Simultaneous non-vanishing for Dirichlet L-functions
Raphael Zacharias
TL;DR
This paper develops new asymptotic formulas for cubic moments of modular L-functions at the central point, extending previous work on correlations of Hecke eigenvalues and algebraic trace functions, with applications to non-vanishing results of Dirichlet L-functions.
Contribution
It introduces an asymptotic formula for a generalized cubic moment of modular L-functions, extending prior correlation work and applying it to non-vanishing results for Dirichlet L-functions.
Findings
Established asymptotic formula for cubic moments of modular L-functions.
Proved positive proportion of characters with non-vanishing L-values.
Extended correlation techniques to new settings.
Abstract
We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular L-functions at the central point s = 1/2 and for prime moduli q. As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet L-functions to derive that for any pair of multiplicative characters modulo q, there is a positive proportion of (mod q) such that and are simultaneously not too small.
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Taxonomy
TopicsAnalytic Number Theory Research · Advanced Algebra and Geometry · Algebraic Geometry and Number Theory
Simultaneous non-vanishing for Dirichlet -functions
Raphaël Zacharias
Abstract
We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized cubic moment of modular -functions at the central point and for prime moduli . As an application, we exploit our recent result on the mollification of the fourth moment of Dirichlet -functions to derive that for any pair of multiplicative characters modulo , there is a positive proportion of such that and are simultaneously not too small.
Contents
- 1 Introduction and statement of results
- 2 Background on automorphic forms
- 3 -adic twists of modular forms
- 4 Bilinear forms involving trace functions
- 5 Proof of Theorem 3.6
- 6 Proof of Theorem 1.3
1 Introduction and statement of results
The zeros of automorphic -functions on the critical line have received considerable attention these last years [BFH90, Duk95, L*+*93, KM99, IS00]. In particular, at the central point , an -function is expected to vanish only for either a good reason or a trivial reason. For example, if is an elliptic curve defined over and is its associated -function, then according to the Birch and Swinnerton-Dyer conjecture, if and only if the group of -points has positive rank. A trivial reason is for instance when the sign of the functional equation is , which is the case if the -function is attached to an odd Hecke-Maass form.
A typical approach in the study of non-vanishing problems is to consider a family of -functions for varying in some finite set and try to give a lower bound for the proportion of such that as . In [IS99], H. Iwaniec and P. Sarnak examined at as ranges over all primitive Dirichlet characters modulo . They proved that at least of the central values are not zero. This proportion has been slightly improved to by H.M. Bui [Bui12] and finally to by R. Khan and H.T. Ngo [KN16] with the restriction to prime moduli .
In [MV02], P. Michel and J. Vanderkam considered simultaneous non-vanishing problems : given three distinct Dirichlet characters of fixed modulus (satisfying some technical conditions), they proved that a positive proportion of Holomorphic primitive Hecke cusp form of weight , prime level and trivial nebentypus are such that the product is not zero for sufficiently large (in terms of ). They derived various arithmetic applications, especially the existence of quotients of of large dimension satisfying the Birch and Swinnerton-Dyer conjecture over cyclic number fields of degree less than 5.
In this paper, we let be a prime number, be arbitrary Dirichlet characters of modulus , a Hecke eigenform for (holomorphic or Maass) and we are interested in the distribution of the values of the two families indexed by Dirichlet characters modulo
[TABLE]
[TABLE]
as . Using mollification method, a technique that has made the success of many of the papers cited in the previous paragraphs, we show that for both families, a positive proportion of these central values of character twists is not zero. We give in fact a more precise statement, saying that they are simultaneously not too small. We point out that there is no additional difficulty by considering fixed of conductors . These simultaneous non-vanish results require the evaluation of the two cubic moments
[TABLE]
[TABLE]
for an integer coprime with .
We now state the main results of this paper :
Theorem 1.1**.**
Let be a prime number, two Dirichlet characters of modulus and . Then there exists an explicit absolute constant and such that for any prime ,
[TABLE]
Theorem 1.2**.**
Let be a prime number, a Hecke cusp of level and spectral parameter satisfying the Ramanujan-Petersson conjecture and . Then there exists an explicit absolute constant and such that for any prime ,
[TABLE]
The new main ingredient in the proof of Theorems 1.1,1.2 is the following result which establishes an asymptotic formula for the moments (1.1),(1.2).
Theorem 1.3**.**
Let be a prime number, be Dirichlet characters of modulus , a primitive Hecke cusp form of level or and trivial nebentypus. Assume that satisfies the Ramanujan-Petersson conjecture, then for any , we have
[TABLE]
[TABLE]
where the implied constant only depends on and polynomially on the Archimedean parameters of (the weight or the Laplace eigenvalue) in (1.4).
Remark 1.4**.**
The case where is of level one and has been announced by S. Das and S. Ganguly and it seems that their method is similar to our.
When is of level with non trivial central character, the proof of Theorem 1.3 is similar but requires a mild extension of [KMS15, Theorem 1.3] for Kloosterman sums twisted by characters. We will return to this question in a coming paper.
Remark 1.5**.**
The asymptotic formula (1.4) is similar to the mixted cubic moment evaluated by S. Das and R. Khan in [DK14]
[TABLE]
As the authors explained, the complex conjugation above was introduced to avoid difficulties connected to the oscillations of Gauss sums. What we show here is that this difficulties are resolved using variants of the methods of [FKM15] [KMS15].
1.1 The mollification method
We show here how to derive Theorems 1.1 and 1.2 from Theorem 1.3. Let be a real number such that . For any multiplicative character , we define the short linear form
[TABLE]
where is the Möbius function. Let denotes the sequence of Hecke eigenvalues of a Hecke cusp form of level one and be the convolution inverse of given by
[TABLE]
For with , we also define
[TABLE]
We finally consider the two mollified cubic moments
[TABLE]
where is the trivial character, and
[TABLE]
Note that (1.7) and (1.8) can be written in the form
[TABLE]
[TABLE]
with
[TABLE]
Since satisfies the Ramanujan-Petersson conjecture, we have for and ,
[TABLE]
where is the divisor function. Hence an immediate consequence of Theorem 1.3 is the following corollary :
Corollary 1.6**.**
For any , the mollified cubic moments (1.7) and (1.8) satisfies
[TABLE]
where the implied constant only depends on and polynomially on in the second expression.
In [Zac16, Theorem 1.2], we established an asymptotic formula for a mollified fourth moment of Dirichlet -functions : for with , we obtained
[TABLE]
where is a degree four polynomial with calculable coefficients.
Similarly, in a paper of preparation “Non-Vanishing of twisted L-function ” by Blomer, Fouvry, Kowalski, Michel, Milićević and Sawin, they obtained
[TABLE]
for with Hence, combining Corollary 1.6 with (1.9) and (1.10) yields
Proof of Theorems 1.1 and 1.2.
We first present the proof of Theorem 1.1. For any , we define the characteristic function
[TABLE]
By Cauchy-Schwarz inequality, we infer
[TABLE]
where the symbol ∗ means that we sum over non trivial characters modulo . On the other hand, we have the lower bound for the left handside in the first line
[TABLE]
where is defined in (1.7) and
[TABLE]
To estimate , note that the condition means that one of the central values is less than . Therefore, if for , is the subsum of restricted to such that , we obtain, by positivity, with for each ,
[TABLE]
using again twice Cauchy-Schwarz inequality and (1.9) (recall that ). Moreover, opening the square in and applying the orthogonality relation yields
[TABLE]
since . Hence, assuming with , we get
[TABLE]
If
[TABLE]
then for any , there exists depending on satisfying . Finally, choosing large enough such that for and the result follows.
We proceed in a similar way. Setting
[TABLE]
we obtain
[TABLE]
where (resp. ) are defined by (1.9) (resp. by (1.10)). As in the previous part, the left handside admits the lower bound
[TABLE]
where is the same as , but with the absolute values inside and with the restriction in the summation to such that . Writing (resp. ) for the contribution of (resp. ), we get with
[TABLE]
Finally, we have
[TABLE]
with
[TABLE]
Hence,
[TABLE]
and the rest of the proof is exactly the same as in the previous case. ∎
Remark 1.7**.**
Let be a primitive Hecke cusp form of prime level satisfying the Ramanujan-Petersson conjecture. The formula (1.4) could be used to prove simultaneous non-vanishing for as runs over non trivial Dirichlet characters modulo provided that it is possible to evaluate a second twisted moment of the form
[TABLE]
where and are coprime with . An asymptotic formula for this moment is given in [BFK*+*14] in the special case where the level is and and for general in the paper of preparation mentioned above (also for level ). The principal difficulty here is that since the level is , we have the solve a shifted convolution problem of the shape
[TABLE]
for Hecke eigenvalues of level .
1.2 Sketch of the proof of Theorem 1.3
After an application of the approximate functional equation to (1.2) and (1.1), which expresses the central value of automorphic -functions as a convergent series, and an average over the characters, we isolate a main term which appears only if (c.f. § 6.1.1-6.1.3).
The treatment of the error term passes by the analysis of sums of the shape
[TABLE]
[TABLE]
where is the classical rank Kloosterman sum, is the twisted version as defined in (3.10), are the Hecke eigenvalues of which satisfy and are parameters such that
[TABLE]
The ultimate goal is to obtain a bound of the form
[TABLE]
for some absolute constant . Using Poisson summation in the three variables in (1.11), or Voronoi formula in the -variable in (1.12) (followed by Poisson on ) allows us to get rid of the cases where the product of the variables is larger than ; namely in § 6.1.6 and 6.2.4, we prove
[TABLE]
Combining these two estimates with the trivial bounds
[TABLE]
we can assume for the rest of this outline that
[TABLE]
We treat these sums differently according to the relative size of the various parameters. If (say) and , we exploit the -sum (resp. the -sum) in (1.11) (resp. in (1.12)) and average trivially over the others. Grouping into a long variable and we need to analyze roughly
[TABLE]
where for any ,
[TABLE]
In [FKM14] and [FKM15], Fouvry, Kowalski and Michel studied these sums when is a fixed cusp form of level and is replaced by the generalized divisor function They treated the problem in a more general case than , namely when the -periodic function arises as Frobenius trace function of a certain constructible -sheaves on the affine line for a prime number different from .
Our second main result is a generalization of their results (see [FKM15, Theorem 1.2] and [FKM14, Theorem 1.15]) for modular forms with nebentypus (possibly trivial) whose level is the same as the “level ” of the trace function and also for the twisted divisor function . The only restriction we made in comparison with their Theorems is that we avoid the case of exceptional trace functions (see Definition 3.19). More precisely, for a smooth and compactly supported function on , we define
[TABLE]
[TABLE]
We prove in Section 5 :
Theorem 1.8**.**
Let be a prime number, a Dirichlet character of modulus , a primitive Hecke cusp form of level , nebentypus and spectral parameter . Let be a non Fourier-exceptional isotypic trace function of conductor , as defined in (3.7),(3.9),(3.19), and a function satisfying see Definition (3.1). Then there exists absolute constants and such that
[TABLE]
for any .
Therefore, Theorem 1.8 provides the desired power saving (set ) in the special case where one of the variable is very small in (1.11) and in (1.12).
Assume now that and for some small real number . From now, we need to take care of the different nature of expressions (1.11) and (1.12). Indeed, for (1.11), the fact of having three free variables allows us to factorize two of them (say ) in such a way that . In this case, we can form a bilinear sum and use a general version of Polyá-Vinogradov (see Theorem 4.2) to obtain a power saving in the error term. The same method also works for (1.12), as long as , because in this case , or . Hence, the critical range for the second sum, i.e. when Polyá-Vinogradov is useless, appears when and and here we apply the general result of Kowalski, Michel and Sawin concerning bilinear forms involving classical Kloosterman sums [KMS15, Theorem 1.3].
1.3 Acknowledgment
This paper is part of my Phd thesis and I would like to thank my supervisor Philippe Michel for his valuable advice. I am also grateful to Satadal Ganguly for its careful reading and its reference to the paper [DK14] of S. Das and R. Khan.
2 Background on automorphic forms
In this section, we briefly compile the main results from the theory of automorphic forms which we shall need in Section 5. Among these are Hecke eigenbases, multiplicative properties of Hecke eigenvalues, the Kuznetsov trace formula and the spectral large sieve inequality.
2.1 Hecke eigenbases
Let be an integer, a Dirichlet character of modulus , and satisfying (mod ). We denote by , and , respectively, the Hilbert spaces (with respect to the Petersson inner product) of holomorphic cusp forms of weight , of Maass forms of weight , of Maass cusp forms of weight , with respect to the Hecke congruence group and with nebentypus . These spaces are endowed with an action of the commutative algebra generated by the Hecke operators . Among these, the with generate a subalgebra of made of normal operators. As an immediate consequence, the spaces and have an orthonormal basis made of eigenforms of . We denote this basis respectively by and .
The orthogonal complement of in is the Eisenstein spectrum (plus a constant if the character is trivial) and it is denoted by . The space is continuously spanned by the Eisenstein series where runs over singular cusps (with respect to ) of .
2.1.1 The Eisenstein series in the special case
Let be a prime number. For some technical reasons, it is convenient for the proof of Theorem 1.8 to see the form of level as a form of level (see the beginning of Section 4.1 and Section 5.5 in [FKM15]). For arbitrary level , the Eisenstein series are usually not eigenfunctions of the Hecke operators. In the special case where , there are exactly four inequivalent cups for which are
[TABLE]
see for example [Iwa97, Proposition 2.6] and all are singular. The main advantage in this situation is that these Eisentein series are eigenforms of the Hecke operators for . More precisely, if with , then we have for ,
[TABLE]
with explicitly
[TABLE]
see [DFI02, (6.16)-(6.17)].
Remark 2.1**.**
In the case , there are exactly two inequivalent cusps and the two Eisenstein series are eigenfunctions of the Hecke operators for with eigenvalues given by (2.1). Moreover, they are also Eisenstein series of level after the normalization by .
2.2 Multiplicative and boundedness properties of Hecke eigenvalues
Let be any Hecke eigenform of level and nebentypus and let be the corresponding Hecke eigenvalues for . Then for , we have
[TABLE]
[TABLE]
Note that if is holomorphic, then by the work of Deligne and Serre, we have the Ramanujan-Petersson conjecture, namely
[TABLE]
The same bound is of course trivial in the special case or and the Eisenstein series are eigenfunctions with eigenvalues given by (2.1). In the case of a Maass cusp form , the best result is due to Kim and Sarnak [K*+*03] and it is given by
[TABLE]
with an analogous bound for the spectral parameter
[TABLE]
where is the Laplace eigenvalue of . However, the conjecture is true on average, in the sense that for every ,
[TABLE]
with an implied constant depending only on [DFI02, Proposition 19.6]. We will also need later similar bound for the fourth-power on average and it is enough for our purpose to restrict to prime numbers not dividing the level
[TABLE]
for any and the constant only depends on . This bound is a consequence of the automorphy of the symmetric square and Rankin-Selberg theory (see for example [Mic07]).
2.3 Hecke eigenvalues and Fourier coefficients
Let be a modular form. For , we write the Fourier expansion as
[TABLE]
[TABLE]
where is a Whittaker function, as defined in [DFI02, § 4]. For an Eisenstein series , we write
[TABLE]
When is a Hecke eigenform, there is a closed relation between the Fourier coefficients and the Hecke eigenvalues ; for and , one has
[TABLE]
In particular, for all ,
[TABLE]
If is primitive, the relations (2.9) and (2.10) are valid for every . We will also need lower bounds for the first coefficient ; we have for any
[TABLE]
see [DFI02, (6.22),(7.16)] and [Mic04, Lemma 2.2 and (2.23)]. For an Eisenstein series , we have
[TABLE]
see [DFI02, (6.23),(7.15)].
2.4 Summation formula, trace formula and the spectral large sieve inequality
2.4.1 Voronoi summation formula
We state a version of Voronoi summation formula for the cuspidal case. For our purpose, it is enough to restrict to modular form of prime level .
Proposition 2.2**.**
Let be a prime number, a Dirichlet character of modulus and a primitive Hecke cusp form of level and nebentypus with associated Hecke eigenvalues . Given an integer coprime with and a smooth and compactly supported function, we set
[TABLE]
Then
[TABLE]
where
[TABLE]
with
[TABLE]
if is holomorphic of weight and
[TABLE]
if is a Maass form of parity and spectral parameter .
Proof.
The proof is contained in [KMV02, Appendix A.3-A.4] and also [Mic07, Lemma 2.3.1] for the holomorphic case. ∎
Finally, we consider the decay properties of the Bessel transforms (see [BFK*+*14, Lemma 2.4]).
Lemma 2.3**.**
Let be a smooth and compactly supported function satisfying
[TABLE]
for any and . In the non-holomorphic case set , otherwise set . Then for any , for any and all , we have
[TABLE]
2.4.2 The Petersson formula
For an integer such that , the Petersson trace formula expresses an average of product of Fourier coefficients over in terms of sums of Kloosterman sums (see [Iwa02, Theorem 9.6] and [IK04, Proposition 14.5]) : for any integers , we have
[TABLE]
where
[TABLE]
and the Kloosterman sum is defined by
[TABLE]
2.4.3 The Kuznetsov formula
Let be a smooth function satisfying and for and every . For , we define the two Bessel transforms
[TABLE]
Then for every integers , we have the following spectral decomposition of the Kloosterman sums,
[TABLE]
where
[TABLE]
see [Iwa02, Theorem 9.4 and 9.8].
2.4.4 The spectral large sieve inequality
Lemma 2.4**.**
Let be an integers and a Dirichlet character of modulus whose conductor is odd and squarefree. Then for any and , we have the Weil bound
[TABLE]
Proof.
The proof is a consequence of the twisted multiplicativity of Kloosterman sums and [KL13, Propositions 9.7 and 9.8]. ∎
Proposition 2.5**.**
Let be an integer, a Dirichlet character of modulus with squarefree and odd conductor and such that . Let , and a sequence of complex numbers. Then for any , each of the following three quantities
[TABLE]
[TABLE]
[TABLE]
is bounded, up to a constant depending only on , by
[TABLE]
Proof.
This result has been proved in [Dra15, Proposition 4.7], except that the bound (2.17) is replaced by , where is the conductor of . The extra factor in the proof of this Proposition appears in [Dra15, Lemma 4.6, (4.20)] and is a consequence of the general estimation [KL13, Theorem 9.3]
[TABLE]
By the hypothesis on the conductor of (that it is squarefree), we can apply Lemma 2.4 whose consequence is the cancellation of the factor in [Dra15, Lemma 4.6, (4.20)] and the rest of the proof is completely similar. ∎
2.5 -functions and functional equations
2.5.1 Dirichlet -functions
Let be a non-principal Dirichlet character of modulus with prime, satisfying and define the complete -function
[TABLE]
where
[TABLE]
It is well known that admits an analytic continuation to the whole complex plane and satisfies the functional equation (c.f. Theorem 4.15, [IK04])
[TABLE]
where is the normalized Gauss sum defined by
[TABLE]
Using (2.19), we can express the central value of a Dirichlet -function as a convergent series (see Theorem 5.3, [IK04]) and thus, extend in an easy way the proof to a product of three -functions.
Lemma 2.6**.**
Let be Dirichlet characters such that . If resp are such that resp , , then we have
[TABLE]
where
[TABLE]
for some entire and even function with exponential decay in vertical strips and satisfying
Remark 2.7**.**
Since and have been fixed, the function depends on only through its parity. Now shifting the -contour to the right in (2.21) and we see that for and any , we have the estimation
[TABLE]
Now moving the -line to , we pass a simple pole at of residu and thus, we obtain for
[TABLE]
2.5.2 Twisted -functions
Let be a prime number, a Dirichlet character of modulus and a primitive Hecke cusp of level and nebentypus . For a non trivial character modulo , we can construct the twisted modular form whose -th Fourier coefficient is given by . This form is again a Hecke eigenform of level with nebentypus and eigenvalues for (see [Iwa97, Chapter 7]), but it is not necessarily primitive because the level of and the conductor of are not coprime. In order to have a good understanding of this twist and a perfect description of the sign of the functional equation of the associated -function , it is convenient to switch into the representation theory language.
A well known principle associates to an automorphic cuspidal representation of , where is the adele ring of , with central character , the adelization of (see [GH11]) and of conductor (as defined in [Gel75, Chapter 5]). Such a representation admits a factorization into irreducible local representations of with central character and where .
Let be the adelization of the character . Then the twisted representation defined by admits the factorization [JL70, Corollary 11.2]. We can attach to an -function which can be represented as an infinite product over the primes of local -functions
[TABLE]
with the property that
[TABLE]
The -function is completed by the factor at the archimedean place and the product can be extended holomorphically to and satisfies a functional equation of the form
[TABLE]
where is the contragredient reprsentation and is called the root number. The representation being ramified only at and possibly at , we have
[TABLE]
The infinite factor is of modulus and depends on only through its parity (see for example [Gel75, Theorem 6.16]). To determine the finite part , we proceed as follows : Since the conductor of , which is equal to the conductor of , is a prime number, the local representation is not supercuspidal and thus corresponds either to an irreducible principal series, or to a special representation (see [Gel75, Remark 4.25]). More precisely, is a special representation when the original character is trivial. In this case, the two quasi character of are unramified and satisfy . If is not the trivial character, then is an irreducible principal series representation with , unramified and is ramified with conductor (the argument is again a consequence of [Gel75, Remark 4.25]). The twist by preserves principal series or special representations is the sense that if , then . Since is not trivial, is ramified and we obtain in both cases (principal or special)
[TABLE]
(see [Gel75, Theorem 6.15]), where for any quasi character of , is the root number appearing in the theory. If is unramified, we have , otherwise, writing for some and unitary with conductor , we have with (see [GH11, Chapter 2, Ex 2.6])
[TABLE]
where is the normalized Gauss sum
[TABLE]
It follows from [GH11, Definition 2.1.7] that
[TABLE]
We summarize all the previous computations in the following more convenient form:
Proposition 2.8**.**
Let be a prime number, a Dirichlet character of modulus and a primitive Hecke cusp form of level and nebentypus with associated Hecke eigenvalues for all . Then for every character modulo such that , the twisted modular form is a primitive Hecke cusp of level and nebentypus with associated Hecke eigenvalues for every . If
[TABLE]
is the associated -function, then there exists a factor such that the product
[TABLE]
extends holomorphically to and satisfies the functional equation
[TABLE]
where and the infinite factors and satisfy and both depends on only trough its parity.
Remark 2.9**.**
The infinite factor presents as a product of Gamma functions
[TABLE]
where are the local parameters at the infinite place (encodes the weight in the holomorphic setting or the Laplace eigenvalue if is a Maass form) and we recall that they depend on at most trough its parity. In any case, a consequence of the work of Kim and Sarnak [K*+*03] toward the Ramanujan-Petersson conjecture is that
[TABLE]
We finally state the analogous of Lemma 2.6 for the product on the critical point (see [Mic07, § 1.3.2] for the general result).
Proposition 2.10**.**
Let be a prime number, a Dirichlet character of modulus and a primitive Hecke cusp form of level and nebentypus with associated Hecke eigenvalues for all . Then for every character modulo such that , , we have
[TABLE]
where
[TABLE]
for some suitable entire even function with exponential decay in vertical strips and satisfying .
Remark 2.11**.**
Shifting the -contour on the right in (2.29) and we obtain that for every and any ,
[TABLE]
By (2.27), moving the -line to , we catch a simple pole at of residue and thus
[TABLE]
3 -adic twists of modular forms
In this section, we fix a prime number, a Dirichlet character modulo , a primitive Hecke cusp form of level and nebentypus and we denote by the Hecke eigenvalues of . For any , we also define the twisted divisor function by
[TABLE]
which, for , appears as Hecke eigenvalues of Eisenstein series of level and nebentypus for a suitable choice of cusp (c.f. § 2.1.1 and (2.1)).
As announced in Section 1.2, for a -periodic function, a crucial step in the proof of Theorem 1.3 requires non trivial estimates for sums of the shape
[TABLE]
[TABLE]
where is a smooth a compactly supported function on . Assuming that for every , we obtain by Cauchy-Schwarz inequality and (2.7),
[TABLE]
with an implied constant depending only on and the spectral parameter and this bound can be seen as the trivial one. Theorem 1.8 improves on (3.4) with a power saving in the -aspect, namely
[TABLE]
for any and with an implied constant depending on (or ) and controlled by some invariant of , called the conductor (see Definition (3.9)). As in [FKM15, Definition 1.1], we make the following definition about the test function .
Definition 3.1** (Condition ).**
Let and be real numbers and let be a sequence of non-negative real numbers. A smooth compactly supported function on satisfies Condition if
The support of is contained in the dyadic interval ; 2.
For all and all integer , we have the inequality
[TABLE]
In practice, the test function is not compactly supported, but rather in the Schwartz class. We give here a simple Corollary of Theorem 1.8.
Corollary 3.2**.**
Let be a prime number, a Dirichlet character of modulus , a primitive Hecke cusp form of level , nebentypus and spectral parameter . Let be a non Fourier-exceptional isotypic trace function of conductor . Let , a sequence of non-negative real numbers and a smooth function on with the property that for any ,
[TABLE]
Then for every and every , we have
[TABLE]
[TABLE]
Proof.
We consider the cuspidal case since the other is completely similar. Applying a partition of unity to leads to the decomposition
[TABLE]
where is a smooth and compactly supported function on satisfying for some sequence of non-negative real numbers and runs over real numbers of the form with . Since has fast decay at infinity, we can focus on the contribution of at the cost of an error of size , (say). Hence, we just need to bound sums of the form
[TABLE]
By Mellin inversion formula, we have for any
[TABLE]
where the function satisfies
[TABLE]
For fixed with , we apply Theorem 1.8 to the inner sum with the function which satisfies condition for some depending on , obtaining the bound
[TABLE]
Using the fact that the Mellin transform satisfies
[TABLE]
for every with an implied constant depending on and , we see that we can restrict the integral to . Hence replacing by its maximal value, maximizing over and average trivially over in the integral yields the desire result. ∎
3.1 Good functions and correlating matrices
We follow the same structure as in [FKM15]. We forget all the specific properties that might have and proceed abstractly in order to deal with a great level of generality. Therefore we consider the problem of bounding the sum for a general -periodic function, assuming only that we know that for some that we think as fixed. The strategy is to estimate an amplified second moment of for varying in a basis of cusp forms of level and nebentypus . After completing the spectral sum, applying Kuznetsov-Petersson and Poisson formula, we have to confront some sums that we call , which we now define.
We denote by or the normalized Fourier transform of given by
[TABLE]
and we let acts on the projective line by fractional linear transformations.
Definition 3.3** (Twisted correlation sum).**
Let . For a multiplicative character modulo and , we define the twisted correlation sum by
[TABLE]
Remark 3.4**.**
Note that for , is well defined up to multiplication by . This is in fact not a problem since only the complex modulus will be considered later (see for example Definition 3.5). We also mention that unlike the definition of correlation sum that the authors made for the original Theorem (see [FKM15, eq. (1.10)]), we have the presence here of a twist by the nebentypus of the modular form . This is because the Kloosterman sums that we obtain after the application of Kuznetsov trace formula are also twsited by .
Assuming , Cauchy-Schwarz and Parseval identity leads to
[TABLE]
Of course this bound will not be satisfactory. According to the square root cancellation philosophy, one expects bound of the form and that the s for which this estimate fails is well controlled. By this, we mean that this set of matrices (called correlation matrices) is nicely structured and rather small. Before giving a precise definition of correlation matrices and good functions, we introduce the following notations concerning algebraic subgroups of .
We denote by the subgroup of upper-triangular matrices, i.e. the stabilizer of 2. 2.
We write for the Weyl element, so that (resp. ) is the set of matrices mapping [math] to (resp. to [math]); 3. 3.
We denote by the subset of parabolic matrices in , i.e. which have a single fixed point in 4. 4.
Given in , the pointwise stabilizer of and is denoted (this is a maximal torus), and its normalizer in (or the stabilizer of the set ) is denoted .
Definition 3.5** (Good functions).**
Let be a prime, an arbitrary function and a Dirichlet character modulo . Let be such that .
We define
[TABLE]
the set of -. 2.
We say that is -good if and there exists at most pairs of distinct elements in such that
[TABLE]
where
[TABLE]
[TABLE]
If is a -good function, it turns out that the set of matrices obtained form the amplification method does not intersect the set of correlating matrices in a too large set. More precisely, the technical result is the following generalization of [FKM15, Theorem 1.9].
Theorem 3.6**.**
Let be a prime number, a primitive Hecke cusp form of level and nebentypus . Let be a function satisfying . Let be a -good function with . Then there exists an absolute constant such that
[TABLE]
[TABLE]
for any .
3.2 Trace functions of -adic sheaves
The functions to which we will apply Theorem 3.6 are called modulo , which we now define formally.
Let be a prime number and be an auxiliary prime. To any constructible -sheaf on and any point , we have a linear action of a geometric Frobenius acting on a finite dimensional -vector space (we can take if is lisse at , where is the geometric generic point [Kat80, § 4.4]). We can thus consider the trace . Because this trace takes value in , we also fix a field isomorphism
[TABLE]
and we consider functions of the shape
[TABLE]
as defined in [Kat90, (7.3.7)].
Definition 3.7** (Trace sheaves).**
-
A constructible -sheaf on is called a if it is a middle extension sheaf (in the sens of [FKM13, Section 1]) whose restriction on any non empty open subset where is lisse is pointwise pure of weight zero.
-
A trace sheaf is called a if in addition, it is a Fourier sheaf in the sense of [Kat88, Definition 8.2.2].
-
We say that is an if it is a Fourier trace sheaf and if for every non empty open subset as in (1), the associated -adic representation
[TABLE]
of the geometric etale fondamental group is an isotypic representation [Kat88, Chapter 2]. We define similarly an .
Definition 3.8**.**
Let be a prime number. A function is called a (resp. ) if there exists a trace sheaf (resp Fourier trace sheaf, isotypic trace sheaf) such that is given by (3.9).
There is an important invariant which measure the complexity of a trace function which we define now.
Definition 3.9** (Conductor).**
Let be a constructible -sheaf on with singularities in . The of is the integer defined by
[TABLE]
where if is lisse at (see for example [Kat80, Section 4.6]). If is a trace function, the conductor of is the smallest conductor of a trace sheaf with trace function .
3.3 Examples of trace functions
We give in this section basic examples of trace functions that we shall need later.
3.3.1 Kummer and Artin-Schreier sheaves
Let be a non trivial additive character and be a rational function with all poles of order . Then for every prime , there exists a trace sheaf of rank 1 on with the following properties :
for every (under the convention that ); 2.
is lisse on with equal to the order of the pole at ; 3.
We have .
Now let be a non-trivial multiplicative character of order and a rational function with zeros and poles of order not divisible by . Then for every prime , there exists a trace sheaf of rank 1 on such that
for every (under the convention that ); 2.
is lisse on with for every ; 3.
.
The sheaf is called an - and we say that has at because . The sheaf is called a and we say that is tamely ramified at the singularities because . We refer to [Del77, Sommes trig.] for the construction and properties of such sheaves and also to [PGdM16, Proposition 1.44-1.45].
3.3.2 Kloosterman sheaves
Let be an integer, be multiplicative characters on The twisted rank Kloosterman sum is the function on defined by
[TABLE]
for every . If , we write instead of . The main result is the existence of Kloosterman sheaves and it is due to Deligne [Kat88, Theorem 4.1.1].
Theorem 3.10** (Kloosterman sheaves).**
For every prime , there exists a constructible -sheaf on , denoted by (or simply ), of rank satisfying the following properties :
For every we have the equality
[TABLE] 2.
* is geometrically irreducible, lisse on and pointwise pure of weight zero;* 3.
* has wild ramification at with , tamely ramified at [math] and has conductor .*
In other words, since the rank is , is an irreducible trace sheaf in the sense of Definition 3.7 .
Corollary 3.11**.**
For every we have the sharp bound
[TABLE]
It will be convenient in Section 6, specially because of the Poisson summation and Fourier inversion formula, to present as a Fourier transform of a function defined in . For this, we let
[TABLE]
and we see that for any and , is given by the formula
[TABLE]
where the function is defined by
[TABLE]
Remark 3.12**.**
There are several ways to extend the function to . One can choose for example the extension by zero. We choose here the middle extension, i.e. that coincides with the trace of the Frobenius at . It is a deep result of Deligne that the estimate (3.11) remains valid for (see [Del80, (1.8.9)]).
3.4 The -adic Fourier transform and the Kummer-Möbius group
There is a deep interpretation of the Fourier transform at the level of sheaves who was discovered by Deligne and developed by Laumon, specially in [Lau87]. To be precise, let be a prime, another prime, a Fourier sheaf and a non-trivial additive character. Then there exists a Fourier sheaf with the property that
[TABLE]
for every and where is the choice of a square root of in which maps to under the fixed isomorphism . In particular,if we choose such that and is the trace function corresponding to , we obtain
[TABLE]
Moreover, the sheaf is geometrically isotypic (resp. geometrically irreducible) if and only if has this property [FKM15, Lemma 8.1].
Finally, taking in account the twist by the character in the Definition 3.3, we adapt the Definition of the Fourier-Möbius group given in [FKM15, Definition 1.15], which is a geometric interpretation of the set of correlating matrices, as follows :
Definition 3.13**.**
Let be prime number and an isotypic trace sheaf on with Fourier transform with respect to a non trivial additive character . Let be a multiplicative character. The Kummer-Möbius group is the subgroup of defined by
[TABLE]
Remark 3.14**.**
Note that Definition 3.13 makes sense in the sense that if are equal in , then and thus
[TABLE]
The key property here is that is indeed a subgroup of .
Proposition 3.15**.**
* is a subgroup of .*
Proof.
Let be the set of geometric isomorphism classes of trace sheaves. To show that is a subgroup, it is enough to prove that the map given by
[TABLE]
defines a right group action because will be the stabilizer of . For this, we will use the fact that we have geometric isomorphisms (we use the notation instead of )
[TABLE]
where denotes the constant sheaf. The first isomorphism implies that the identity matrix acts trivially. For the second part, note that for and , we have
[TABLE]
and
[TABLE]
Combining the above equality with the second isomorphism in (3.15) leads to
[TABLE]
Hence we obtain
[TABLE]
which completes the proof of this Proposition. ∎
We will also need the following fact about the conductor of .
Lemma 3.16**.**
Let be a trace sheaf and . Then
[TABLE]
Proof.
Since the Kummer sheaves are of rank one and tamely ramified at the singularities (c.f. § 3.3.1), we have for any (by definition (3.14) of the action of on ),
[TABLE]
see [Kat80, 4.6 (iv)]. Moreover, if (resp ) denotes the number of singularities of (resp. of ), the tensor product satisfies (see for example [PGdM16, Proposition 1.23])
[TABLE]
which completes the proof since or depending on whether or not. ∎
Here is a key Lemma.
Lemma 3.17**.**
Let be a prime number , and an isotypic trace sheaf on with corresponding isotypic trace function . Then for
[TABLE]
we have the inclusion
[TABLE]
Proof.
Let be the -adic Fourier transform of . The proof of this Lemma is a consequence of the Riemann hypothesis over finite field and we sketch here how it works.
As in [FKM15, Theorem 9.1], the Lefschetz trace formula can be used to derive that for any , we have
[TABLE]
with
[TABLE]
and
[TABLE]
and where is the dual sheaf, is the first étale cohomology group with compact support. Here is an open subset where the sheaf is lisse and can be taken such that
[TABLE]
where we used Lemma 3.16 for the last inequality. Using [FKM15, Proposition 8.2] which is a consequence of the Grothendieck-Ogg-Schavarevich formula and again Lemma 3.16, we obtain
[TABLE]
Now according to the bounds (see [FKM15, Proposition 8.2 (1), (8.4)-(8.5)])
[TABLE]
we obtain
[TABLE]
and the result follows. ∎
Let be an isotypic trace sheaf with corresponding trace function and with . Suppose that its Fourier-Möbius group satisfies condition in Definition 3.5 for some with . Then by Proposition 3.17, choosing and we have the inclusion , so is -good and Theorem 1.8 follows from Theorem 3.6. The following Proposition, which is a simple variant of [FKM15, Lemma 9.3] and follows from classification of Artin-Schreier sheaves, shows that contains parabolic elements if the Fourier transform contains in its irreducible component a sheaf of the form .
Lemma 3.18**.**
Let be a prime and an auxiliary prime. Let be a non trivial -adic additive character of and let be a finite direct sum of copies of . Then for , we have a geometric isomorphism if and only if , where for any , denotes the unipotent radical of the Borel subgroup of fixing .
Proof.
If , then by definition of the action (3.14), we have geometrically
[TABLE]
Conversely, assume that . We show first that we also have Indeed, writing (resp. ) for the inclusion of (resp. the projection in) the -th component, we can choose such that the morphism
[TABLE]
is not the zero map. Since the rank is one on both side, we conclude that it is an isomorphism. Hence we obtain
[TABLE]
The sheaf has wild ramification at and is tame at this point (see § 3.3.1), so we must have , which implies that and thus . The conclusion follows now from the classification of Artin-Schreier sheaves. ∎
It turns out that if contains parabolic elements, then contrary to the original Theorem of [FKM15], the size of its intersection with the matrices constructed from the amplification method may be too large. We therefore formulate the last definition :
Definition 3.19**.**
An isotypic trace sheaf is called Fourier-exceptional if the irreducible component of its Fourier transform is of the form for some and some (possibly trivial) additive character .
With all these definitions, we can now formulate the last result which, together with Theorem 3.6, immediately implies Theorem 1.8.
Theorem 3.20**.**
Let be a prime number, and a non-exceptional isotypic trace sheaf, as in Definition 3.19, on with . Let be the corresponding isotypic trace function. Then there exists absolute constants and such that is -good as in Definition 3.5.
Proof.
By Proposition 3.17, there exists such that
[TABLE]
which is a subgroup of . If the order of is coprime with , then we proceed as in the first paragraph of the proof of [FKM15, Theorem 1.14].
We now show that under the non-exceptional assumption, the order of cannot be divisible by . Assume by contradiction that it is the case and fix an element of order . Then is necessarily parabolic, so it has a unique fixed point . Let be such that
[TABLE]
and define for . Since geometrically we have for any trace sheaf , we see that we have a geometric isomorphism
[TABLE]
Suppose first that is ramified at some , then by the above, is also ramified at and we obtain by Lemma 3.16
[TABLE]
and in this case is -good for trivial reasons.
Assume now that is lisse on . Since is geometrically isotypic, the same is true for and the geometrically irreducible component of also satisfies . Using [FKM13, Lemma 5.4, (2)] with and , we have either
[TABLE]
and in this case we are done as before, or is geometrically isomorphic to some Artin-Schreier sheaf for some additive character . It follows that is geometrically isomorphic to a direct sum of copies of and thus, by definition of , we have a geometric isomorphism
[TABLE]
which contradicts the fact that is not Fourier-exceptional. ∎
4 Bilinear forms involving trace functions
We begin with a classical result.
Proposition 4.1** (Polyá-Vinogradov method).**
Let be a prime number and be a Fourier trace sheaf on with corresponding trace function modulo . Let be a smooth and compactly supported function on and be a real number. Then for any , we have
[TABLE]
where the implied constant depends on and the conductor of .
Proof.
The first bound is clear. For the second, an application of Poisson summation formula gives
[TABLE]
where is the Fourier transform defined in (3.7) and is the analytic Fourier transform
[TABLE]
Since has fast decay at infinity, the sum is essentially supported on and thus, using the fact that the infinity norm of is bounded in terms of the conductor of (c.f. (3.16)) and estimating trivially the right handside of (4.1) yields the second bound. ∎
A more elaborate treatment of the Polyá-Vinogradov method can be used to obtain bounds for bilinear sums [FKM14, Theorem 1.17].
Theorem 4.2**.**
Let be an isotypic trace function modulo associated to an isotypic -adic sheaf such that does not contain a sheaf of the form in his irreducible component. Let be parameters and two sequences of complex numbers supported on and respectively.
We have
[TABLE]
with
[TABLE] 2.
We have
[TABLE]
In both estimates, the implicit constants depend only, and at most polynomially, on the conductor of
The above theorem beats the trivial bound and gives a power saving in the error term as long as and for some . In the critical case where , we have the powerful result of Kowalski, Michel and Sawin, which still saves a small power of , but has been proved in the special case of classical Kloosterman sums [KMS15, Theorem 1.3] and [BFK*+*14, Theorem 5.1].
Theorem 4.3**.**
Let be a prime number and an integer coprime with . Let be such that
[TABLE]
Let be a sequence of complex numbers and be an interval of length . Then for any , we have
[TABLE]
with
[TABLE]
where the implied constant in (4.3) only depends on and .
5 Proof of Theorem 3.6
This section is devoted to the proof of Theorem 3.6. For the cuspidal case, we will indicate the necessary changes in [FKM15, Sections 4,5,6] due to the level and the presence of a nebentypus. Finally, we will explain how to adapt [FKM14, Section 2] and put together with Section 5.1 to obtain the conclusion of Theorem 3.6 in the Eisenstein case.
5.1 The cuspidal case
5.1.1 The amplification method
Let be a prime number, a Dirichlet character of modulus and such that . Let be a -normalized primitive Hecke cusp form of weight (resp. with Laplace eigenvalue ) if is holomorphic (resp. if is a Maass form) of level and nebentypus . For some technical reasons, it is convenient to view as a modular form of level (see the beginning of § 2.1.1) under the isometric embedding (with respect to the Petersson inner product)
[TABLE]
which can be embedded in a suitable orthonormal basis of modular cusp forms of level , i.e. either or . The strategy is therefore to estimate an amplified second moment of the sum where runs over a basis of and .
To be precise, let and a sequence of coefficients supported on . For any modular form , we let
[TABLE]
For an Eisenstein series , we set
[TABLE]
where for any singular cusp , is given by (2.1). Since the original form is of level and will be at the end a small power of , we cannot choose the standard coefficients for a prime , but rather the less obvious amplifier found by Iwaniec in [DFI94],
[TABLE]
Since we will apply the trace formula, it is also better to consider the Fourier coefficients instead of the Hecke eigenvalues in the definition of . For this, we define
[TABLE]
and note that for primitive, it is related to the original sum by the simple relation (c.f. (2.10)),
[TABLE]
We then let
[TABLE]
where for any with
[TABLE]
and we refer to [FKM15, Section 3.2] or [BHM07, (2.9)] for the choice and properties of the test function . The key Proposition is the following [FKM15, Proposition 4.1].
Proposition 5.1**.**
Assume that is such that is -good. Let be a smooth compactly supported function satisfying Condition . Let be the sequence of complex numbers defined by (5.1).
For any , there exists such that for any and any integers satisfying
[TABLE]
we have the bound
[TABLE]
provided
[TABLE]
and where the implied constant depends on and polynomially on the archimedean parameter of .
Theorem 3.6 can be deduced from Proposition 5.1 exactly in the same way as in [FKM15, Section 4.2]. The only changes is to use (5.2) to pass from to and then (2.11) for the upper bound on . Finally, since for any prime different from we have the elementary relation
[TABLE]
we obtain the lower bound
[TABLE]
simply using the prime number Theorem. Hence it remains to prove Proposition 5.1.
Expanding the square in and (choosing the variables for those comming from the amplifier), we get a first decomposition of and
[TABLE]
depending on weither (the diagonal term) or not. For the diagonal term, we have the following lemma which is the analogous of [FKM15, Lemma 5.1].
Lemma 5.2**.**
Assume . For any , we have
[TABLE]
where the implied constant depends only on .
Proof.
We consider which decomposes as a sum of the holomorphic, Maass and Eisenstein contributions
[TABLE]
We treat only since the others contributions are the same and even simpler. For instance, we have
[TABLE]
with
[TABLE]
By definition of the coefficients (c.f. (5.1)), the case appears when , (or the inverse) and . We write according to the different possibilities and we estimate the three quantities individually. We first have by Cauchy-Schwarz inequality and (2.8),
[TABLE]
where the implied constant only depends on . For the second case, we have using (c.f. (2.2)), Hölder and again (2.8),
[TABLE]
Using the inequality , we treat in the same way . The rest of the proof is exaclty the same as [FKM15, Lemma 5.1], except that we must use Proposition 2.5 for the spectral large sieve (possible since the conductor of is either or a prime ) instead of the original version of Deshouillers-Iwaniec [DI82, Theorem 2, (1.29)]. ∎
Now comes the contribution of the such that . We first change the complex conjugate in and appearing in the decomposition (5.7) (c.f. (2.3)). By the primiality condition, we use the multiplicativity of the Hecke eigenvalues (2.2) followed by the relation (2.9) to obtain
[TABLE]
Once we have done this, we apply the Petersson trace formula (2.15) to in (5.7), obtaining
[TABLE]
where corresponds to the diagonal term . Similarily, we apply Kuznetsov formula (2.16) to and since there is no diagonal term, we write
The treatment of the diagonal term is contained in [FKM15, Lemma 5.3], with the appropriate changes using (2.7) for the coefficients of the amplifier,
[TABLE]
with an implied constant depending only on .
5.1.2 The off-diagonal terms
This is the most important case of and and thus we write explicitly the quantities to study. For an arbitrary function, we write
[TABLE]
in order to have
[TABLE]
where is the Bessel function. We transform the sum as
[TABLE]
where
[TABLE]
and
[TABLE]
with
[TABLE]
As in [FKM15, Section 5.4], we truncate the parameter in by writing where denotes the contribution of ther terms with for some and correspondingly
[TABLE]
It turns out that with the choice
[TABLE]
the contribution of is negligible (see [FKM15, (5.9)]), so we focus on the complementary sum, which is given by
[TABLE]
In particular, the above expression is zero if .
Recall that we factored the product as . Since we allow and to be square of primes, there are more different type of factorization to consider. We distinguish three types.
Type I (balanced case) : this is when both and are and , so and are either primes (type ()) or square of primes (type ) with in each case. 2.
Type II (unbalanced case) : this is when satisfies , i.e. is of type and . 3.
Type III (unbalanced case) : this is when satisfies , so is of type and
Assuming the harmless condition
[TABLE]
we obtain by (5.13) :
Lemma 5.3**.**
Suppose that is of Type III and that (5.15) is satisfied. Then we have the equality
[TABLE]
It remains to deal with the types I and II. The goal now is to transform the sums to connect them with the correlation sums defined in (3.3) for suitable matrices . This is the content of [FKM15, Section 5.5] and it is achieved using principally twisted multiplicativity of the Kloosterman sums and Poisson summation formula. The only difference here is the appearance of the nebentypus when we open the Kloosterman sum. We also mention that it is in this treatment that we use the fact that the level is and not . The result is that for any , we have the identity
[TABLE]
where is the Fourier transform of and
[TABLE]
Remark 5.4**.**
Observe that which is coprime with . Hence the reduction of modulo provides a well defined element of .
5.1.3 Analysis of
The first step in the analysis of (5.16) passes by the study of the Fourier transform of . This is the content of [FKM15, Sections 5.6-5.7] and it is contained in Lemmas 5.7,5.9. One of the consequences is that it allows to truncate the -sum in to
[TABLE]
(see [FKM15, (5.21)]). The final strategy is to separate the terms in (5.16) (with the restriction (5.18) on ) according to whether
[TABLE]
or not, i.e., as to whether the reduction modulo of the matrix is in the set of -correlation matrix or not. We thus write
[TABLE]
where is the subsum of (5.16) where we restrict to the variables such that the reduction modulo of belongs to and is the contribution of the remaining terms. According to (5.14), (5.12) and (5.10), we also write
[TABLE]
and
[TABLE]
Lemma 5.5**.**
With the above notations, we have
[TABLE]
for any where the implied constant depends on for and on for .
Proof.
This is the content of [FKM15, pp. 625-626], with minimal changes due to the different nature of pairs of type I and II. ∎
To conclude the proof of Proposition 5.1, it remains to evaluate the contribution corresponding to the matrices whose reduction modulo is a correlating matrix, i.e. such that . The final lemma is the following :
Lemma 5.6**.**
Under the assumption
[TABLE]
we have
[TABLE]
where the implied constant depends on .
Proof.
The proof is [FKM15, Sections 6.1,6.3,6.5] (recall that here there are no parabolic elements). Various arguments use the fact that the discriminant of certain binary quadratic form is not zero. For example, if is a toric matrix, then we need to have and we cannot say that is squarefree since we allow square of primes in the amplifier. This is not a problem here because if , then we would get (see (5.17))
[TABLE]
This means that has only one distinct eigenvalue, so its is necessarily scalar since not parabolic by assumption. But for scalar, we have , wich is not possible by (5.19) and (5.13).
Another argument uses the fact that in a situation where (c.f. [FKM15, Section 6.3, p.p 629]). Again, and are not necessarily coprime here so we cannot argue in the same way. However, if , then since , we obtain and the matrix takes the form
[TABLE]
Since , this matrix is parabolic with single fixed point , which contradicts the fact that does not contain parabolic elements. ∎
5.2 The Eisenstein case
We recall the notations from Section 1. For prime, a Dirichlet character modulo and , we set
[TABLE]
and
[TABLE]
for a non Fourier-exceptional isotypic trace function and satisfying condition . Since appears as Hecke eigenvalues (for ) of the Eisenstein series (the cusp ) lying in the continuous spectrum of the Laplacian on the space of modular forms of level (and thus also of level after a normalization) and nebentypus (see § 2.1.1 and Remark 2.1), we may estimate an amplified second moment of by embedding in the Eisentsein spectrum and using Kuznetsov trace formula as in the cuspidal case.
For , we define as in (5.1)
[TABLE]
and for a cuspidal form, we set
[TABLE]
For an Eisenstein series , we let
[TABLE]
and we also write so that it corresponds to the Eisenstein series having as Hecke eigenvalues. Since the trace formula involves Fourier coefficients instead of Hecke eigenvalues, we define as in Section 5.1
[TABLE]
with the relation
[TABLE]
Remark 5.7**.**
Actually, the relation (5.21) is true if we restrict the -summation in to . However, we could consider directly this restriction at the beginning since the error to pass from one to the other is given by
[TABLE]
Using the lower bound for given by (2.12) and (c.f. [BHM07, (2.21)]), we obtain exactly as in Section 5.1 (see Proposition 5.1),
[TABLE]
In order to apply (5.22), the following Lemma gets a suitable lower bound for the amplifier when is close enough to (see [FKM14, Lemma 2.4]).
Lemma 5.8**.**
For large enough, we have
[TABLE]
uniformly in satisfying
[TABLE]
Proof.
Observe that since has at most three divisors, we have and thus
[TABLE]
It is therefore enough to prove the result for . But this is a consequence of the elementary relation
[TABLE]
valid for , and the prime number Theorem. ∎
The above Lemma combining with the average bound (5.22) allows us to deduce a first upper-bound for short averages of twists of Eisenstein series. For this, we introduce the notations
[TABLE]
and
[TABLE]
so that Theorem 3.6 claims that
[TABLE]
for any and some depending on .
Proposition 5.9**.**
For any , there exists , depending only on , such that for any we have
[TABLE]
where the implied constant depends only on .
Proof.
Using Lemma 5.8 and (5.22), we obtain
[TABLE]
and we conclude as in [FKM15, Section 4.2] by choosing
[TABLE]
and which depends on . ∎
The last step is to derive a pointwise bound for . For this, we separate the variables in the twisted divisor function and using a partition of unity, we can decompose into sums of the shape
[TABLE]
where the parameters belongs to the set
[TABLE]
and are smooth and compactly supported functions on satisfying for every . It follows that
[TABLE]
The relation between and and an average of is given through the Mellin transform (see [FKM14, Lemma 2.1]).
Lemma 5.10**.**
Given and , we define
[TABLE]
Then for every , we have
[TABLE]
Proof.
Using Mellin inversion formula for and , we can write
[TABLE]
where denote the Mellin transform of the smooth functions and
[TABLE]
Note that this sum can be expressed as a twist of Eisenstein series, namely
[TABLE]
with
[TABLE]
For the smooth function satisfies condition with
[TABLE]
where the implied constant is absolute. Thus by a change of variables, we get
[TABLE]
Because we have the estimations
[TABLE]
with an implied constant depending on and , we can truncate the integral (5.28) to , for a cost of by taking large enough in term of and using the trivial bound for
[TABLE]
∎
5.2.1 Conclusion
We are now in position to obtain the conclusion of Theorem 3.6 in the Eisenstein case. Indeed, fix and take as in Proposition 5.9. By (5.25), it is enough to estimate for . Now let such that we have the estimate (5.26) of Lemma 5.10. We thus get
[TABLE]
We split the above integral into integrals over intervals of length . For such interval centered at , we obtain by Proposition 5.9, the value (5.27) and Cauchy-Schwarz inequality, the bound
[TABLE]
(the function is increasing). Finally, taking the maximal value yields the desire result.
6 Proof of Theorem 1.3
6.1 The Eisenstein case
It is natural to separate the sum in (1.1) into even and odd primitives characters because they have different gamma factors in their functional equations. We will treat only the case of even characters since the odd case is completely similar. We therefore consider
[TABLE]
6.1.1 Applying the approximate functional equation
Applying the approximate function equation provided by Lemma 2.6, we decompose (6.1) into two terms
[TABLE]
with
[TABLE]
and
[TABLE]
where the symbol ∗ over the sum means that we restrict to and the function is defined in (2.21). In particular, since we sum over even characters, this function is constant on the average and we write instead of
Remark 6.1**.**
The function has rapid decay at infinity by Remark 2.7, so the sums (6.2)-(6.3) are essentially supported on . It follows that the sum over is trivially bounded by , so we can remove as it suits us the contribution of or for an error of size .
6.1.2 Average over the primitive and even characters
We need to average the sum over the characters in (6.2)-(6.3). For this, we use some orthogonality relations asserting that for any prime and any integer corpime with , we have (c.f. [IS99, (3.2)-(3.4)])
[TABLE]
and for
[TABLE]
where the supscript means that we sum over such that . In (6.2), we remove the contribution of (see Remark 6.1) and after applying (6.4), we get with
[TABLE]
For (6.3), we remove the contribution of and note that for we have, opening the Gauss sum and using (6.5),
[TABLE]
The second expression in the above parenthesis is easily computed as a Gauss sum. For the first term, we have
[TABLE]
where the twisted rank Kloosterman sum is defined by (3.10) (see also (3.12)). Hence we see that (6.7) equals
[TABLE]
Now opening the Gauss sum and using (6.8), we obtain for every ,
[TABLE]
Finally, applying (6.9) in (6.3) with yields (recall Remark 6.1) with
[TABLE]
We will evaluate each of these two terms ((6.6)-(6.10)) separately and find that a main term appears only in when . The others will contribute as an error term.
6.1.3 The Main Term
The main contribution comes from in (6.6). Indeed, assuming , we obtain by the Remark 2.7
[TABLE]
When (mod ) with , we write the congruence equation in the form with . Therefore, we get that the contribution of is at most
[TABLE]
We conclude with
[TABLE]
which gives the desired main term of Theorem 1.3 provided
[TABLE]
6.1.4 The error term
In this section, we analyze the expression (6.10) and will find that it contributes as an error term. Applying a partition of unity to in order to locate the variables and we obtain with
[TABLE]
where the functions are smooth and compactly supported on and the s runs over real numbers of the form , . By the fast decay at infinity of , we can assume that
[TABLE]
Hence it remains to bound sums of the shape (6.12). It is also convenient to separate the variables in the test function . This can be done using its integral representation (2.21) and we refer to [Zac16, Section 4.1] for more details. We keep the same notation , but with the factor removed, and also for the functions , i.e.
[TABLE]
with
[TABLE]
Note finally that the trivial estimate is
[TABLE]
6.1.5 Polyá-Vinogradov bound
We show here that (6.13) is very small if we assume that one of the parameters is greater than . Indeed, since the argument is the same, we suppose that . In this case, for fixed we focus on the -sum
[TABLE]
By Remark 3.12, we can add the contribution of for an error of size (since ). Hence, applying Proposition 4.1 with the Fourier trace sheaf
[TABLE]
we get
[TABLE]
Finally, averaging trivially over and in (6.13) yields
[TABLE]
Since , we obtain
Proposition 6.2**.**
Assume that for some Then for any , we have
[TABLE]
with an implied constant depending only on .
6.1.6 Applying Poisson summation in the three variables
In this section, we obtain an estimate for which is satisfactory if the product of the three variables is greater that . This can be done using successive applications of Poisson summation formula. Before this, we just note that by Proposition 6.2, we can assume that for , which allows us to ignore the primality condition in (6.13) since we also have .
We begin with the -variable. In (6.13), we write the Kloosterman sum as the Fourier transform of the function defined in (3.13). Hence, an application of Poisson summation in and Fourier inversion formula gives (recall that )
[TABLE]
Since by Definition for , we obtain
[TABLE]
We continue with the -variable. As before, since the argument of is non zero modulo , we can express the Kloosterman sum as suitable Fourier transform, namely (see (3.13))
[TABLE]
Using exactly the same argument as before, after replacing by (6.17) in (6.16), we get
[TABLE]
Applying Poisson in the -variable yields
[TABLE]
Hence
[TABLE]
It remains to do Poisson in the -variable. Let , we denote by the Dirac function on defined by if and zero else. Then the exponential map
[TABLE]
is the additive Fourier transform of the Dirac function . It follows that the sum in (6.18) equals
[TABLE]
Summarizing all the previous computations yields the bound
[TABLE]
Finally, using the fact that all these Fourier transform have fast decay at infinity, we see that the above sum is essentially supported on and thus, a trivial estimate leads to
[TABLE]
6.1.7 Estimation of using Theorems 1.8 and 4.2
We return to expression (6.13). The combination of (6.15) and (6.19) shows that it remains to deal with the case where the product is of length about . The strategy is the following : if one of the variables is very small, then we factorize the two others to form a new long variable and apply Theorem 1.8 for the twist of Eisenstein series. If all are not too small, then it is possible to factorize two variables and form a bilinear sum in such a way that an application of Theorem 4.2 is beneficial.
We prove in this section :
Proposition 6.3**.**
Let and write for the remaining parameter, i.e. . Then for every , we have
[TABLE]
Proof.
To fix the ideas, we assume that
[TABLE]
and we leave it to the reader to ensure that the other cases treated with minimal changes.
We first focus on the -sum in (6.13) and write it in the form (recall that so the primality condition is satisfied)
[TABLE]
We show now how to transform this expression in order to obtain the same as in Corollary 3.2. To simplify notations, we define
[TABLE]
Using Mellin inversion on and in (6.21) leads to
[TABLE]
Making the change of variables
[TABLE]
and we see that the above integral takes the form
[TABLE]
where for any and , we defined
[TABLE]
Because the Mellin transforms satisfy (c.f. (6.14))
[TABLE]
with an implied constant depending on , and , the function satisfies
[TABLE]
uniformly in . Since we want to estimate the inner sum in (6.23) using Theorem 1.8 and then average trivially over the -integral, we also need to control the function with respect to the -variable. By (6.25), for any , we have uniformly on and with an implied constant depending only on ,
[TABLE]
Note the identity
[TABLE]
Hence, splitting the integral depending on whether or not and we get
[TABLE]
Therefore, for any , we obtain that (6.23) is bounded, up to a constant which depends only on , by
[TABLE]
We now apply Corollary 3.2 with the Schwartz function and with the sheaf
[TABLE]
having trace function (6.22). Note that since is invariant under permutation of the triple , we have by (3.12) a geometric isomorphism
[TABLE]
and hence is not Fourier-exceptional since by Fourier inversion, its -adic Fourier transform is a rank irreducible sheaf. It follows that for any , we can estimate (6.26) by
[TABLE]
Choosing , maximizing the above quantity by setting , replacing the obtained bound in (6.21) and finally, averaging trivially over in (6.13) yields the first estimate of (6.20)
[TABLE]
For the second bound, we group together the variables in (6.13) and we obtain
[TABLE]
with
[TABLE]
Applying Theorem 4.2 (1) with and gives
[TABLE]
as wishes. ∎
6.1.8 Conclusion of the Eisenstein case
Write with and let be a parameter. If or , we use the trivial bound (6.15) or the estimate (6.19) to obtain
[TABLE]
We therefore assume that we are in the range
[TABLE]
Let be an auxiliary parameter. As we already see, there is no loose of generality assuming that . Suppose first that
[TABLE]
In this case, we apply (6.27) which, combining with (6.30) and (6.31) gives
[TABLE]
provided
[TABLE]
which condition we henceforth assume to hold.
Suppose now that we are in the case
[TABLE]
The estimate (6.29) leads to
[TABLE]
The first term is clearly smaller than by (6.32). For the second, note that and thus, by (6.30)
[TABLE]
It follows that
[TABLE]
under the assumption that
[TABLE]
Finally, the combination of (6.30), and (6.33) gives
[TABLE]
and hence
[TABLE]
provided
[TABLE]
which is more restrictive than (6.34). To finalize the computations, we just note that the second condition in (6.32) and (6.35) are simultaneously satisfied as long as , which gives the correct exponent of the error term in Theorem 1.3.
6.2 The cuspidal case
We consider as in Section 6.1 the average over primitive and even characters (recall that the nebentypus is trivial)
[TABLE]
6.2.1 Applying the approximate functional equation
Using Proposition 2.10, we can write (6.36) in the form
[TABLE]
with
[TABLE]
[TABLE]
where we recall that depends on only through its parity. Since we assume that satisfies the Ramanujan-Petersson conjecture, we have . Hence, proceeding as in § 6.1.2 for the average over the characters and writing , we find
[TABLE]
where
[TABLE]
and
[TABLE]
6.2.2 The main term
The extraction of the main term is done is a similar way as in § 6.1.3. We just conclude with
[TABLE]
Note that the error terms are (c.f. Theorem 1.3) if we assume that
[TABLE]
6.2.3 The error term
Applying a partition of unity to (6.38), removing the test function using its integral representation (see § 6.1.4) and we are reduced to analyze sums of the shape
[TABLE]
where are smooth and compactly supported functions on such that for all and are reals numbers with the standard restriction due to the fast decay of at infinity
[TABLE]
Note that the trivial bound is
[TABLE]
Moreover, if , then an application of Polyá-Vinograov method in the -variable (see Proposition 4.1 and § 6.1.5) leads to
[TABLE]
Hence we can suppose from now on that in such a way that the condition under the summation in (6.38) is automatically satisfied.
6.2.4 Application of Poisson/Voronoi summation formula
The first step is to apply Voronoi summation formula in the -variable. To get in a good position, we write the Kloosterman sum for in the form
[TABLE]
Note that this definition can be extended to with the value
[TABLE]
It follows that after writing in the form (6.42) and adding the contribution of for negligible error term (of size at most ), we get
[TABLE]
Assuming we are dealing with the plus case and applying Voronoi formula (c.f. Proposition 2.13) to the inner sum in (6.43), we obtain
[TABLE]
Changing the order of summation, making the change of variable (recall that ) allows us to write
[TABLE]
By Poisson formula and since is the Fourier transform of the function defined by (3.13), we see that the -sum in (6.44) is equal to
[TABLE]
Replacing this identity in (6.44) yields
[TABLE]
with the same expression for the minus case , but with replaced by in the exponential. Because of the fast decay of and at infinity (c.f. Lemma 2.3), the -sum (6.45) is essentially supported on and . In this range, we use the estimate and we apply Lemma 2.3 with (recall that satisfies R-P-C) to bound by . Adding the contribution of , estimating this extra factor trivially and executing the complete -summation gives
[TABLE]
Therefore, as in § 6.1.6, we obtain
[TABLE]
6.2.5 Estimation of using bounds for bilinear forms and Theorem 1.8
We finally state the analogous of Proposition 6.3 which is an immediate application of Theorem 4.2 (1)-(2), Theorem 4.3 and Corollary 3.2.
Proposition 6.4**.**
For any , the quantity defined in (6.40) satisfies
[TABLE]
where the implied constant depends on and polynomially on in the last bound and the third bound is valid in the case where , and c.f. (4.2).
6.2.6 Conclusion of the cuspidal case
Fix a parameter and write , with . By the trivial bound (6.41) and (6.46), we can assume that
[TABLE]
otherwise we get We now let be sufficiently small auxiliary parameters and we distinguish four cases :
Assume that In this case we apply the fourth estimate of Proposition 6.4 and we get by (6.47)
[TABLE]
provided
[TABLE] 2.
If , the first bound of Proposition 6.4 yields
[TABLE]
The first term is less than since . For the second, we have (use (6.47) and ). Thus it is less than under the assumption that
[TABLE]
The third term is at most if
[TABLE] 3.
Suppose that In this configuration, we apply the third bound and we obtain
[TABLE]
Using (6.47) and allows us to bound the above expression by
[TABLE]
provided
[TABLE] 4.
Assume that the second bound gives
[TABLE]
if we assume that
[TABLE]
Finally, the combination of conditions (6.48) and (6.49) forces and (6.50)-(6.51) are simultaneously satisfied as long as , which gives the correct exponent of the error term in Theorem 1.3.
Remark 6.5**.**
The treatment carried out in Section 5.1 remains almost identical if is level Hecke cusp form. The only change we have to make is to replace the exponent by in the fourth bound of Proposition 6.4, which is due to the original Theorem [FKM15, Theorem 1.2] for small level compared with . However, it does not improve the final exponent since (6.50)-(6.51) is more restrictive and independent of (6.48)-(6.49).
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