Arithmetic statistics of modular symbols
Yiannis N. Petridis, Morten S. Risager

TL;DR
This paper proves conjectures about the statistical behavior of modular symbols, including their moments and distribution, using analytic techniques involving Eisenstein series, advancing understanding of their asymptotic properties.
Contribution
It establishes average growth of moments and the Gaussian distribution of modular symbols, refining previous conjectures with new analytic proofs.
Findings
Proved asymptotic growth of first and second moments of modular symbols.
Established Gaussian distribution of normalized modular symbols with cusp restrictions.
Used analytic properties of Eisenstein series twisted by modular symbols.
Abstract
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve -functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
| Experimental Shift | ||
| 1 | -0.440048 | -0.440 |
| 3 | -0.244592 | -0.246 |
| 5 | -0.153710 | -0.153 |
| 15 | -0.041745 | -0.040 |
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Arithmetic statistics of modular symbols
Yiannis N. Petridis
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
and
Morten S. Risager
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
Abstract.
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve -functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.
2010 Mathematics Subject Classification:
Primary 11F67; Secondary 11E45, 11M36.
The second author was supported by a Sapere Aude grant from The Danish Council for Independent Research (Grant-id:0602-02161B). The first author would like to acknowledge the hospitality of the University of Copenhagen, while this paper was written.
1. Introduction
Modular symbols are fundamental tools in number theory. By the work of Birch, Manin, Cremona and others they can be used to compute modular forms, the homology of modular curves, and to gain information about elliptic curves and special values of -functions. In this paper we study the arithmetical properties of the modular symbol map
[TABLE]
Here is a holomorphic cusp form of weight for the group , and denotes the homology class of curves between the cusps and .
For our purposes it is convenient to work with the real-valued, cuspidal one-form . The finite cusps are parametrized by , so for we write
[TABLE]
Note that the path can be taken to be the vertical line connecting to .
If is equivalent to under the -action such that for some we write the map (1.1) as
[TABLE]
where in the last expression we have replaced by any .
Mazur, Rubin, and Stein [MR16, Ste15] have recently formulated a series of conjectures about the value distribution of . We now describe these conjectures. Let be an elliptic curve over of conductor with associated holomorphic weight 2 cusp form . We write the Fourier expansion of at as
[TABLE]
It is a fundamental question in number theory to understand how often the central value of , vanishes, when runs over the characters of .
Mazur, Rubin, and Stein used raw modular symbols. For these are defined by
[TABLE]
This corresponds roughly to taking in (1.2) to be the real or imaginary part of the 1-form . See Remark 2.1 for the precise statement. Modular symbols and the central value of twists of the corresponding -function are related by the Birch–Stevens formula e.g. [Pol14, Eq. 2.2], [MTT86, Eq 8.6] that is
[TABLE]
for a primitive character of conductor (here the choice of corresponds to the sign of ). To understand the vanishing of at Mazur, Rubin, and Stein were led to investigate the distribution of modular symbols and theta constants. In this paper we investigate modular symbols but not theta constants.
Mazur, Rubin, and Stein studied computationally the statistics of (raw) modular symbols. Since has period , the same is true for the modular symbols: They observed the behavior of contiguous sums of modular symbols, defined for each by
[TABLE]
Based on their computations they defined
[TABLE]
and arrived at the following conjecture.
Conjecture 1.1* (Mazur–Rubin–Stein).*
As we have
[TABLE]
They added credence to this conjecture with the following heuristics. If we cut off the paths in (1.2) for modular symbols at height , then
[TABLE]
because the left-hand side is a Riemann sum for the integral. The heuristics involves interchanging this limit with the limit as .
In another direction Mazur and Rubin investigated the distribution of for as . Define the usual mean and variance by
[TABLE]
where is Euler’s totient function. They conjectured the following asymptotic behavior of the variance.
Conjecture 1.2* (Mazur–Rubin).*
There exist a constant and constants for each divisor of , such that
[TABLE]
Moreover,
[TABLE]
The constant is called the variance slope and the constant the variance shift. In Conjecture 1.2 the symmetric square -function is normalized such that 1 is at the edge of the critical strip.
Moreover, the numerics suggest that the normalized raw modular symbols obey a Gaussian distribution law.
Conjecture 1.3* (Mazur–Rubin).*
Let . The data
[TABLE]
has limit the standard normal distribution.
In this paper we prove average versions of Conjectures 1.1 and 1.2 when we average over . We only work with squarefree. The restriction to squarefree may not be necessary. On the other hand we can only work with averages over and not individual .
Moreover, we prove a refined version of Conjecture 1.3. We can restrict the rational number to lie in any prescribed interval and we can restrict to rational numbers with a fixed number.
To prove these results we specialize to more general results on modular symbols for cofinite Fuchsian groups with cusps. Here is a statement of our results for , when is squarefree.
Theorem 1.4**.**
Conjecture 1.1 holds on average. More precisely we have
[TABLE]
Remark 1.5*.*
In fact we can restrict the summation to for a fixed divisor of . See Corollary 8.1 and the discussion following it.
Theorem 1.6**.**
Conjecture 1.2 holds on average. More precisely let , and let be given by (1.4). Then
[TABLE]
where
[TABLE]
Here , are explicitly computable constants given by (8.12).
Theorem 1.7**.**
Let be an interval of positive length, and consider for the set . Then the values of the map
[TABLE]
ordered according to , have asymptotically a standard normal distribution.
Remark 1.8*.*
Putting in Theorem 1.7 we prove Conjecture 1.3. The difference in normalization, i.e. the appearance of in the denominator, is irrelevant as explained in Remark 7.9.
Theorems 1.4, 1.6, and 1.7 will follow rather easily from the following general theorems for finite covolume Fuchsian groups with cusps. Let , be cusps of , not necessarily distinct. Let , where is a holomorphic cusp form of weight . We do not assume that has real coefficients. Fix scaling matrices and for the two cusps. Define
[TABLE]
see Proposition 2.2. Note that
[TABLE]
We denote the denominator of by . We order the elements of according to the size of and define
[TABLE]
We define general modular symbols as
[TABLE]
Theorem 1.9**.**
Let . Let be the Fourier coefficients of at the cusp . There exists a such that, as ,
[TABLE]
Theorem 1.10**.**
Let be the Petersson norm of . There exists an explicit constant , depending on , , , which we call the variance shift, such that
[TABLE]
where
[TABLE]
The constant is the variance slope.
The formula for is explicit but complicated, see (7.4). It depends on , the period and data from the Kronecker limit formula for the Eisenstein series for the cusps and .
Theorem 1.11**.**
Let be an interval of positive length. The values of the map with
[TABLE]
ordered according to have asymptotically a standard normal distribution, that is, for every with we have
[TABLE]
as .
Remark 1.12*.*
We have obtained different but related normal distribution results for modular symbols in [PR04, Ris04, PR05, PR09]. One difference between these papers and the current one is in the ordering and normalization of the values of . The orderings in these papers were more geometric (action of on ) and less arithmetic, and the modular symbols used closed paths in . However, in Theorem 1.7 we need to combine statistics from various cusps, since not all rational cusps are equivalent to for . Moreover, we allow to restrict to a general . This is a new feature, making our current results significantly more refined.
The expression for the variance shift in Theorems 1.6 and 1.10, see also (7.4), is very explicit. This is an unexpected facet. The analogue in [PR04, Theorem 2.19] involves also the reduced resolvent of the Laplace operator, which is much harder to understand.
Remark 1.13*.*
To prove Theorem 1.11 we study the asymptotic th moments of the modular symbols for all , see Theorem 7.5. We make no effort to optimize the error bounds for the moments in Theorems 1.9, 1.10, and 7.5, but they can all be made explicit in terms of spectral gaps.
Remark 1.14*.*
An important tool in this paper and in [PR04] is non-holomorphic Eisenstein series twisted with modular symbols . These were introduced by Goldfeld [Gol99b, Gol99a] and studied extensively by many authors, see e.g. [O’S00, DO00, JO08, JOS16, BD16] and the references therein. For their definition see (2.11). In [PR04] we used the Eisenstein series twisted with the th power of modular symbols as a generating series itself to study the th moment of modular symbols. In this paper we need to understand the th Fourier coefficient of , which involves twists by the th power of . The reason why the results here are more arithmetic is because the Fourier coefficients of Eisenstein series and of twisted Eisenstein series encode arithmetic data and modular symbols. To isolate the th coefficient we use inner products with Poincaré series.
Remark 1.15*.*
The structure of the paper is as follows. In Section 2 we introduce the generating functions (Dirichlet series) for the powers of modular symbols. We also introduce Poincaré series and Eisenstein series twisted by powers of modular symbols.
In Section 3 we analyze for , and conclude with the statement that is equidistributed modulo .
In Section 4 we study Eisenstein series twisted by modular symbols, or rather a related series . We prove the analytic continuation for and study the order of the poles and leading singularity at . The crucial identity is Eq. (4.4) that allows us to understand inductively using the resolvent of the Laplace operator .
In Section 5 we find explicit expressions for the functional equations of and , see Theorems 5.2 and 5.4.
In Section 6 we study the analytic properties of the derivatives for . When we find the residue at , and the whole singular part when . Finally, we identify the order of the pole and the leading singularity for all .
In Section 7 we prove Theorems 1.9, 1.10, and 1.11 for general finite covolume Fuchsian groups with cusps. We use the method of contour integration.
In Section 8 we specialize the general results to for squarefree . This leads to the proofs of Theorems 1.4, 1.6, and 1.7.
Remark 1.16*.*
In a recent preprint [KS17] the authors prove that, when through primes, the limiting behavior in Conjecture 1.1 holds. Their method is different from ours.
Acknowledgments
We would like to thank Peter Sarnak for alerting us to [MR16]. Also we would like to thank Barry Mazur and Karl Rubin for useful comments, and for providing us with numerical data.
2. Generating series for powers of modular symbols
From now on we allow to be any cofinite Fuchsian group with cusps. All implied constants in our estimates depend on and . In this section we define a generating series for modular symbols, and explain how it can be understood in terms of derivatives of Eisenstein series with characters.
2.1. Modular symbols
The modular symbols defined by
[TABLE]
are independent of , and independent of the path between and . Fix a set of inequivalent cusps for . For such a cusp we fix a scaling matrix , i.e. a real matrix of determinant 1 mapping to , and satisfying . Here is the stabilizer of in , and is the standard parabolic subgroup. We have for parabolic since is cuspidal.
For any real number and every cuspidal real differential 1-form we have a family of unitary characters defined by
[TABLE]
Note that this character is the conjugate of the one we considered in [PR04]. We also need the antiderivative of . We define
[TABLE]
We expand at a cusp . Let us assume that , where , and
[TABLE]
where . Then
[TABLE]
and
[TABLE]
We consider the line integral
[TABLE]
By [JO08, Eq. (3.3), (3.5)] we have the uniform bound
[TABLE]
for . Consequently, we have the estimate
[TABLE]
Remark 2.1*.*
If be a cusp form for with real Fourier coefficients at infinity, then . Since it follows that
[TABLE]
where . Consequently, taking covers both cases of raw modular symbols.
2.2. The generating series
Let be cusps and fixed scaling matrices. Then we define
[TABLE]
It is easy to see that is well-defined for the double coset containing . Note also that .
Proposition 2.2**.**
Let . Then there exists a unique
[TABLE]
satisfying
[TABLE]
Proof.
We imitate [Iwa02, p. 50]. Assume
[TABLE]
and , . We may assume that and . The matrix has lower left entry . If this is zero then and . If not, then by [Iwa02, Eq. (2.30)] we have , which implies that
[TABLE]
Therefore and since this implies that . ∎
From Proposition 2.2 we conclude the following result:
Corollary 2.3**.**
Any determines a unique number and unique cosets , and satisfying and .
From Corollary 2.3 it follows that there exists a unique pair such that and . We can therefore put an ordering on the elements of by putting the lexicographical ordering on the i.e.
[TABLE]
For we define by . For we define
[TABLE]
We suppress from the notation. We notice that if corresponds to as in Proposition 2.2 we have
[TABLE]
which we will also refer to as a modular symbol. The map does not grow too fast in terms of :
Proposition 2.4**.**
The following estimate holds
[TABLE]
Proof.
We use (2.3) with and a fixed . Writing the lower row of as we may assume that . Writing , we use the elementary inequalities
[TABLE]
from which the result follows. ∎
We can now define the main generating series:
Definition 2.5**.**
For we define
[TABLE]
By Proposition (2.4) and [Iwa02, Prop 2.8] we see that is absolutely convergent for , and uniformly convergent on compacta of . It is the analytic properties of this series that will eventually allow us to prove our main results.
2.3. Relation to Eisenstein series
To explain how relates to Eisenstein series we recall that the generalized Kloosterman sums are defined by
[TABLE]
For we define
[TABLE]
where the sum is over for This is a version of the Selberg–Linnik zeta function. When , that is when the character is trivial, we omit it from the notation. Using [Iwa02, Prop. 2.8] we see that for this is an absolutely converging Dirichlet series. Note that
[TABLE]
where and are related as in Proposition 2.2. It therefore follows that
[TABLE]
Proposition 2.6**.**
For any cusps and any we have
[TABLE]
Proof.
This follows easily from (2.4), and the following basic properties of Kloosterman sums: By inspection we see that
[TABLE]
Also, we have
[TABLE]
as seen by inverting in the definition of the Kloosterman sums. ∎
We now recover as Fourier coefficients of Eisenstein series twisted by modular symbols. For we define Poincaré series
[TABLE]
When , i.e. in the case of the usual Eisenstein series we will often omit the subscript . When we omit to specify in the notation, we have set . For it is known that and that it admits meromorphic continuation to , see [GS83, p. 247]. The usual Eisenstein series has Fourier expansion at a cusp given by (see e.g [Sel89, p. 640–641])
[TABLE]
where
[TABLE]
As usual if and is [math] otherwise.
The derivatives of and in are given by
[TABLE]
It follows that the series can be understood by understanding derivatives of Eisenstein series.
We consider the th derivative in of at , which we denote by . It is easily seen that, when , we have
[TABLE]
This series is absolutely and uniformly convergent on compact subsets of , as can be seen by (2.3) and using the standard region of absolute convergence of the Eisenstein series . We note that for
[TABLE]
Here the Fourier expansion is computed by termwise differentiation of the Fourier expansion of at the cusp , which is allowed. Hence, the generating series for the modular symbols appear as Fourier coefficients of , see (2.8) and (2.9).
2.4. Automorphic Poincaré series with modular symbols
While the generating series for the modular symbols appear as Fourier coefficients of , the series are not automorphic when . They are higher order modular forms. Properties of such forms has been studied extensively in many papers such as [CDO02, DS06, DD09, DS08, Sre09, IO09, Dei11, BD12].
Since , we see that our is indeed a linear combination of the Eisenstein series twisted by powers of and with :
[TABLE]
For background on see Remark 1.14. For our purpose it is convenient to consider a related function, which has the advantage of also being automorphic: Recall (2.1). Let . For we define
[TABLE]
Using (2.2), and by comparison with the standard Eisenstein series, we see that the function is absolutely and uniformly convergent on compact subsets of . It follows immediately that is -automorphic, and holomorphic for in this half-plane.
We consider also
[TABLE]
so that is the th derivative in of at . As usual when we omit it from the notation.
We now explain how and are related. We arrange our Eisenstein and Poincaré series in column vectors indexed by the cusps as follows:
[TABLE]
and
[TABLE]
We define the diagonal matrix with diagonal entries
[TABLE]
so that
[TABLE]
Let be the diagonal matrix with diagonal entries the antiderivatives . It follows from (2.13) by differentiation at that for we have the vector equations
[TABLE]
Hence, we can freely translate between and .
3. The generating series
In this section we discuss the analytic properties of for . In order to do so we first need some general bounds on Eisenstein series and Poincaré series with modular symbols.
3.1. Bounds on Eisenstein series
We first discuss a construction of Eisenstein series of weight for , generalizing the construction for weight [math] in [CdV83, Section 2]. This approach is useful to estimate Eisenstein series in the cusps for .
For we define . Let
[TABLE]
be the Laplace operator of weight and the closure of acting on smooth, weight functions such that are square integrable. We fix a fundamental domain of , and notice that is a fundamental domain for . However, the spectral analysis of the -Laplacian remains the same, since the manifold is isometric to . Recall the decomposition of as for sufficiently large, see [Iwa02, p. 40].
Lemma 3.1**.**
Let . Let be a smooth function that is identically for and identically [math] for . Then for and there exists a unique function satisfying the eigenvalue equation
[TABLE]
and such that
[TABLE]
Moreover, the -norm in (3.1) is .
Proof.
If such a solution exists we denote the left-hand side of (3.1) by . We apply to deduce
[TABLE]
where
[TABLE]
is a compactly supported function, is independent of , and is holomorphic in .
We now define by (3.2), and apply to it the resolvent of , which is defined for with . This produces a unique function . The standard inequality for the operator norm of the resolvent
[TABLE]
allows to estimate . ∎
Lemma 3.2**.**
Let . Then for cusps we have
[TABLE]
In particular
[TABLE]
Proof.
We use
[TABLE]
and the estimate [Iwa02, corollary 3.5], which can be made uniform for , . We estimate the right-hand side of (3.4) on the compact part of and on the cuspidal zones
[TABLE]
see [Iwa02, p. 40]. For different cusps and , the matrix has nonzero bounded from below by , see [Iwa02, Section 2.6]. We have . For the terms cancel. ∎
As usual we define the Eisenstein series of weight by
[TABLE]
Using Lemma 3.2 we see that
[TABLE]
This equation and Lemma 3.1 show that agrees with the construction of Lemma 3.1. Therefore, the conclusions of Lemma 3.1 hold for the Eisenstein series of weight in the region , .
We can also estimate defined in (2.12) using (2.2) and Lemma 3.2. We write
[TABLE]
If , and we see that the contribution of the identity term decays exponentially at the cusp i.e. and is at the other cusps. When , we notice that in the cuspidal zone for , the expansion (2.1) shows that decays exponentially. We can deduce that for all cusps we have . These estimates show that is square integrable uniformly in the strip for or .
Lemma 3.3**.**
Let , , . Moreover, assume that . Then the -invariant functions
[TABLE]
belong to . In fact
[TABLE]
where the implied constant depends only on , and .
Proof.
We use the result from Lemma 3.1 and the notation in its proof. We have
[TABLE]
We need to analyze also
[TABLE]
It suffices to concentrate on the cuspidal sector for , since vanished elsewhere. We have
[TABLE]
We use (3.5). When , we see that
[TABLE]
decays exponentially, otherwise it is bounded by . We easily see that is bounded. ∎
3.2. Meromorphic continuation and bounds on
It is well known that the inner product of an automorphic function and is directly related to the th Fourier coefficient of at the cusp . We first use this to find an integral expression for for .
Lemma 3.4**.**
Let and . The Dirichlet series has the integral representation
[TABLE]
Proof.
We unfold and, using (2.5), we find
[TABLE]
where we have used [GR07, 6.621.3, p. 700]. The result follows from the above computations and (2.7). ∎
We can now use the integral expression in Lemma 3.4 to find the analytic properties of .
Lemma 3.5**.**
For any cusps , and the Dirichlet series
[TABLE]
admits meromorphic continuation to . For the continuation is holomorphic at , while for the continuation has a simple pole with residue
[TABLE]
For , and bounded away from , the following estimate holds:
[TABLE]
Proof.
It is well known that the Fourier coefficients of admit meromorphic continuation to and the meromorphic continuation of the functions follows from (2.6) and (2.7).
To analyze further when , we use Lemma 3.4 when . Using Proposition 2.6 we see that it suffices to consider . We let , where . With this choice of Stirling’s formula gives that the quotient of Gamma factors is . The claim about growth on vertical lines follows from Lemma 3.3. For the residue at has as a factor. But this vanishes by unfolding.
For we examine (2.6). Using the well-known fact that has a pole of order at with residue , the claim for the residue of follows. For the growth on vertical lines, we observe that in the region , and away from the spectrum of , see [Sel89, Eq. (8.5)–(8.6), p. 655]. The result follows from Stirling’s formula. ∎
Remark 3.6*.*
The analytic properties of for have been studied in [GS83].
Remark 3.7*.*
If is the full modular group and with trivial scaling matrices then so that in this case
[TABLE]
Here is the standard Ramanujan sum. We notice that in this case the bound in Lemma 3.5 on for is far from optimal.
3.3. Equidistribution of .
We can use Lemma 3.5 to observe that is equidistributed modulo 1. The generating series of for is . Proposition 2.6 allows us to understand its behavior through . We define
[TABLE]
Using contour integration and the polynomial estimates on vertical lines for in Lemma 3.5 we deduce that, for some depending on the spectral gap for ,
[TABLE]
These are the Weyl sums for the sequence . As usual if and is [math] otherwise and similarly for a set , if and is [math] otherwise. Good studied the asymptotics of such Weyl sums in [Goo83].
To state our results we need to introduce norms on -functions on : For let
[TABLE]
where denotes the th Fourier coefficient of . We then have the following result:
Theorem 3.8**.**
The sequence is equidistributed modulo , i.e. for any continuous function we have
[TABLE]
If, moreover, then
[TABLE]
Proof.
The first claim follows from Weyl’s equidistribution criterion. For the second we write the Fourier series of , which is absolutely convergent, and interchange the summation over and over . We have
[TABLE]
∎
Remark 3.9*.*
In Theorem 3.8 the Sobolev norm can be replaced by for any , at the expense of a worse exponent . This can be done by making the appropriate changes in the contour integration argument leading to (3.6). Note that if we do so the exponent depends both on the spectral gap and on .
Remark 3.10*.*
Usually equidistribution modulo 1 of a sequence is stated as
[TABLE]
In Theorem 3.8 we are only looking at a subsequence of the left-hand side, namely only equal to for some . It is possible to consider the whole sequence by noticing that it follows from (3.6) that
[TABLE]
This improves on the trivial bound in [Iwa02, (2.37)].
4. Eisenstein series with modular symbols
In this section we analyze further the series defined in (2.12), when . We will omit from the notation and simply write . We will show that this function admits meromorphic continuation and prove -bounds for it. Let
[TABLE]
Define
[TABLE]
Remark 4.1*.*
These definitions are motivated by perturbation theory. Even if we are not using perturbation theory directly, as in our previous work [PR04, PR13], it is useful to have the following in mind.
Let be a closed smooth -form rapidly decaying at the cusps but not necessarily harmonic. We define unitary operators
[TABLE]
We also define
[TABLE]
where the automorphic Laplacian is the closure of the operator acting on smooth functions with . This ensures that acts on the fixed space . It is then straightforward to verify that
[TABLE]
We notice that does not depend on the cusp . We observe that , if is harmonic. From now on we assume to be harmonic. We remark that as a function of is a polynomial of degree 2, and that and defined in (4.1) and (4.2) are the first and second derivative of at .
The eigenvalue equation for and (2.13) imply that
[TABLE]
A formal differentiation of this eigenvalue equation leads to the formula in Lemma 4.2 below. This lemma is the main ingredient in understanding the meromorphic continuation of :
Lemma 4.2**.**
The function satisfies the relation
[TABLE]
when , where the last term should be omitted for .
Proof.
Since we get
[TABLE]
so that . Using the product rule we have that
[TABLE]
where we have used that . We recognize the first term as
[TABLE]
The other two terms give
[TABLE]
This proves that
[TABLE]
Now we automorphize this equation over , and use that . We notice that for any two differential forms , we have for the action of on functions
[TABLE]
Using that is invariant under , we arrive at the result. ∎
From Lemma 4.2 we find that, if and are square integrable, then
[TABLE]
where is the resolvent of . We recall that
[TABLE]
is a bounded operator on satisfying (3.3) and
[TABLE]
where is the projection to the eigenspace of for the eigenvalue [math], and is holomorphic in a neighborhood of .
Before we can prove the meromorphic continuation of we need a small technical lemma about Eisenstein series:
Lemma 4.3**.**
For the function is square integrable over . More precisely we have
[TABLE]
Proof.
We have
[TABLE]
where
[TABLE]
are the raising and lowering operators as in [Fay77, Eq. (3)]. It follows that
[TABLE]
Applying this with we see that it suffices to study and . We analyze the first and notice that the analysis of the latter is similar. We have , where is the Eisenstein series of weight , see [Roe66, Eq. (10.8), (10.9)]. Using that decays exponentially at all cusps the claim now follows easily from Lemma 3.1.
∎
We are now ready to prove the meromorphic continuation of .
Theorem 4.4**.**
Let . The function admits meromorphic continuation to . For and the functions and are smooth and square integrable. Moreover, we have
[TABLE]
Proof.
We use induction on . For Lemma 4.3, Eq. (4.4), and the mapping properties of the resolvent imply that is meromorphic when and, using (3.3), we easily prove the bound . Recall that for a twice differentiable function with we have and
[TABLE]
see [Roe66, Satz 3.1]. Since is in , we use Lemma 4.2 for and elliptic regularity to conclude that . By (4.7) the function is smooth as well. Using Lemma 4.2 for again we deduce that . We find from (4.7) and the estimate that it suffices to estimate and . We use (4.8) and Lemma 4.2 to conclude
[TABLE]
where we have used Cauchy-Schwarz and Lemma 4.3. This proves the case .
Assume now that the claim has been proved for every . Then for the function is meromorphic, smooth, and square integrable. Hence, by (4.4), the mapping properties of the resolvent, and (3.3), we find that is square integrable and satisfies the bound
[TABLE]
By elliptic regularity and Lemma 4.2 it follows that . Therefore, the function is also smooth. We now use (4.7), (4.8), and Lemma 4.2 to see that
[TABLE]
This completes the inductive step. ∎
Lemma 4.3 and Theorem 4.4 validate the conditions for (4.4), so we can now conclude the following fundamental recurrence relation for .
Corollary 4.5**.**
For the following identity holds
[TABLE]
when , where the last term should be omitted for .
Proposition 4.6**.**
For all and we have
[TABLE]
Proof.
Applying Stokes’ theorem, as in e.g. [PR04, p. 1026], we find that is a self-adjoint operator. Alternatively, we notice that is the infinitesimal variation of the family of self-adjoint operators given in Remark 4.1. Therefore, for we have
[TABLE]
since is a differentiation operator, see (4.1). The case involves , which is not square integrable. This is easily compensated by the fact that is cuspidal. ∎
From Corollary 4.5 it is evident that has singularities at the spectrum of . We now describe the nature of the pole at .
Proposition 4.7**.**
The functions and are regular at .
Proof.
We have
[TABLE]
We note that is regular since has a constant residue and is a differentiation operator. It then follows from (4.5) that can have at most a simple pole at . By (4.5) and Proposition 4.6 we see
[TABLE]
It follows that is regular at . ∎
Theorem 4.8**.**
Let denote the Petersson norm of . If is even, then has a pole at of order . The leading term in the corresponding expansion around , that is, the coefficient of is the constant
[TABLE]
If is odd, then has a pole at of order at most .
Proof.
The case simply describes the well-known pole and residue of the standard Eisenstein series. For the result is Proposition 4.7.
To run an inductive argument we assume that the claim has been proved up to some odd. Then is even and we see from (4.5) that can have at most a pole of order . The leading term comes from
[TABLE]
which is zero by Proposition 4.6. So has pole of order at most .
On the other hand by inductive hypothesis has a pole or order with leading coefficient
[TABLE]
Here we have again used (4.5).
It follows from Corollary 4.5 that has a pole of order with leading coefficient
[TABLE]
We observe that . The order of the pole and leading singularity of agrees with the claim of the theorem.
We now prove that has at most a pole of order . We use Corollary 4.5 for . Since annihilates the leading term in , as it is a constant, see (4.10), the function
[TABLE]
can have at most a pole of order . This order of singularity at is attained only if is not identically zero. But it is indeed identically zero by Proposition 4.6. Hence, has at most a pole of order . By (4.5) and the inductive hypothesis on it is straightforward that has at most a pole of order . This concludes the inductive step. ∎
5. Functional equations
Selberg’s theory of Eisenstein series [Iwa02, p. 84-94] gives that satisfies the functional equation
[TABLE]
where the scattering matrix is determined by (2.5). Recall also that
[TABLE]
In this section we show that and have analogous properties.
Recall the weighted -spaces in [Mül96, p. 573]. We choose a smooth function with for and for . For we define
[TABLE]
The resolvent of the Laplace operator defined on for and admits meromorphic continuation to if we restrict the domain to a smaller function space. Müller in [Mül96, Theorem 1] showed that
[TABLE]
can be defined as a bounded operator on weighted -spaces for away from its poles. This is achieved by first continuing meromorphically the resolvent kernel (automorphic Green’s function) to . The analytic continuation of the resolvent kernel to satisfies the limiting absorption principle:
[TABLE]
see [Lan85, p. 352].
We choose so that is decaying exponentially at the cusps.
Lemma 5.1**.**
Let . The function admits meromorphic continuation to . Moreover, we have
- (i)
, 2. (ii)
, 3. (iii)
, , and 4. (iv)
.
Proof.
The Eisenstein series twisted by modular symbols in (2.11) has Fourier expansion
[TABLE]
with
[TABLE]
see Jorgenson and O’Sullivan [JO08, Eq. (2.4)]. Here if and is [math] otherwise. We quote their work [JO08, Thm 2.2] for the meromorphic continuation of the series for . Since is a linear combination of for , it admits meromorphic continuation to . The function is related to the for through (2.14). Therefore, admits meromorphic continuation to as well.
Furthermore, we need bounds for the Fourier coefficients of for all cusps. Jorgenson and O’Sullivan [JO08, Thm 2.3] proved the following: for in a compact set , there exists a holomorphic function such that for all we have
[TABLE]
It is easy to see that the derivatives of decay exponentially in : we can use the integral representation of the -Bessel function [GR07, 8.432.1, p. 917] to see that
[TABLE]
or, alternatively, use repeatedly [GR07, 8.486.11, p. 929]. Combining with (5.3) we see that, for every , the function is smooth and
[TABLE]
Since is a linear combination of for , its derivatives satisfy the same upper bounds. By (2.14) the functions are also smooth. This proves claim (i).
For the claims (ii), (iii) we need also to control the derivatives of . For , we use (2.2). For we have exponential decay by (4.3). We conclude that
[TABLE]
The claim (ii) follows by taking , and the claim (iii) by noticing that and in (4.6) are expressed in terms of and . Finally, claim (iv) follows from (4.7) using the cuspidality of at all cusps. ∎
5.1. Functional Equation for
Theorem 5.2**.**
The meromorphic continuation of the vector-valued automorphic function satisfies the functional equation
[TABLE]
with meromorphic matrices given by
[TABLE]
if and is the standard scattering matrix.
Proof.
For notational purposes we define to be [math]. Using Corollary 4.5 we get
[TABLE]
for , and . We can extend the validity of this equation to , since the resolvent is applied to a function belonging to , which follows from Lemma 5.1. Furthermore we see that is holomorphic outside the poles of .
The proof of (5.4) is a relatively obvious generalization of [PR13, Prop. 3.1]. To justify the arguments below we quote Lemma 5.1. We proceed by induction. For the claim of the theorem is the standard functional equation for the vector of Eisenstein series.
Assume the result for . We define the matrix indexed by the cusps by (5.4). Then
[TABLE]
where we have used (5.2) and (5.5). For the first term above we now use the inductive hypothesis and finally (5.5) to see that it equals
[TABLE]
This completes the inductive step. ∎
Selberg proved [Sel89, Eq. (8.5)–(8.6), p. 655] that for with bounded away from the spectrum the function is bounded. We now show how this generalizes to .
Lemma 5.3**.**
Fix . For with bounded away from the function is bounded, i.e.
[TABLE]
Proof.
Considering (5.4) it suffices to show that for every we have
[TABLE]
For (5.6) we use Stokes’ theorem, as in the proof of Proposition 4.6, and bound
[TABLE]
By Theorem 4.4, Lemma 4.3, and Cauchy–Schwarz this is bounded by a constant times .
For (5.7) we note that we can move in front of , since is a multiplication operator. Since by Lemma 3.1, and , see Theorem 4.4, the result follows using Cauchy–Schwarz. ∎
5.2. Functional equation for
Let be a natural number. Using (2.15) and (2.14) we can find the functional equation for as follows:
[TABLE]
Setting we see that
[TABLE]
If we set
[TABLE]
then
[TABLE]
We can rewrite this to see that
[TABLE]
We emphasize that does not depend on . We also note that if there is only one cusp we have .
Looking at the -entry of (5.8) and its Fourier expansion (2.10) at the cusp we get for the zero Fourier coefficients:
[TABLE]
This gives . We summarize the results for .
Theorem 5.4**.**
The vector of Eisenstein series twisted by modular symbols satisfies the functional equation
[TABLE]
where
[TABLE]
and the are given by (5.4). Moreover, , where is the zero Fourier coefficient of , see (2.10).
Remark 5.5*.*
The functional equations for can be deduced also from the functional equations for , see [JO08, Thm. 7.1]. However, the explicit expressions for in Theorem 5.4 are new. In this work we need the integral representation of in Section 6 below.
The matrices satisfy the functional equation
[TABLE]
cf. [JO08, Th. 2.2]. For this is due to Selberg and for it follows by comparing the coefficient of in (5.9). We notice also that for with bounded away from the spectrum we have
[TABLE]
as follows from Lemma 5.3.
6. Analytic properties of the generating series
In this section we use the results from Sections 3, 4, 5 to study the analytic continuation of for and .
6.1. Meromorphic continuation
If we obtained the meromorphic continuation of in Lemma 3.5. For we consider first the case . From (2.8), and Theorem 5.4 we find that
[TABLE]
The right-hand side is meromorphic by Theorem 5.2. If we deduce from Lemma 3.4 and (2.13) that, for , ,
[TABLE]
where
[TABLE]
We differentiate to get
[TABLE]
The differentiation is allowed and the right-hand side is meromorphic for by Theorem 4.4, Lemma 3.3 and the fact that is bounded for . Using Proposition 2.6 we can deal also with . To summarize we have proved the following result:
Theorem 6.1**.**
For any cusps and any integers , the function admits meromorphic continuation to .
6.2. The first derivative.
We now study in more detail the analytic properties of .
Theorem 6.2**.**
The function has a simple pole at with residue
[TABLE]
For bounded away from , and we have
[TABLE]
Proof.
Using Proposition 2.6 the claim for follows from the case . So we can assume that . Consider (6.2) when . For fixed, the functions is holomorphic as long as , so we must analyze the three expressions
[TABLE]
To analyze (6.3) we get by Lemma 3.3
[TABLE]
There is a pole of the Eisenstein series at , which gives rise to a residue for (6.3)
[TABLE]
To see that this vanishes we unfold the integral as in the Rankin method. The integrand contains the factor , with , and we notice that, as varies over the cosets , the sets cover the strip
To analyze (6.4) we note that by Theorem 4.4, Proposition 4.7 and (3.5) the term is holomorphic at and, for bounded away from the spectrum, satisfies
[TABLE]
Finally, we analyze (6.5). Since the Eisenstein series has a pole at we find that the integral has a simple pole at with residue
[TABLE]
where we have unfolded using (2.12) and (2.1). We notice also that by Lemma 3.3 we have
[TABLE]
Using the above analysis of the three integrals we can finish the proof for as follows. Observing that we get the residue at for . For the growth on vertical lines we choose and use Stirling’s asymptotics to get
[TABLE]
The bound on vertical lines now is obvious when we choose .
As far as is concerned we use (6.1) with , which leads to analyze and . We start by noticing that by Lemma 5.3 they are both bounded for . With the help of the Stirling asymptotics on the quotient of Gamma factors we easily prove the bound on vertical lines for .
Since is the standard scattering matrix it is well known that has a simple pole with residue . Using Theorem 5.2 and Proposition 4.7 we see that, if has a pole, it must be a simple pole with residue a constant times . This vanishes by Proposition 4.6 so is regular at . The conclusion follows. ∎
6.3. The second derivative
We will now describe the full singular part of at . We denote the constant term in the Laurent expansion of by , i.e.
[TABLE]
as . For , the function can be described in terms of the Dedekind eta function (Kronecker’s limit formula). For general groups the function is given in terms of generalized Dedekind sums, see e.g. [Gol73].
Theorem 6.3**.**
The function has a pole of order 2 at . The full singular part of the Laurent expansion at equals
[TABLE]
where
[TABLE]
For bounded away from , and we have
[TABLE]
Proof.
Using (6.1) we see that equals
[TABLE]
We consider each of the three terms separately:
We start by noting that since , the first term has singular part
[TABLE]
To analyze the second term we note that by Theorem 5.2 we have
[TABLE]
which is regular by Proposition 4.7 and Proposition 4.6.
To analyze the third term we note that by Theorem 5.2
[TABLE]
and analyze the contribution of the two summands. Since is regular at (Proposition 4.7), the first summand has at most a first order pole. The corresponding residue is zero by Proposition 4.6, so the first summand is regular.
It follows that the singular part of equals the singular part of . It follows that the singular part of the third term of (6.8) equals the singular part of
[TABLE]
The result follows using (6.7), (4.2), and standard values of [GR07, 8.366].
The bound on vertical lines follow from Theorem 5.3 and Stirling’s asymptotics. ∎
Remark 6.4*.*
We remark that Theorem 6.3 allows us to write the singular expansion of exclusively in terms of data of Rankin–Selberg integrals and periods. Indeed, writing
[TABLE]
as , we have
[TABLE]
so that the singular expansion of equals
[TABLE]
6.4. Higher derivatives
Theorem 6.5**.**
If is even the function has a pole at of order . The leading term in the singular expansion around equals
[TABLE]
If is odd the function has a pole at of order less than or equal to . For bounded away from , and we have
[TABLE]
Proof.
By (6.1) we must understand the leading expansion of each for . The claim about the order of the pole for all , and the leading singularity for even follows from (5.4), Theorem 4.8, and Proposition 4.6.
The claim on bounds on vertical lines follow from (6.1), Stirling’s formula, and Lemma 5.3. ∎
Theorem 6.6**.**
Let . Then we have:
- (i)
The function has a pole at of order strictly less than . 2. (ii)
For bounded away from , and we have
[TABLE] 3. (iii)
All coefficients in the singular expansion of are bounded independently of .
Proof.
Using Proposition 2.6 it suffices to treat the case . Considering (6.2) we see that claim (i) about the orders of the pole follows from Theorem 4.8 and the fact that as is seen by unfolding.
Claim (ii) follows from the bound (6.6) (with ) valid when combined with Lemma 3.3, Theorem 4.4 and the fact that is bounded independently of and for , see the discussion after (3.5).
For claim (iii) we note that the constants in all singular expansions are linear combinations of
[TABLE]
where is one of the coefficients in the singular expansion of . For the function is constant so, in particular, is square integrable. For we note that
[TABLE]
for some and sufficiently small . Here is the circle centered at with radius . The radius is chosen so that there are no other singularities of inside apart from . It follows from Theorem 4.4 that is square integrable. By using Cauchy–Schwarz we see that (6.9) is bounded independently of . See again the discussion after (3.5). ∎
7. Distribution results
We are now ready to prove Theorems 1.9, 1.10 and 1.11. Since we have identified the behavior of the generating functions at and on vertical lines in Section 6, we can use the well-known method of contour integration to deduce the asymptotics of .
7.1. First moment with restrictions.
In this subsection we study sums of the form
[TABLE]
for smooth functions or indicator functions . Hence we are studying a (partial) first moment of the modular symbol but with restrictions on imposed by .
Remark 7.1*.*
We present a variant of the Mazur–Rubin–Stein heuristics: By Theorem 3.8 is equidistributed on . If it had been possible to extend the function to a continuous function of , this would give the asymptotics of (7.1) immediately. Using (2.1) it would be tempting to define the modular symbol for all by
[TABLE]
We cannot do so as the series is not convergent, even if Wilton’s classical estimate, see [Wil29], [Iwa97, Thm 5.3], shows that it just barely fails to converge conditionally.
By termwise integration against we would get the result
[TABLE]
This series converges to a continuous function as is easily seen from Hecke’s average bound, [Iwa97, Thm 5.1].
If instead we consider, for a fixed ,
[TABLE]
then this function does indeed define a continuous function on , and we can use equidistribution with as a test function. If we do so, and then let we arrive again at (7.2). However, it is not easy to justify that one can interchange the limits and . On the other hand Mazur, Rubin and Stein have numerics suggesting that (7.2) is indeed the correct limit.
The above heuristics gives the correct answer. This is the content of Theorem 7.2 below. For a formal series
[TABLE]
we have a linear functional (distribution) from the set of smooth functions on given by
[TABLE]
where denotes the th Fourier coefficient of .
Recall the norm (3.7). We are now ready to prove the main result of this section:
Theorem 7.2**.**
Let be a smooth function on with . Then there exists a such that
[TABLE]
where is the formal series given by
[TABLE]
Proof.
The generating series of is . Writing we have by Proposition 2.6, Theorem 6.2, and a complex integration argument that
[TABLE]
from which the result follows. ∎
Let . Approximating by smooth periodic functions we can conclude the following result, which makes rigorous the heuristic conclusions in Remark 7.1.
Corollary 7.3**.**
Let . There exists a such that
[TABLE]
7.2. The variance
In this subsection we study the second moment of the modular symbols, i.e. the variance. Following Mazur and Rubin we denote the variance slope by
[TABLE]
Recall (6.7). We also define the variance shift by
[TABLE]
The following theorem follows directly from Theorem 6.3 and a complex integration argument.
Theorem 7.4**.**
There exists a such that
[TABLE]
We deduce from Theorem 7.4 and Theorem 3.8 that
[TABLE]
7.3. Normal distribution
In this subsection we show that the value distribution of modular symbols (appropriately normalized) obeys a standard normal distribution, even if we restrict to any interval.
Theorem 7.5**.**
Let be a function on satisfying , and let . Then there exist such that
[TABLE]
Proof.
We use Proposition 2.6, Theorem 6.5, and Theorem 6.6. We apply a complex integration argument in a strip of width around to deduce that
[TABLE]
We insert this in
[TABLE]
and use to get the result. ∎
Remark 7.6*.*
The result (7.5) can be strengthened to the following: There exists a polynomial such that
[TABLE]
The degree of is strictly less than if either or is odd, and exactly for even and .
In Theorem 7.4 we identify this polynomial when and .
Using a standard approximation argument based in Theorem 7.5 we arrive at the following corollary:
Corollary 7.7**.**
Let be an interval and let . Then
[TABLE]
The above corollary allows us to renormalize the modular symbol map and determine the distribution of the renormalized map using the method of moments:
Corollary 7.8**.**
Let be an interval of positive length. Then the values of the map
[TABLE]
ordered according to have asymptotically a standard normal distribution, i.e. for every we have
[TABLE]
as .
Proof.
Using summation by parts we find from Corollary 7.7 and Theorem 3.8 that
[TABLE]
which is the th moment of the standard normal distribution. The result follows from a classical result due to Fréchet and Shohat [Loè77, 11.4.C].
∎
Remark 7.9*.*
From the above proof we see that the asymptotic moments do not change if you replace with for any constant . Therefore Corollary 7.8 also holds if we normalize accordingly.
8. Results for Hecke congruence groups
In this section we translate the distribution results of Section 7 to the case of Hecke congruence groups , where is a squarefree integer. In this case the cusps of and their scaling matrices can be described as follows, see [DI82, Section 2.2]): a complete set of inequivalent cusps of are given by with . Notice that if then is equivalent to the cusp at infinity. Write . We may take
[TABLE]
for the corresponding scaling matrix. It follows that
[TABLE]
Using this and definition (1.5) we easily see that
[TABLE]
and for we have
[TABLE]
Therefore,
[TABLE]
where is as in (1.2).
8.1. First moment with restrictions
From Theorem 7.2 and Theorem 3.8 we now deduce the following corollary.
Corollary 8.1**.**
Let , and let be a smooth function on with . Then there exists such that
[TABLE]
where is the formal series
[TABLE]
with the Fourier coefficients at infinity of .
Summing Corollary 8.1 over all positive divisors and using that
[TABLE]
we may remove the divisibility condition on and conclude that
[TABLE]
We can also remove the condition , by summing according to . Using that we find that
[TABLE]
Using partial summation we find that
[TABLE]
By an approximation argument, where we approximate by appropriate smooth functions, we find
[TABLE]
as . This completes the proof of Theorem 1.4.
8.2. The variance
Similarly to the analysis above we can use Theorem 7.4, Corollary 7.3, and Theorem 3.8 to conclude the following corollary.
Corollary 8.2**.**
There exists a such that
[TABLE]
Recall now the mean and variance from (1.3).
Lemma 8.3**.**
Assume is a Hecke eigenform. Then .
Proof.
Since is an eigenfunction of all Hecke operator with eigenvalue it is easy to see that for any rational we have
[TABLE]
If is a fixed integer, then we deduce that
[TABLE]
Using Möbius inversion and the Eichler bound for weight holomorphic cusp forms on congruence groups [Eic54], i.e. , we find that
[TABLE]
Another application of Möbius inversion, and the well-known lower bound [HW79, Thm. 329] give . ∎
We define the variance shift by
[TABLE]
Using Lemma 8.3 we see that
[TABLE]
Using this and Corollary 8.2 we deduce that
[TABLE]
as . Here we have used (3.6) for the asymptotics of the last sum.
8.2.1. Relation with the symmetric square -function
We explain how to relate , and to the symmetric square -function. We recall the definitions but refer to [ILS00, Sec. 2-3] for additional details. Assume that is a Hecke eigenform normalized with first Fourier coefficient equal to 1, and let be its th Hecke eigenvalues. Let
[TABLE]
be the Rankin–Selberg -function. It is known that admits meromorphic continuation to with a simple pole at with corresponding residue . We have
[TABLE]
where
[TABLE]
and is the Riemann zeta function with the Euler factors at removed. The symmetric square -function of is defined by
[TABLE]
Using these definitions, the formula , and that has a simple pole at with residue , we find that
[TABLE]
This verifies the variance slope of Conjecture 1.2.
To express in more arithmetic terms the constant we notice that
[TABLE]
is the constant term in Laurent expansion at of
[TABLE]
by definition of . By unfolding we see that the last integral equals
[TABLE]
Hence we conclude that
[TABLE]
where
[TABLE]
To understand the part of involving we recall the Atkin–Lehner involutions [AL70]. We also recall that we assume that is squarefree. For every there exists an integer matrix of determinant of the form
[TABLE]
with . It is straightforward to verify that normalizes , that is, . Since is assumed to be a Hecke eigenform it follows from Atkin–Lehner theory that
[TABLE]
with the Atkin–Lehner eigenvalues. It follows easily that
[TABLE]
Lemma 8.4**.**
The Atkin–Lehner involutions permute the Eisenstein series. More precisely, for every we have
[TABLE]
Proof.
Let . Then a direct computation shows that
[TABLE]
so is an admissible scaling matrix for the cusp . Since the Eisenstein series is independent of the choice of scaling matrix we find that
[TABLE]
where we have used that normalize .
∎
Combining Lemma 8.4 and (8.10) we find that
[TABLE]
It follows that
[TABLE]
Using the definition of i.e. (8.9), we see that
[TABLE]
Combining (8.5), (8.6), (8.8), and (8.11) we find the following expression for the variance shift. We have
[TABLE]
where
[TABLE]
This completes the proof of Theorem 1.6.
8.3. Numerical investigations
As an example we consider the elliptic curve 15.a1. We computed and using lcalc in Sage [Sag17] and the data from [LMF17]111See lmfdb.org/L/SymmetricPower/2/EllipticCurve/Q/15.a/. These numbers should be accurate to at least decimal places. This allows to estimate the value of the variance shift and compare with the experimental values in Table 1. We would like to thank Karl Rubin for providing us with the experimental values of the variance shift. Notice that these are the opposite of what appears in [MR16], because our modular symbols are purely imaginary.
8.4. Normal distribution
With the description of the modular symbols and from (8.2), (8.3), and (8.4), we can apply Corollary 7.7 to compute the moments of . Since has , we may use and Remark 7.9 to conclude the following corollary.
Corollary 8.5**.**
Let be any interval of positive length, and consider for the set . Then the values of the map
[TABLE]
ordered according to have asymptotically a standard normal distribution.
This completes the proof of Theorem 1.7.
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- 5[Cd V 83] Yves Colin de Verdière. Pseudo-laplaciens. II. Ann. Inst. Fourier (Grenoble) , 33(2):87–113, 1983.
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