functions, the advantages of considering Rankin-Selberg -functions had already been noticed by Serre, see [Poi77a] p. 150. Miller in [Mil02] to show that for there is no in such that is either or . As explained in [CL19, Sect. 9.3] and [Chea], the simple inequality (2.4.1) is very constraining generally in motivic weight : for suitable the bilinear form is positive definite on for , and there is an explicit finite list of elements of such that whenever is in with , there is no in with . It has however some limitations: as we shall see, the list is quite large, and far from optimal. For instance, it does not seem possible to exclude this way the possibility999An intuitive reason for that is that there actually exists a in with very close weights, namely , with . See the discussion in [CL19, Sect. 9.3.19] for many other examples (and how to deal with this case differently), which allow to develop some intuition.
. Nevertheless, Inequality (2.4.1) will be extremely helpful to us in §3.3 and Sect. 4. Inequality (2.4.2) was first observed by Taïbi. In the case , it may be seen as an effective form of Harish-Chandra’s finiteness theorem. We will often use it to show that there is at most one in with given ; note that such a has to be self-dual if it exists, as we have .
2.4.2. A general method
For in , set if is self-dual, and otherwise. In this section, we will develop a method trying to answer in the negative the following question.
Question 2.4**.**
Fix an integer , and for each elements in and in . Does there exist distinct representations , …, in with and for each ?
To do so, assume we are given an integer and for each , elements in , in , and an integer . In other words, we fix a quadruple
[TABLE]
with in , in and in . To the choice of and of a test function , we associate the symmetric bilinear form on the standard Euclidean space , defined by the formula
[TABLE]
where is the standard orthonormal basis of . We will discuss the practical numerical evaluation of (i.e. of , and ) in §2.4.3. Set .
Problem 2**.**
Fix a test function and a quadruple as in (2.4.3). Determine wether the map takes a negative value on .
The relationship between Question 2.4 and this problem (which does not involve automorphic representations) is the following. Suppose for each (the general will play a role only later). Assume there are distinct in with and for each . Denote viewed as an Euclidean space with orthonormal basis . As the are distinct we actually have for all . We also have
[TABLE]
In other words, the linear map defined by is an isometry satisfying for all . If we are able to find an element with , then the element contadicts Proposition 2.2: we have answered Question 2.4 in the negative.
From now on we thus focus on Problem 2. We fix an arbitrary quadruple as in (2.4.3) and a test function . To simplify the notations we also set and . Let us introduce, for each non-empty :
- •
the subspace of , the intersection and its interior . We have .
- •
the minimal eigenvalue of the Gram matrix of the restriction of to , and the corresponding eigenspace .
We also denote by the minimum of on .
Proposition 2.5**.**
Fix a test function and a quadruple as in (2.4.3). Let be the set of non-empty such that intersects . Then is non-empty and we have .
Proof.
The minimum of on the compact is reached in for some . By Lemma 2.6 below applied to and , any local minimum of on is an eigenvector for and we have . We have , and the other inequality for any is obvious. ∎
Lemma 2.6**.**
Let be an Euclidean space with scalar product , its unit sphere, a symmetric bilinear form on and the (symmetric) endomorphism of satisfying for all in . Assume that the map has a local minimum at the element in . Then is an eigenvector of whose eigenvalue is the minimal eigenvalue of .
*Proof. * Set . We have when goes to [math] in . As is a local minimum of , this shows for all in . So is an eigenvector of . Denote by be the corresponding eigenvalue. Assume has an eigenvalue , and choose in with . We have and for all , a contradiction.
Example 2.7**.**
Assume and set (\beta_{\mathcal{Q}}^{F}(e_{i},e_{i}))_{1\leq i,j\leq 2}={\tiny\left(\begin{array}[]{cc}a&b\\ b&c\end{array}\right)}. We have and . We may assume and are , otherwise Problem 2 is solved. For , the eigenvalue is if and only if the determinant is . In this case, we have and the eigenspace is a line: we easily check that this line meets if and only if . Proposition 2.5 implies that assuming and , or equivalently , we have .
Lemma 2.8**.**
Fix a test function and a quadruple as in (2.4.3). Assume , , as well as for some indices . Then any element in with satisfies .
*Proof. * Set . Consider the set ; then is convex and we have for . Fix with . An inspection of Formula (2.4.4) shows that for any real numbers we have
[TABLE]
The set
[TABLE]
is a compact interval in with end points and . By assumption we have , and so the minimum of (2.4.6) on (2.4.7) is reached for . If we assume then belongs to , a contradiction.
This lemma leads to the following considerations. Start with a quadruple with the property for . Assume we have a partition
[TABLE]
with the property that for each , and each , we have . Consider the new quadruple where for each we define (resp. ) as the element (resp. ) with (this does not depend on the choice of such an ), and set We have a natural inclusion
[TABLE]
sending to for each . This embedding is an isometry for the standard Euclidean structures on both sides, and it follows from Formula (2.4.4) that we have for all and all test functions . Lemma 2.8 shows then (the inequality being obvious):
Corollary 2.9**.**
Let and be as above, and fix a test function with . We have if and only if , and if these inequalities hold we have
[TABLE]
Remark 2.10**.**
Assume we have two quadruples of the form and with for each . Choose a test function with . Then we have for all in by Formula (2.4.4). This shows . In particular, implies .
2.4.3. A digression on numerical evaluation
Before discussing the natural algorithm that follows from Propositions 2.5 and Corollary 2.9, let us discuss the numerical evaluation of the bilinear form . Given a test function , we will have to be able to compute with enough and certified precision the quantities
[TABLE]
It amounts to computing certain indefinite integrals. Numerical integration routines of computer packages such as PARI allow to compute approximations of such integrals, with increasing and in principle arbitrarily large accuracy. Although these routines have been very useful in our preliminary computations, and experimentally return highly accurate values when properly used, it would be delicate to rigorously bound the differences between these computed values and the exact ones. This is why we proceed differently.
In this paper, we only use Odlyzko’s function with parameter . This is the function defined by , where is twice the convolution square of the function : see [Poi77b, Sect. 3] and [CL19, Sect. 9.3.17]. These functions satisfy (POS), , and Proposition 9.3.18 of [CL19] provides alternative closed formulas for all the other quantities in (2.4.8) (see Proposition 4.4 for similar expressions). Each is a sum of a linear combination of a few special values of the classical digamma function and of its derivative , and of a simple rapidly converging series with given tail estimates [CL19, (3) p.127]. Using these formulas and estimates, we implemented functions in Python using Sage [S*+*14] to compute certified intervals containing the real numbers (2.4.8) for . See [CT19b] for the source code. For interval arithmetic, Sage relies on the Arb library http://arblib.org/. Our computations only use the four operations, the exponential and logarithm functions, the constant , the function (acb_digamma in this library), and its derivative (a special case of acb_polygamma).
Remark 2.11**.**
Fix an integer . For suitable , the restriction of to is positive definite (see e.g. Lemma 9.3.37 and Proposition 9.3.40 in [CL19], as well as [Chea]). By Corollary 2.3, it is important to be able to enumerate, for , all the (finitely many) effective elements in satisfying . We use for this the Fincke-Pohst algorithm enumerating the short vectors in a lattice. Using interval arithmetic as explained above we can obtain rational lower bounds for the coefficients of the Gram matrix of , and since we are only interested in effective elements of we can work with this rational Gram matrix instead. Unfortunately PARI’s qfminim does not (yet?) include an exact variant of the Fincke and Pohst algorithm for Gram matrices with integral entries. For this reason we reimplemented the first (simple) algorithm of Fincke-Pohst [FP85] using only exact computations, adding the condition of effectivity in the recursion to avoid unnecessary computations. Of course in practice this algorithm always leads to the same conclusions as PARI’s qfminim algorithm, if the latter is properly used. See [CT19b] for our source code.
2.4.4. The algorithm
The following algorithm tries to solve Problem 2 using the method discussed in §2.4.2.
- Input:
A quadruple as in 2.4.3. 2. Output:
(if the algorithm terminates) A triple with , a non empty , and with . 3. Step 1.
Choose a real number and compute an approximation of the Gram matrix . We do this using the formulas (2.4.4) of §2.4.2 and the expressions of [CL19] for the quantities (2.4.8) with discussed in §2.4.3. 4. Step 2.
Choose a nonempty subset of and compute an approximation of the minimal eigenvalue of the Gram matrix , as well as an approximate corresponding eigenvector . For doing so, we apply PARI’s qfjacobi function to (an implementation of Jacobi’s method). 5. Step 3.
If we have and for all , return , and and go to Step 4. Otherwise, go back to Step 2 and change the subset . If all the have been tried, go back to Step 1 and change the parameter . 6. Step 4.
Check rigorously, using interval arithmetic as discussed in §2.4.3, that we have indeed . If it fails go back to the second part of Step 3.
Let us comment this algorithm and discuss the unexplained choices involved:
- •
The choice of in Step 1 is based on some preliminary experiments, and it seems quite hard to guess a priori a range for the best ones. In our applications, we will choose in .
- •
The loop consisting of Steps 2 and 3, for a given , can be very long if is large, as there are possibilities for . In practice, we order the subsets by increasing cardinality, often restrict to of small cardinality. In practice again, the eigenspace is just a line.
- •
Whenever we reached Step 4, the rigorous check with interval arithmetic of the inequality never fails in practice. This single check is enough to justify that the algorithm does show that takes a negative value on . This is the most important remark regarding this algorithm. In particular, we do not have to justify any of the computations done in Steps 1, 2 and 3 before: all is fair in order to find a candidate . Of course, the experimental fact that the last check in Step 4 never fails just reflects that the computations made with PARI are highly accurate.
In the end, a charm of this algorithm is that even if the loop of Steps 1, 2 and 3 can be very long, once we get the candidate we just have to store it, and then inequality can be rechecked instantly.
2.4.5. Final algorithm
For our applications in §2.4.6, §4 and §3.3, it will be convenient to apply Algorithm 2.4.4 in the following slightly more restrictive context.
Set up. We fix in , in , and an integer . We fix as well a known set of elements of and our aim is to show that there does not exist distinct elements in with and for each . By known we mean that we assume given and 101010Let us mention that, at present, the authors are not aware of the existence of any non self-dual element in , so in practice will always actually have for in . for all . We denote by the set of triples in such that there are exactly elements in with .
Algorithm. Set . Assuming it is convenient choose a bijection
[TABLE]
and write . Set also . This defines a quadruple . We now apply Algorithm 2.4.4 to . In Step we obviously may, and do, restrict to subsets containing , i.e. of the form with , via the identification (2.4.9).
Output. When this algorithm terminates, it produces such that . For , set , the sum being over the with . Assume there are distinct elements in with and for each . Then for the element of we have (see Remark 2.10 for the first inequality), contradicting Proposition 2.2.
Remark 2.12**.**
In the case , this method just amounts to applying Corollary 2.3. In the case , it amounts to applying Scholium 9.3.26 of [CL19], by the discussion of Example 2.7. The case of arbitrary can thus be viewed as a generalisation of these criteria loc. cit. See [CT19b] for our source code in PARI of the algorithm above.
2.4.6. An illustration
Algorithm 2.4.5 can be used to give another proof of the Chenevier-Lannes classification theorem [CL19, Thm. 9.3.3] mentionned in §1.3 of the introduction, which is both very fast (a few seconds of computations) and systematic. Although this alternative proof shares many steps with the one loc. cit., it bypasses the geometric criterion involving Satake parameters explained in §9.3.29 therein (and does not rely at all on any computation of Satake parameters of known elements in ). Such an improvement, although not decisive here, will be crucial in the proof of Theorem 3, because at present we only know rather few Satake parameters for the known elements of of dimension (see however [BFvdG17] and [Még18]).
For the convenience of the reader, and in order to illustrate our new method, let us now explain the aforementioned proof of [CL19, Thm. 9.3.3] in the most complicated case of motivic weight . So we want to prove that there is a unique in of motivic weight , namely (for which we have ). We refer to the working sheet in [CT19b] for the numerical verifications used below.
Step 1. We first observe that is positive definite on the lattice for (Lemma [CL19, 9.3.37]). Using the PARI qfminim command, or better Remark 2.11, we may and do list all the effective elements in satisfying
[TABLE]
for . We retain furthermore only those satisfying and containing . The resulting list has elements. If in has motivic weight , then is in by Corollary 2.3. We will study each of these possibilities for mostly case by case.
Step 2. Denote by be the number of elements in with . We want to bound for each in by applying Inequality (2.4.2) of Corollary 2.3. For this we check that for all in we have , unless belongs to the subset , in which case we only have (here is still ). This shows for in , and for in .
Step 3. Fix in . We want to show . Assume , i.e. that there exist distinct in with . We apply Algorithm 2.4.5 to , (see Remark 2.10), and to the known set . For and it returns for instance an element close to
[TABLE]
We verify (using interval arithmetic, see §2.4.3) that we have up to : this contradicts Proposition 2.2. The algorithm produces a similar element in the case , with replaced by , and we have then up to . Nevertheless, it does not seem to produce any contradiction in the remaining case , even if we let vary. To deal with this last we add to the known element of , whose Archimedean -parameter is (which is “close” to ). The algorithm returns for the element and we verify that we have up to , which is indeed .
Step 4. We have proved so far for all . In particular, any in with is self-dual. Fix in . We now apply Algorithm 2.4.5 to , , and to the same set as above (with ). Using the nine in , it yields a contradiction in each case! Actually, if we restrict to subsets with in Step 2 of the algorithm (in other words, if we only apply the Scholium of [CL19] mentioned in Remark 2.12) we already get a contradiction for all but the elements mentioned in Table 1 below. These remaining cases were exactly the ones dealt with using the geometric criterion involving Satake parameters explained in [CL19, §9.3.29]. In these cases, our algorithm produces contradictions for subsets of size , such as the ones gathered in Table 1. This concludes the proof.
2.4.7. Another illustration: a strengthening of a vanishing result in [CvdG18]
As another example, let us show that for all odd , there is no cuspidal selfdual algebraic level automorphic representation of with . We apply for this Algorithm 2.4.5 to , and to the set of cuspidal automorphic representations generated by level cuspidal eigenforms for . Note that we have for and , and otherwise. We obtain a contradiction in each case using and . This shows for all by [CvdG18, Lemma A.2].
3. Effortless computation of masses in the trace formula
Let be a split classical group over such that admits discrete series. In other words, belongs to one of the three families
[TABLE]
In this section, we explain how to implement the strategy explained in §1.4 in order to determine the masses for . In §3.1, we first make elementary observations that will allow us to replace by a concrete set of equivalence classes of polynomials, and to rewrite the elliptic terms accordingly. In §3.2, we restrict further by an explicit subset containing all conjugacy classes such that . It turns out that finding a subset significantly smaller than is possible in the case of special orthogonal groups, using spinor norms considerations. In the last §3.3, we finally prove Theorems 6 and 7, by discussing how to produce sets of dominant weights satisfying a variant of conditions (P1) and (P2) alluded in §1.4.
3.1. Conjugacy classes and characteristic polynomials: elementary
observations
Let be one of , or . We shall denote by the dimension of the standard (or tautological) representation of , so is respectively or . (Do not confuse with the introduced in §1.4).
The indexing set for the sum defining the elliptic part of the geometric side in Arthur’s -Lefschetz trace formula [Art89] recalled in (1.4.3), is the set of conjugacy classes of semi-simple elements which are -elliptic (i.e. belongs to an anisotropic maximal torus of , in particular the eigenvalues of have absolute value ) and such that the conjugacy class of in meets the compact support of the smooth function we put in the trace formula, in our case the characteristic function of . In particular, the characteristic polynomial of such a , a monic polynomial of degree in , belongs to and has all its complex roots of absolute value . Using a celebrated theorem of Kronecker, these conditions imply that the roots of are roots of unity, hence that the semi-simple element has finite order. This explains the discussion of Formula (1.4.3) in §1.4, and the indexing set of finite order elements of taken up to conjugacy by in the sum (1.4.5).
Definition 3.1**.**
Let be the set of polynomials in having degree , which are products of cyclotomic polynomials and in which has even multiplicity (or equivalently, with ).
If is a class in , then all the elements have the same , and we will denote by this polynomial. It is an element of by the above discussion. We have thus defined a map
[TABLE]
It is well-known that if two semi-simple conjugacy classes in the classical group have the same characteristic polynomial , then they are equal, except in the case , and and are conjugate under . In particular, for the fiber111111Beware that the map is not surjective in general. For instance, Corollary 3.6 shows that for any prime , there is no order element in , as is not a square. has at most element if or , and [math] or elements otherwise. The following elementary lemma (see [Taï17, Remark 3.2.11]) shows that this latter case does not create complications:
Lemma 3.2**.**
For and with , we have .
Thus we may write
[TABLE]
with
[TABLE]
and
[TABLE]
with in the first case and in the second case. This also implies where is the non-trivial outer automorphism of induced by . This invariance is fortunate also because Koike and Terada’s simple (and most importantly very effective for small weights) formulas [KT87] for traces in algebraic representations apply to irreducible representations of symplectic and orthogonal (rather than special orthogonal) groups. Equivalently, their formula gives in terms of , but not in the second case above if .
There is another obvious invariance property of masses. For or , the element of is in the center of , and preserves . Formula (1.4.4) thus shows:
Lemma 3.3**.**
For or , and , we have .
For and as above, we have , , as well as by the lemma. A consequence is that if the restriction of to the center of is non-trivial.121212
From the perspective of the strategy discussed in §1.4, this vanishing is in agreement with the vanishing of for all dominant weights of or such that , which is a consequence of the property for orthogonal in : see [Art13, Thm. 1.5.3] and [CR15, Prop. 1.8]. Define the following equivalence relation on : if or . Assuming we may thus finally write
[TABLE]
where denotes the size of the equivalence class of .
To sum up, for the purpose of implementing our strategy introduced in §1.4 we can replace the indexing set by , which is computable, and we may as well restrict to dominant weights such that , and even to a set of representatives for the orbits under in the even orthogonal case.
3.2. Conjugacy classes and characteristic polynomials in the orthogonal
case: spinor norms
As announced in Remark 1.1, it turns out that in the orthogonal cases we can further reduce the set parametrizing conjugacy classes. Let be the subset of equivalence classes containing a finite order element in whose -conjugacy class meets . In particular contains the set of such that . A priori it may happen that is smaller than . Using the analysis in [Taï17, §3.2.2] and Jacobson’s hermitian analogue of the Hasse-Minkowski theorem [Jac40], one can argue that for any . Since this fact is rather unfortunate for our strategy, we leave the details to the interested reader.
We now focus on special orthogonal groups. Proposition 3.7 below gives an explicit subset of such that its preimage under the map char (3.1.1) contains . In contrast with the symplectic case we will see that in general, owing to the fact that special orthogonal groups are not simply connected.
Remark 3.4**.**
The second author had already observed that there was such a restriction on classes satisfying in [Taï17, Remark 3.2.8], unfortunately without giving details or proofs …He was also unaware of related previous work of Gross and McMullen: [GM02, Theorem 6.1] is similar to Proposition 3.7. Unfortunately we could not deduce Proposition 3.7 from the results of [GM02], so we give a slightly different proof below, relying on the Zassenhaus formula for spinor norms.
Let be a commutative ring, a projective -module of finite constant rank and a quadratic form. We say that is non-degenerate if the associated -bilinear form is a perfect pairing on . We say that is regular if either is non-degenerate, or is odd and Zariski-locally on the half-discriminant of is invertible: see [Knu91, Ch. IV §3] (who rather uses the terminology semi-regular in this case). When is a free -module and is non-degenerate, we denote by the class of the determinant of a Gram matrix of , where denotes the subgroups of squares in .
First we recall a few definitions from [Con14, Appendix C] or [Knu91, Ch. IV]. Assume and regular. Associated to are (reductive) groups schemes and over . The group is the group of even degree invertible elements in the Clifford algebra which stabilize the submodule under conjugation. This conjugation action gives a morphism , with kernel the central (invertible scalars in the Clifford algebra). See e.g. Propositions C.2.8 and C.4.6 of [Con14] for these properties and the fact that factors through the special orthogonal group. The Clifford norm morphism is defined in (C.4.2) and (C.4.4) loc. cit. The restriction of to the central is . The group can be defined as the kernel of the Clifford norm: see the proof of Lemma C.4.1 loc. cit. for the case even and the proof of Proposition C.4.10 loc. cit. and the paragraph following it for the case odd. We have [Knu91, (6.2.3) p.231] an exact sequence of sheaves in groups on the Zariski site of
[TABLE]
and thus an exact sequence of sheaves in groups on the fppf site of
[TABLE]
The (not so) long exact sequence in cohomology associated to the second short exact sequence above gives the spinor norm . If , which will always be the case in this paper, the fppf exact sequence gives the isomorphism , and we will implicitly consider the spinor norm in this last group. The spinor norm of is then represented by where is any lift of ; such a lift exists by (3.2.1) and . The spinor norm is additive: if and stabilizes and then we have .
Theorem 3.5** (Zassenhaus).**
Let be a field of characteristic different from . Let be a finite-dimensional vector space over , endowed with a non-degenerate quadratic form . Let and write the characteristic polynomial of as with . Then the spinor norm of is represented by in .
Proof.
This is just a reformulation of the main theorem of [Zas62]. Here is a short argument for the convenience of the reader. Using the orthogonal decomposition , and the additivity of spinor norms, we may assume , so , and either or . In the case , we have as unipotent elements are squares, and we conclude by the classical formula (use an orthogonal basis of ). The arguments so far have used that the characteristic of is , but the following ones will not.
Assume , write in , and choose a lift of . Write its Jordan decomposition in , with semi-simple and unipotent. There is a decomposition , where each is a -stable hyperbolic plane on which the two eigenvalues of are . Using the natural isomorphism between and the graded tensor product of the Clifford algebras of the (see e.g. [Knu91, IV. Prop. 1.3.1]) we easily sees that there is a pair in such that: , the Clifford norm of (or equivalently, of ) is , and the trace of (or equivalently, of ) in the spin representation of is . The spinor norm of is thus represented by . Note that although the spin representation may not be defined over , its trace is. Indeed, this representation is defined as the tautological morphism and is a central simple algebra over , whose reduced trace is -valued. Since we have for each as , the spinor norm of is represented by . ∎
We deduce the following discriminant formula, which can also be proved directly (see [GM02, Proposition A.3]).
Corollary 3.6**.**
Under the same assumptions, assume moreover that . Then is represented by .
Proof.
The discriminant of is the spinor norm of , or equivalently of . We conclude by applying the theorem to and . ∎
Proposition 3.7**.**
Let be a free -module of rank endowed with a regular quadratic form , , and a monic polynomial of degree in with . Assume that for every prime there exists having characteristic polynomial . If then the integer is square, and if then the integer is a square.
Note that the existence of for all primes implies that has integer coefficients.
Proof.
Fix a prime . Since and the element can be lifted to an element of by (3.2.1), so the spinor norm of lies in the image of in . Together with Theorem 3.5, this implies that
[TABLE]
lies as well in the image of . Assuming , it follows that the integer has an even valuation at each prime , so is a square. Assume now . In particular, is even and we have . By Corollary 3.6 applied to the orthogonal of in , we have in or equivalently:
[TABLE]
But have seen that the left-hand side is in the image of . So the integer has an even valuation at each prime , and is a square.∎
The astute reader will notice that the second statement could also be proved by considering and . In odd dimension this would require the introduction of and groups to define the spinor norm (see complications in [Con14, Appendix C]).
Definition 3.8**.**
For or let be the subset of consisting of all polynomials of the form , where is a product of cyclotomic polynomials with , which satisfy
- •
* if the positive integer is not a square, and*
- •
* if the positive integer is not a square.*
For denote (Definition 3.1).
The positive integers for may be computed inductively in terms of the prime decomposition of : see [GM02, Theorem 7.1]. In terms of the notation defined in §3.1, Proposition 3.7 asserts:
Corollary 3.9**.**
If satisfies , then we have .
This constraint is very useful, particularly in the even case, as the following table shows. In practice, we will see that it is almost sharp: see Remark 3.10. Observe that is stable under the equivalence relation introduced in the end of §3.1.
3.3. Non-existence of level one regular algebraic automorphic cuspidal
representations
In this paragraph, we prove Theorems 6 and 7 of the introduction. To implement the strategy explained in §1.4, taking into account the reduced Formula (3.1.2) and Corollary 3.9, it remains to actually produce, for as many “small rank” groups as possible in the families , and , sets of dominant weights for satisfying the following properties:
- (P1’)
For all we have and .
- (P2’)
For , the matrix has rank .
Of course (P2’) implies so our aim is roughly to produce as many dominant weights satisfying (P1’) as possible. See also Footnote 12 for an important remark regarding the condition on in (P1’).
Notations: For an integer we denote by the (finite) set of all dominant weights of such that: , , as well as in the case . For a dominant weight of , there is a unique effective element with and such that the multi-set of weights of (as defined in §2.1) coincides with (viewed of course as the multi-set ). The representation is multiplicity free, and for any having weights we have .
In order to produce we will first use the inequality (2.4.1) in Corollary 2.3. We choose big enough, and for every , and for all parameters , we compute . Whenever we find a negative value (certified using interval arithmetic as explained in §2.4.3), we know that by Corollary 2.3 and thus we add to . In other words, we choose
[TABLE]
for the set . For our purpose this simple method is already very effective. Table 4 below displays all groups for which it works, i.e. for which, using the given , the set satisfies the rank condition (P2’).
See [CT19b] for the Sage program which checks that each set in the table satisfies (P1’) (using Corollary 2.3 and interval arithmetic) and inductively computes, for each group appearing in the table:
- (1)
all masses , 2. (2)
, and for all dominant weights in any desired range (only limited by computer memory).
Note that we obtain in particular, independently of the computation of masses in [CR15] and [Taï17], the existence of self-dual elements of having regular weight and motivic weight . In §4 we will prove that there is no other such element in . The only case which was obtained in [Taï17] and that we cannot recover using this much simpler method is . The case of is new. For , considering all dominant weights in , we only find a set of cardinality , whereas we have . Higher motivic weights do not seem to provide any new non-existence results. Similarly this method does not allow us to conclude either in the case of .
To go further we use the algorithm explained in §2.4.5 to find larger sets satisfying (P1’). More precisely, for a large enough and each dominant weight we applied this algorithm with , , and taking for 131313We actually have , explained by the equality . the set of known elements of having motivic weight found above. As before we try various . Using this refined method we obtain the following three new cases:
- •
For we have and we found a subset of cardinality satisfying (P1’) and (P2’).
- •
The case is easier: we have and we found a subset of cardinality satisfying (P1’) and (P2’).
- •
For we have and we found a subset of cardinality satisfying (P1’) and (P2’). (Restricting to is not enough, as it yields a set of dominant weights of cardinality which does not satisfy (P2’)).
Again our program checking rigorously that these sets satisfy (P1’) and the inductive computation of masses and of the numbers can be found at [CT19b]. This concludes the proof of Theorem 6.
Remark 3.10**.**
Let be as in Theorem 6. An inspection of the masses found above shows for all , except for polynomials in the case , and others in the case . This shows that the spinor norms constraints established in §3.2 are almost sharp.
This second method only gives us these three additional cases for which we can compute all masses by solving a linear system. For example the set has elements, but we are not even close to producing ’s with enough elements: we were only able to produce a subset having elements satisfying (P1’). To overcome this scarcity of dominant weights satisfying (P1’), we computed a lot of masses for , namely for all in a certain subset of , using the method of [Taï17], i.e. by computing orbital integrals directly, and then we computed the remaining ones by solving a linear system.
To describe the set explicitly, for and a prime write
[TABLE]
where the first product is over all integers coprime to , and . Then if and only if for any prime number and any coprime to we have and
[TABLE]
For such a polynomial the computation using the method explained loc. cit. of the orbital integrals (1.4.4) occurring in the mass is purely a combinatorial matter and does not require any bilinear algebra. To be more precise, in general computing an orbital integral using the method loc. cit. involves enumerating totally isotropic subspaces stable under a given unipotent automorphism in (possibly degenerate) symplectic or hermitian spaces over a finite field, enumerating isomorphisms between such triples , and/or computing the complete invariants attached by Wall [Wal63] to isomorphism classes of such triples with non-degenerate; we restrict to cases where no such computation is necessary. Although these easier cases have the obvious benefit of being much easier to implement, the second advantage here is that these orbital integrals are computed (by a computer) in a matter of seconds. In contrast, there are relevant orbital integrals for for which the implementation of [Taï17] does not terminate in any reasonable time.
Denoting , we have . We found a subset of dominant weights for of cardinality satisfying (P1’) and the analogue of (P2’) for . This concludes the proof of Theorem 7.
4. Classification results in motivic weights and
This section is in the natural continuation of Sect. 2, of which we shall use freely the notations. We have decided to postpone it here because, in a few places, we will use below existence or inexistence results of certain self-dual regular elements of , results which have been proved in Sect. 3.
4.1. Motivic weight
We now prove Theorem 3 along with the following supplementary result.
Proposition 4.1**.**
Let be an effective element of containing with multiplicity . Let be the subset of elements in with .
- (1)
If then we have and for some non-self-dual . 2. (2)
If then belongs to an explicit set of elements and is of symplectic type.
The set of possible mentioned above can be found in [CT19b]. They all satisfy .
Proof of Theorem 3 and Proposition 4.1.
For the restriction of the symmetric bilinear form to is positive definite. As explained in Remark 2.11, using interval arithmetic we obtain rational lower bounds (we take them in ) for the coefficients of its Gram matrix in the basis . Applying the Fincke-Pohst algorithm, we obtain the set of all effective elements in containing and satisfying . By Corollary 2.3, contains all the elements such that there exist two distinct elements in of motivic weight and with .
For each in , we systematically applied Algorithm 2.4.5 to , , and to the set of known elements of having motivic weight , and various . For all but one , namely the one of assertion (1), it led to a contradiction with Inequality (2.3.7). Let us be more precise about the choices of and of the subset that we can make a posteriori in order to reach these contradictions more quickly (see also the source code [CT19b] for a working sheet). We first replace for the rest of the proof the above by its subset whose elements have motivic weight . We now have and . If we apply Algorithm 2.4.5 with all in and all subsets with , we obtain in a few seconds (on a personal computer) a contradiction for all but elements of . Using then all in and all with for those elements, the algorithm finds again in a few seconds a contradiction in all but cases. Two of these six are eliminated in a minute about using and all the subsets with . The remaining elements have the form with in the following list: . In the case we use and all the subsets with . To give an example, the algorithm produces in about minutes a linear combination close to
[TABLE]
with up to . In the case , we use similarly and . The case is quite harder to discard. After many tries, we found a contradiction using and a certain element subset of : see the source code in [CT19b] for the details. So far, we have thus proved the following:
(a) For any there is at most one element of with motivic weight and . In particular any such is self-dual.
Despite our efforts, we could not find a contradiction in the case of the last element . We have however for . By Corollary 2.3, this shows:
(b) For , there are at most elements of of motivic weight and with .
Note that we have proved assertion (1) except for the non self-duality assertion. To go further, we determine the set of effective elements in containing and satisfying for . For this we proceed as in the first paragraph of the proof and obtain an explicit set with elements. By Corollary 2.3, contains all the elements such that there exists in of motivic weigth with . For each in we applied Algorithm 2.4.5 to , (we restrict to self-dual elements), , to the set of all known elements of having motivic weight , for various choices of and subsets . We obtained contradictions with Inequality (2.3.7) for all but elements of . We refer to [CT19b] for an explicit list of triples leading to a contradiction in each case (checked using interval arithmetic). It would be tedious to explain here in details which and we did choose to find these triples: this is unnecessary anyway as all the necessary information for our proof is contained in the aforementioned list! We nevertheless refer to [CT19b] for the log file of our computations (which took several months).
Among the aforementioned “resistant” elements of , six of them are multiplicity free: and . These six regular weights have dimension , and we know from the results of Sect. 3 that there is no self-dual with these Archimedean components. An inspection of the list of remaining elements reveals that for any in :
- (i)
contains with multiplicity ,
- (ii)
contains for some ,
- (iii)
for any , the multiplicity of in is at most one.
Assertion (i) concludes the proof of Theorem 3. Assertion (b) above and the fact that is not in imply assertion (1) of Proposition 4.1. By (ii) and (iii) above, for any in there is some which occurs in with multiplicity . In particular, such a has no -equivariant nondegenerate symmetric pairing. This shows that any self-dual with is of symplectic type by [Art13, Theorem 1.4.2], and proves assertion (2) of Proposition 4.1. ∎
Remark 4.2**.**
For a given , it seems hard to us to guess a priori what will be the best choices of and (or ) to plug into Algorithm 2.4.5 for the purpose of reaching a contradiction with Inequality (2.3.7). Athough the authors have developped their own intuition and artisanal methods to find good and , they are mostly based on numerical experiments. In the same vein, in the cases where we did not find any contradiction, it seems difficult to prove that there cannot be any, as it is always possible to let vary and increase the size of . However, based on the large number of experiments we made, we find it likely that it is not possible to discard any of the elements of the remaining list by changing or .
4.2. Motivic weight
The following lemma is the first step in the proof of Theorem 5.
Lemma 4.3**.**
Let . Let be a self-dual level cuspidal algebraic regular automorphic representation of over of motivic weight . Then belongs to the following list:
- •
* for ,*
- •
* for ,*
- •
* for ,*
- •
* for .*
In particular we have .
Proof.
Let be as in the lemma and set . This is a multiplicity free effective element of containing and with . There are only finitely many such elements, with in all cases. Moreover, is orthogonal as is even (see the last paragraph of §2.2). By [Art13, Theorem 1.5.3] loc. cit. we have thus . Since this gives an extra constraint on .
A straightforward computer-aided enumeration gives us the list of the effective multiplicity free elements in containing , and satisfying , and . We applied Algorithm 2.4.5 to each such , , , to the set of elements of having motivic weight and dimension . Except in the four cases given in the statement, we obtained a contradiction with Inequality (2.3.7). It is actually enough to choose in and to restrict to the subsets with . We refer to [CT19b] for an explicit list of triples leading to a contradiction in each case (checked using interval arithmetic). ∎
Proof.
(Of Theorem 5) Using Theorem 6 we may compute, for any effective, multiplicity free, element containing , and with , the number of self-dual in with (this uses for and for ). Remarkably, we find only three such , namely the ones in the statement of Theorem 5. We conclude by Lemma 4.3. ∎
4.3. Classification results conditional to
By we shall mean here: for all , the zeros of satisfy . Assuming (GRH), Proposition 2.2 holds more generally for any test function satisfying and for all and in (a condition weaker than ). This condition is trivially satisfied by the function , where is the function recalled in §2.4.3 and is a positive real number (these are the classical functions of Odlyzko “under (GRH)”). In order to apply Algorithms 2.4.4 and 2.4.5 with instead of , we need the following variant of [CL19, Prop. 9.3.18]. We set and .
Proposition 4.4**.**
Let be a real number. For any integer we have
[TABLE]
with and . Moreover, we also have
[TABLE]
with , as well as and
[TABLE]
*Proof. * We follow the proof of [CL19, Prop. 9.3.18] and omit the straightforward details. For real numbers , set . A computation almost identical to p. 276 loc. cit. shows that we have . On the other hand, by [CL19, Prop. 9.3.8] we have for any integer and . This shows the first two formulas. Set for in . By p. 275 loc. cit. we have . We conclude by the relations and .
Upper bounds for the tails of the series and are given in [CL19, (3) p. 277].
*Proof. * (Of Theorem 4) In this proof, whenever we apply Algorithms 2.4.4 & 2.4.5 we do it using instead of . Applying Algorithm 2.4.5 to the element of Proposition 4.1 (1), and the set of known elements of with motivic weight , we obtain a contradiction with Inequality (2.3.7) with and (three elements in the list of Thm. 3). It thus only remains to show that for any of the elements of the list of Proposition 4.1 (2), there is no selfdual in with . For each in , we applied Algorithm 2.4.5 to , and the set above, using various . We found a contradiction in all but one cases. More precisely, we may reach all these contradictions but one using , of size , and not containing any of the elements of with motivic weight (see [CT19b] for a working sheet). The two remaining elements of are then
[TABLE]
For we eventually found a contradiction using and a certain subset with elements ! (see loc. cit.) The remaining case is the one of the statement of Theorem 4.
5. Siegel modular cusp forms for
In this section, we explain how to use our classification Theorems 3 & 5 to prove Theorem 2. Along the way, we will also reformulate much more precisely the Key fact 1 stated in the introduction.
5.1. Brief review of Arthur’s results for
Fix be an integer. We denote by the set of isomorphism classes of discrete automorphic representations of with for all primes . Recall that the Langlands dual group of “is” ; it has a tautological (often called standard) representation of dimension . Let be in . For each prime , the Satake parameter of will be viewed following Langlands as a semi-simple conjugacy class in . Similarly, the infinitesimal character of will be viewed as a semi-simple conjugacy class in the Lie algebra of (and most of the time, as the collection of its eigenvalues).
Let denote the set of level global Arthur parameters for . An element of is by definition a finite collection of distinct triples , for in , with a collection of integers satisfying , and with a level self-dual cuspidal automorphic representation of over which is orthogonal if is odd, symplectic otherwise. It suggestive to view as the isobaric automorphic representation of over defined as
[TABLE]
We often simply write for short 141414For typographical reasons we also replace the symbol with if we have , and by if we have and .
[TABLE]
To any in , viewed as in (5.1.1) as an irreducible admissible representation of over , we may attach a collection of Satake parameters (semi-simple conjugacy classes in ), as well as an infinitesimal character (a semi-simple conjugacy class in ). We shall say that is algebraic when the eigenvalues of are in . In this case, the only one that we shall need to study here, all the are algebraic (see §2.1).
Assume is algebraic. Using the local Langlands correspondence for the , we may attach to a morphism , uniquely defined up to -conjugacy, with the property
[TABLE]
(Recall the notation from §2.1) By §2.1, note that is trivial on , and in particular, is bounded (it is thus an Archimedean Arthur parameter). If is a representation of , and is an integer, it will be convenient to write for the representation of .
Arthur’s first main result [Art13, Thm. 1.5.2] attaches to any in a unique in such that we have for all place of (see also [Taï17, Lemma 4.1.1]). Arthur’s second main result is a converse statement, the so-called multiplicity formula, on which we shall focus from now on and until the end of this section.
Fix in . We assume that is algebraic for our purposes. There are both a local and a global ingredient in the multiplicity formula.
We start with the global one. Write with if, and only if, is even. Define as the abelian group generated by the symbols for , and by the symbols for all , with relations and (note ). This is an elementary abelian -group of order Arthur defines a global character of this group in [Art13, p. 48], that we now recall. For each consider the sign
[TABLE]
The term here is the Rankin-Selberg root number already encountered in §2.3, a (purely Archimedean) sign that we already explained how to compute loc. cit. It is necessarily by [Art13, Theorem 1.5.3] if and are both orthogonal or both symplectic. As the adjoint representation of is isomorphic to , Arthur’s definition reads (see e.g. [CL19, Sect. 8.3.5] for more details):
[TABLE]
We now describe the local ingredient. Fix a maximal compact subgroup of and the Lie algebra of . Arthur associates to a finite multi-set , also called an Arthur packet, of unitary irreducible admissible -modules. One important property he shows is that we have for all . Moreover, is equipped with a map
[TABLE]
where denotes the centralizer of the image of in .
Remark 5.1**.**
The map depends on the choice of an equivalence class of Whittaker datum for . From now on we fix a global Whitaker datum for such that is unramified with respect to , for each prime , in the sense of Casselman and Shalika. Up to conjugating if necessary by the outer action of , its Archimedian component can belong to any of the two classes of Whittaker data for .
We can now state Arthur’s multiplicity formula. Fix an algebraic in . There is a natural group embedding (“local-global” map). Choose in and assume for simplicity that it has multiplicity one in this multiset (this assumption will be satisfied in the cases that we will consider below). Then there is a in such that if, and only if, we have
[TABLE]
Moreover, if these equalities are satisfied then the multiplicity of in the automorphic discrete spectrum of is equal to . There is a slightly more complicated statement when we do not assume has multiplicity one in . This multiplicity one property will always be the case in our applications (see §5.2). It is believed but not known that it holds in general, although Moeglin and Renard have a number of results in this direction.
Remark 5.2**.**
It is important to remark that (5.1.4) trivially holds when we have for some cuspidal of , because the group is trivial.
5.2. Lowest-weight modules: results of Arancibia-Moeglin-Renard and of Moeglin-Renard
For with we denote by the holomorphic, unitary, lowest weight -module of (lowest) weight . 151515These modules have been classified by Enright, Howe and Wallach: they exist if, and only if, we have , with and , and they are unique up to isomorphism if they exist. The precise meaning here for "lowest" or "holomorphic" is a convention that we may fix as in [MR, §3] to fix ideas, nevertheless this choice will play no role in the sequel as we shall see. We are interested in for the following classical reason. Let us denote by the vector-space of vector-valued Siegel modular forms of weight for , and by its subspace of square-integrable forms. We have
[TABLE]
Assume is a Hecke eigenform in . Then generates an element in with . Better, (resp. ) is exactly the number of in with counted with their global discrete (resp. cuspidal) multiplicity.
An important property to have in mind is that the eigenvalues of the infinitesimal character of are [math] and the elements for . Note that these integers are distinct if, and only if, we have . This is also exactly the condition under which is a (holomorphic) discrete series. If is a Hecke eigenform in as above, the shape of the infinitesimal character of implies that is always algebraic. Moreover, is an element of the Arthur packet .
Conversely, let us fix until the end of §5.2 a global Arthur parameter
[TABLE]
in such that the eigenvalues of are [math] and the , with (in particular, is algebraic). In order to apply the Arthur multiplicity formula, we want to know under which condition on the module belongs to , whether it has multiplicity one in this multi-set, and if so, we want to know . We shall consider only the two following special, but important, cases.
5.2.1. Vector-valued case, with .
This situation is studied at length in [CR15, Chap. 9] and [CL19, Sect. 8.4.7]. In this case, is a discrete series, the eigenvalues of are distinct, and we have by a general result of Wallach. We have (a singleton) and is regular for all in , so we have for by §2.2. The parameter is necessarily an Adams-Johnson parameter (see e.g. [CR15, §3.8, App. A],[CL19, Sect. 8.4.15],[Taï17, §4.2.2]), and the main result of [AMgR18] shows that coincides with the packet that Adams and Johnson associate to in [AJ87] (any element of this packet having multiplicity one). Arancibia, Moeglin and Renard also prove the expected form of the map . As was observed in [CR15, §9] (see also [CL19, Sect 8.5.1]), this packet contains if and only if we have , and in this case the corresponding character is given by the formula, for all in :
[TABLE]
where is the number of odd integers such that is a weight of . Note that the quantity does not change if we replace odd with even in the definition of , as we have for odd. This property expresses the fact that the character above does not depend on the choice of the Whittaker datum in Remark 5.1. All in all, we have explained fully, and much more precisely, the Key fact 1 of the introduction.
5.2.2. Scalar-valued case, arbitrary genus
In this case we have in with , and we rather write for . If we have we are in the case of §5.2.1, so from now on we assume . The eigenvalues of the infinitesimal character of are now [math] and the elements for : the eigenvalue [math] has thus the multiplicity , and for the eigenvalues have multiplicity .
We will use as a key ingredient the recent local results of Moeglin and Renard [MR] 161616Note that those authors call what we call respectively. , that we will specialize in what follows to this level situation. The first main result of [MR] is that belongs to if, and only if, we are in one of the two cases called and below. In both cases they show that has multiplicity in and they determine . We use the letter for the case reminiscent of Ikeda lifts, and the letter for those related to the Howe (or theta) correspondence. The formula for given in [MR, Prop. 18.3] depends on the class of , which is represented there by a certain sign loc. cit. and that we represent the same way here (it may be either one of : see Remark 5.1). We will express below only the restriction of to , which is the information we need for applying the global multiplicity formula (5.1.4). In all cases we will see in particular that this restriction does not depend on , i.e. on , hence neither on the choices discussed in the beginning of §5.2 that we made (or rather didn’t) to define : changing of choice amounts to replace with by [MR].
Preliminary general notations and remarks
– Recall we have already defined a partition according to the parity of for in . We now define as the subset of elements in such that [math] is a weight of . We clearly have , and . Set
[TABLE]
– Say that an algebraic cuspidal automorphic representation of satisfies (R) if its weights are , its nonzero weights have multiplicity , the multiplicity of the weight [math] of is , and if and have multiplicity at most in . We shall see below that all the satisfy property (R), and that at most one of them is not regular.
– To decipher the formulas of [MR, Prop. 18.3], we have found it useful to observe, for , and , the following equalities of signs:
[TABLE]
This first is the product of alternating signs starting with ; it only depends on . The second follows from the first by replacing with .
Case (I)
This corresponds to case (i) of [MR, Théorème 7.1]. We have (a singleton), the weight [math] of has multiplicity , , , and the numbers , where is in , is a positive weights of , and with , fill the length segment (hence are distinct). The representation is regular for each , with weights , hence satisfies for by §2.2. Moreover, for the representation does not have [math] among its weights.
In this case we must have so is generated by the with . Fix such an , necessarily in . For any sign we define as the number of integers with such that is a weight of . The first assertion of Proposition 18.3 of [MR] (we are in case (1) of §18 loc. cit.), together with Formula (5.2.2), show that is given by the formula:
[TABLE]
Indeed, consider the sequence of alternating signs starting with , i.e. set . Formulas (5.2.2) show that, for any in and any positive weight of , the sign of Proposition [MR, Prop. 18.3] is the product of the with in the interval of length . Note that when is even this product is ; when is odd, and thus , this product coincides with . The formula shows that we have if and only if with . Formula (5.2.3) follows, as for in the sign is by definition the product, over all the positive weights of , of .
Note that when is odd we have , as is regular and does not have the zero weight, so . As a consequence, Formula (5.2.3) does not depend on .
Case (H)
This corresponds to case (ii) in [MR, Théorème 7.1]. According to Theorem 7.2 loc. cit. there are two subcases:
(H1) There is in with , and contains .
(H2) There is in with , and contains .
Note that is not unique in general, so we fix any satisfying (H1) or (H2). We set in case (H1) and in case (H2). An inspection of shows that in case (H2) we must have , that is (hence ). In both cases we may write
[TABLE]
We have , and the eigenvalues of contributing to are the for in case (H1), and the same ones except in case (H2). It follows that is an Adams-Johnson parameter for the compact group , and in particular, is multiplicity-free. This implies:
– satisfies (R) for all , and is regular for .
In particular, for , we have if is even, and contains if is odd (see §2.2). Moreover, either is regular or we are in the case (H1) (see Remark 5.3) and in one of the two following situations:
– , [math] is a double weight of , and contains twice,
– is odd, , , [math] is a triple weight of , and contains twice and once.
Remark 5.3**.**
Assume we are in the case (H2). Then the weight [math] of has multiplicity , since the eigenvalue does not contribute to . In particular we have and is regular. Moreover, we also have . Indeed, implies , hence , which contradicts (H2).
We now describe the restriction of the character to . For any in and any sign we define an integer as follows. If we are in case (H1), then is the number of integers with such that is a weight of (as in case (I)). If we are in case (H2), we first consider the decreasing sequence where is omitted (this makes sense as we have by Remark 5.3), and rather define as the number of integers with such that is a weight of . In all cases we have by property (R):
[TABLE]
– Assume first we have and . For we have:
[TABLE]
Indeed, this follows from [MR, Proposition 18.3] by a similar argument as in Case (I). The only difference is to replace the alternating sequence of signs defined in Case (I) by the length alternating sequence starting with but with the index omitted in case (H2); in other words, we still set in Case (H1), and in Case (H2) we set for , and otherwise. The same arguments as in case (I) show Formula (5.2.5) and its independence on .
For in , in which case we must have , we rather find
[TABLE]
Indeed, we are in the situation (2) of §18 loc. cit. and in the notations there we have and [MR, Remarque 18.4]. The congruence , the equality , and (5.2.4) show that (5.2.6) does not depend on the sign .
– Assume is in (se we are in case (H1) by Remark 5.3). We have if and only if is regular by §2.2. By [MR, Proposition 18.3] (including the formula for in Case (2) there with ) we find:
[TABLE]
Again, these two formulas do not depend on by (5.2.4) and .
– We are left to consider the case . We must have and . We want to give the value of for in . We may write with and set . By [MR, Proposition 18.3] (including the formula for there and Remark 18.4 (ii)), we have:
[TABLE]
Observe that we also have , as in case (H2), hence by Formula (5.2.2). As is regular of odd dimension, and contains , we have . This implies
[TABLE]
which may also be written . The relation , from (5.2.4), and the trivial identity for , show that Formula (5.2.8) does not depend on .
– We still assume as above. It only remains to give . It will depend on the following property on :
[TABLE]
Similarly to (5.2.7) we find
[TABLE]
These values do not depend on by (5.2.4): the integer is if holds, and otherwise.
Remark 5.4**.**
- (1)
The cases (I) and (H) are disjoint for , and for case (I) is a special case of (H1), and the two formulas (5.2.3) and (5.2.5) for are identical. 2. (2)
The parameter does not always determine : when is even, parameters of type (H1) in weight coincide with parameters of type (H2) in weight .
5.3. Proof of theorem 2
Fix , and assume either that is scalar or . Arthur’s multiplicity formula, as well as the multiplicity one results of [AMgR18] and [MR], show that two Hecke eigenforms in with same Hecke eigenvalues, or equivalently with the same standard parameter, are proportional. It follows that is the number of possible standard parameters of Hecke eigenforms in . In what follows we enumerate these parameters in the case . We thus fix a Hecke eigenform in and denote by its standard parameter. As in §5.1 we write
[TABLE]
Notation: – Assume are positive odd (resp. even) integers and that there is a unique self-dual regular in with weights , then we shall denote by (resp. ) this unique element . Similarly, when are even positive integers and there is a unique self-dual in with weights [math] and , then we shall denote by this element. The ’s are symplectic and the ’s are orthogonal. These notations are compatible with the ones introduced in §1.3, and we have for instance . We shall also denote by the trivial representation of , and by and the two cuspidal representations of generated by the two normalized eigenforms in .
– We will denote by the subset of whose elements are either of motivic weight , or of motivic weight with the weight having multiplicity , or regular self-dual of motivic weight . By the classification theorems ([CL19, Thm. F], Theorems 3 and 5), there are elements in , all regular self-dual, and according to the notations above we have
[TABLE]
[TABLE]
5.3.1. Case with and .
We have . We apply §5.2.1. In this case, each is regular of motivic weight , there is a unique with odd, and we have . In particular, all the are in and is either , or . It is a boring but trivial exercise to enumerate all the in with these properties and such that the eigenvalues of are distinct and . We find exactly such parameters. The possible are then exactly the ones in this list satisfying the Arthur multiplicity formula (5.1.4), using Formulas (5.2.1), (5.1.3) and (5.1.2). We find that only of these do satisfy this formula, and obtain Table 5, as well as the part of Table 6 concerning the case . All those computations can be done easily with the help of a computer: see [CT19b] for a PARI code doing it. They can also be made by hand as follows.
We only treat the case , the two other ones being similar. Note that for such that is symplectic, we have , and either or with . Assume first that is symplectic for all . We have , so if we have then Arthur’s multiplicity formula simply reads . If we have , it rather asserts , i.e. (note for ). This justifies the existence of the in Tables 5 and 6 containing . Assume now there is , necessarily unique, such that is orthogonal. We will show that this case cannot happen. We have either or , and . If we have and we compute , so there is in . For weight reasons we must have , with if , and otherwise. This implies . Note that is if has an odd number of weights , and is otherwise. This shows for and , a contradiction. We finally exclude the last possible case and as we have , and .
5.3.2. Scalar-valued case with and .
We apply §5.2.2. We are either in case (I), (H1) or (H2).
Case (I). Assume first we are in case (I), in particular we have , and the are regular for all in . For each in , we have , and in particular has motivic weight : it belongs to the list . By inspection, has thus to be or . As the weight [math] has multiplicity in and we have . There is then a unique in such that, if denotes the smallest positive weight of , we have ; we must have . We have both and , which implies . By an inspection of , this forces and . But this implies that the positive weights of the with in are . Only the trivial representation has this property in . This shows and that the unique possibility for is
[TABLE]
We recognize the standard parameter of the genus Ikeda lift of [Ike01], a well-known element of , hence does exist. Alternatively, Arthur’s multiplicity formula (5.1.4) is satisfied as we have by (5.2.3), so is indeed the standard parameter of an eigenform in . The shape of and the Zharkovskaya relation 171717This asserts that if is an eigenform in , and if in is non zero, with the Siegel operator, then is an eigenform and the standard -function of and satisfy . imply that this eigenform has to be cuspidal.
Case (H1). Assume we are in case (H1). We will show again . Write as in the definition of (H1).
Lemma 5.5**.**
The representation is in for all , and we have .
*Proof. * As a general fact, all the satisfy condition (R) of §5.2.2. They have motivic weight and their nonzero weights have multiplicity . Moreover is regular for . It follows that all the are in , except perhaps in the case . But for each and each positive weight of we have . Thus implies and , so : this is absurd as there is no nonzero Siegel modular form for with odd weight and genus. So is in with motivic weight , and the unique possibilities are thus or since [math] is a weight of . Assume we have . Then contains so is odd by (H1), is even, and we have . The inequality implies then , and . We have proved
[TABLE]
with , and has the eigenvalues and [math] twice. But the are in with motivic weight , hence in , and we have for in . The only possibility is thus . We have with , but by Formula (5.2.5): the multiplicity formula is not satisfied.
Note that implies by (H1), hence . Write again
[TABLE]
where is a certain -dimensional parameter with weights and [math] twice, and . Each with is then regular of motivic weight . The list of all possible with these properties is easy to determine: see Proposition 9.2.2 of [CL19]. For we find . For we have and . For we have
[TABLE]
For , we have possibilities for , namely the ones in [CL19, Thm. E].
Lemma 5.6**.**
We have .
*Proof. * Assume that , then by inspection and we denote so that and . For any we have and so , i.e. is even. Thus Arthur’s multiplicity formula (5.1.4) implies
[TABLE]
the first equality being (5.2.8). For those epsilon factors are and we have so this formula does not hold. This rules out and . The four other parameters for are ruled out the same way using .
We will now prove that, apart from the case , none of the remaining come from a cuspidal modular form. We will need first to recall some results on orthogonal automorphic forms and theta series. For each integer we fix arbitrarily an even unimodular lattice of rank and denote respectively by and its orthogonal and special orthogonal group schemes over . We refer to [CL19, Sects. 4.4 & 6.4.7] for the basics of the theory of level automorphic forms for and (beware that these group schemes are rather denoted by and loc. cit.). By results of Arthur [Art13] and Taïbi [Taï19], any discrete automorphic representation of or has a standard parameter in , the latter being defined exactly as in the case of (see §5.1) but with the condition instead of .
For , we denote by the set of isomorphism classes of even unimodular lattices of rank . The vector-space is in a natural way the dual of a space of level automorphic forms for . Any Hecke eigenform in generates a discrete automorphic representation of (with trivial Archimedean component and for each prime ), which has a standard parameter in . Moreover, Siegel theta series provide a linear map
[TABLE]
for all (see e.g. [CL19, §5.1], in particular for the conventions for ), with (here denotes the Siegel operator). For in , the degree of is the smallest integer with ; the form is then cuspidal and we have for . If in is an eigenform with degree , and for , then the Eichler commutation relations show that is an eigenform in , and there is a simple relation due to Rallis [Ral82, §6] between the Satake parameters of and that of (see [CL19, Sect 7.1]). Concretely, if is square integrable (e.g. cuspidal), this relation is the equality for , for . Last but not least, we have the following result, a consequence of [Ral84, Thm. I.1.1] and [MW94, Lemme I.4.11] that we learnt from [MR, §16.2].
Lemma 5.7**.**
Let be an eigenform in of degree . If we have and , then is square integrable and .
(Note that we have , hence the last assertion.) We finally go back to our analysis of case (H1), setting . The spaces and have respective dimension , and the standard parameters of their eigenforms turn out to be exactly the , and parameters discussed above for and , by [CL19, Cor. 7.2.7 & Thm. E]. This reference determines as well the degree of each eigenform (see [CL19, Thm. 9.2.6], note that most of these degrees had already been found before by Nebe and Venkov): this is the smallest integer such that is a summand of (hence ), unless we have and . For we have thus as well as by the necessary condition of (H1). By Lemma 5.7, the automorphic representation is thus the (necessarily unique) discrete automorphic representation of with parameter , and it is not cuspidal since we have . In the remaining case we have , and , and again is discrete but not cuspidal if we have . We conclude since for we recover the form found in case (I).
Case (H2). We are going to show that there are exactly two Siegel eigenforms in this remaining case, both for , of respective genus and , and parameters
[TABLE]
Lemma 5.8**.**
We have , odd and even.
*Proof. * As we are in case (H2) we must have and , and so . Assume is non trivial. We have by the Chenevier-Lannes theorem, so , and is odd. This contradicts the last condition of (H2). So is trivial, is odd by the last condition in (H2), hence is even as we are in full level .
Write , with . The eigenvalues of corresponding to are the integers with , with the even number omitted (we shall call those eigenvalues the "weights" of for short). Each is regular algebraic of motivic weight , hence in the list . We are now led to do a simple enumeration exercise: for every odd , enumerate all possible , with in for each , and with weights where the even integer is excluded and satisfies .
Lemma 5.9**.**
*Assume and , then we have , , and , as well as with , and . *
*Proof. * Assume first . The only in with , and having a weight , are and . It follows that among the three consecutive integers , either or is not a weight of , hence must be : a contradiction as is even.
An inspection of shows then , and . As is odd we have . Assuming , contributes to the weights and of . There is thus a unique in with , and for the representation is symplectic with motivic weight . This shows . But there is an odd number of integers with : a contradiction. We have proved .
As is an eigenvalue of , we have either or there exists some with and not having the weight . The only remaining possibility in this latter case is for some . But there is no in with and having a weight : a contradiction with the presence of the weight or of . We have proved , i.e. .
The weights of are thus and [math] twice. Those possible are easily determined (see [CL19, Prop. 9.2.2] or [CL19, Thm. E]): there are possibilities, all of them containing some with , except
[TABLE]
We exclude this case using Arthur’s multiplicity formula. We have with . We have , but by Formula (5.2.10) we have : a contradiction.
We are now able to conclude the proof. Assume first that is a summand of . We must have
[TABLE]
which trivially satisfies Arthur’s multiplicity formula by (5.2.5). In this case there is thus an eigenform with parameter ; this is necessarily cuspidal as we have for odd. As we will explain in §5.4, this is actually the form constructed by Freitag in the last section of [Fre82].
So we may assume that is not a summand of . Consider the double weight [math] of . An inspection of shows then that there are two elements in , say with . Better, Lemma 5.9 implies that we have , , and either or . We apply Arthur’s multiplicity formula at the element . We have by Formula (5.2.10). This implies , which is equivalent to
[TABLE]
where is the set of elements in such that is symplectic and with . We have for , as is even and , which imposes . Among the symplectic representations with such motivic weight, only and have a negative epsilon factor. As a consequence, at least one summand of is among
[TABLE]
Observe that this implies and that such a summand always contributes the weights and to , so that we have .
Assume first , hence , and by Lemma 5.9. Then is a summand of , but the weight occurs in both and , a contradiction. We have proved . The congruence (5.2.9) implies then , which leaves only the two cases or by the inequality . In the case the only possibility is thus that contains , hence is equal to the latter for weights reasons, and
[TABLE]
Arthur’s multiplicity formula is satisfied for this by Formulas (5.2.5), (5.2.8) and (5.2.10). There is thus an eigenform in with parameter , necessarily cuspidal as its weight is odd.
We are left to study the case . In this case we focus on the weight of . It cannot come from a summand in the list (5.3.2). It must thus come from a summand of which does not contribute to any weight in , in particular does not have any weight in . If has motivic weight , an inspection of shows , which leads to
[TABLE]
If has motivic weight , then we have , is a summand of , so and are not weights of , and the only possibility is and
[TABLE]
In both cases, does not satisfy the multiplicity formula at the element with : we have by (5.2.5) and . This concludes the proof of Theorem 2.
5.4. Complements: theta series constructions
Recall that for any integer we denote by the set of even unimodular lattices in the standard Euclidean space , and by the finite set of isometry classes of such lattices.
Our first complement concerns the question of the surjectivity of the linear map of Formula (5.3.1), also called the Eichler basis problem, for . This surjectivity was proved in [CL19, §1.3] in the case .
Corollary 5.10**.**
Assume or . Then is surjective for all . In particular, the Siegel operator is surjective as well for all .
*Proof. * We have the relations and . The corollary follows thus from the case , and from the vanishing for and , implied by Theorem 2.
In other words, the Eichler basis problem holds for all for those three values of . We stress that this is not a general phenomenon: as was observed in [CL19, Cor. 7.3.5] the map is not surjective for .
Remark 5.11**.**
For , the surjectivity of and the determination of for all by Table 6 allow to determine for all . In particular, we have for and , for and , and for and .
Our second complement concerns the concrete construction via theta series of the four weight Siegel modular eigenforms of respective genus and given by Table 6. Consider again the standard Euclidean space with . For any finite-dimensional continuous representation of the compact orthogonal group over the complex number, we denote by the complex vector space of -equivariant functions ; this is a space of automorphic forms for the orthogonal group scheme introduced after Lemma 5.6 (see also [CL19, Sect. 4.4.4]). For any integers , we denote by the representation of on the space of harmonic polynomials of degree on in the sense of [Bö89, §XI]. The construction of Siegel theta series with harmonic coefficients gives rise to a linear map (see [Bö89, §XI] and [CL19, Sect. 5.4.1])
[TABLE]
mapping any -eigenform to a Siegel eigenform (Eichler’s commutation relations) or to zero. The following proposition is suggested by Rallis’s theory [Ral82] and the fact that the standard parameters of the four weight Siegel eigenforms are respectively .
Proposition 5.12**.**
- (i)
For each , the form is in the image .
- (ii)
The form is in the image of .
*Proof. * Böcherer181818Important contributions to this problem have been made by Siegel, Weissauer, Kudla-Rallis (Siegel-Weil formula), and also by Freitag, Garrett, Piatetski-Shapiro, Rallis and Waldspurger. gives in [Bö89, Thm. 5] a necessary and sufficient condition for these properties to hold in terms of the order of vanishing of the standard -function of at , with in case (i) and in case (ii). A case-by-case analysis reveals that this criterion holds true in all five cases. We refer to [CT19b] for the details of this simple, but rather tedious, verification. The only non-trivial necessary ingredient is the non-vanishing at of the Godement-Jacquet -function of and , that was proved in [CL19, Prop. 9.3.39].
In the companion paper [CT19a], we study the maps in a much more elementary way. Note that a harmonic polynomial of weight on is just the datum of a -multilinear alternating form on . For any element in , and any Niemier lattice in , the -multilinear alternating form is invariant under the orthogonal group of . This actually forces to vanish outside the -orbit of the Leech lattice by [CT19a, Prop. 4.1]. A curious consequence of Proposition 5.12 is thus that for , there is a nonzero, -invariant, -multilinear alternating form on the Leech lattice! A computation using the -page of the fortunately confirms this property, and reveals furthermore that there is a unique such form of to scalar, and none for the other values of . (The existence of such a form for is well-known, and follows from the fact that the Leech lattice is orientable, which means that any element in has determinant ). In other words, has dimension for (and [math] otherwise). The main result of [CT19a] is a direct proof of the non-vanishing of the map for these four values of . The non-vanishing of , hence of , and had already been observed in the past by Freitag, in the last section of [Fre82]. The Mathieu group and certain oriented rank sublattices of the Leech lattice play an important role in our argument for . We also prove differently loc. cit. that the standard parameter of the line of Siegel eigenforms in the image of is the one given in Table 13. All of this fully confirms Corollary 1 and Proposition 5.12 (i), and show the following.
Corollary 5.13**.**
The linear map in (5.4.1) is an isomorphism for all .
Case (ii) of Proposition 5.12 also implies the nonvanishing of . Let us simply mention that we actually have using a computation similar to that of [CL19, Cor. 9.5.13]. The space is actually generated by two -eigenforms, with respective standard parameters and .
5.5. Remarks on the case
Let and be non-negative integers satisfying . In this case we have by [Wei83, Satz 3]. We may thus apply Arthur’s endoscopic classification to study .
Proposition 5.14**.**
We have whenever , and this dimension vanishes unless is divisible by .
*Proof. * For any eigenform in with standard parameter , we are in case (H1) as , and with as . In particular is even, is in , and we have where is such that is an Adams-Johnson parameter for the compact group . By the Arthur multiplicity formula, the characters and coincide on . Consider the element defined as the product of and of in the case . Formulas (5.2.5) and (5.2.8) imply . Indeed, one way to argue is to use the interpretation of the signs loc. cit. given right after Formula (5.2.5), which shows . On the other hand, we have . We claim that for and any integer we have the equality
[TABLE]
Indeed, we may assume symplectic (otherwise ), in which case we have and and thus , and we conclude as is even. This shows in particular , and together with, proves that is divisible by if is nonzero.
To prove the asserted equality of dimensions, it is enough to show that the fact that the multiplicity formula holds for implies that for any genus it also holds for the parameter , still in weight and case (H1). We may index the summands of with the same set as for , with the same for , the same , and the same for . There is an obvious bijection between and matching all and . Via this bijection the characters and coincide, as for all we have by (5.5.1). We conclude as the characters and trivially coincide as well.
Of course this proposition is coherent with the known properties of Siegel modular forms for :
[TABLE]
by [Res75], [Fre75] (first equality), [Fre77] (second equality, see also [How81]).
Corollary 5.15**.**
Assume .
- (1)
The Siegel operator is bijective. 2. (2)
If , the linear map is an isomorphism .
*Proof. * The first equality in (5.5.2) means that is injective. By the equality of dimensions of Proposition 5.14, this implies that is bijective. (In the case , the surjectivity of also follows from the second equality in (5.5.2), as is generated by theta series). This proves the first assertion.
For the second, it is obvious that is injective for , hence for all as well by the relation . The surjectivity of follows from the second equality of (5.5.2) for . The surjectivity of , and the surjectivity of proved in (1), implies the remaining surjectivity of .
This corollary seems to be new for .
6. Tables
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