Hecke Eigenforms Research Articles

On a variance of Hecke eigenvalues in arithmetic progressions

Let a ( n ) be the eigenvalue of a holomorphic Hecke eigenform f under the n th Hecke operator. We derive asymptotic formulae for the variance ∑ b = 1 q | ∑ n ≤ X n ≡ b ( mod q ) a ( n ) | 2 when X 1 / 4 + ε ≤ q ≤ X 1 / 2 − ε or X 1 / 2 + ε ≤ q ≤ X 1 − ε , that exhibit distinct behavior. The analogous problem for the divisor function will be studied as well.

Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.

Explicit bounds for sums of squares

For an even integer k, let r2k(n) be the number of representations of n as a sum of 2k squares. The quantity r2k(n) is appoximated by the classical singular series �2k(n) ≍ n k 1 . Deligne's bound on the Fourier coefficients of Hecke eigenforms gives that r2k(n) = �2k(n) + O(d(n)n k 1 2 ). We determine the optimal implied constant in this estimate provided that either k/2 or n is odd. The proof requires a delicate positivity argument involving Petersson inner products.

Automorphic forms for elliptic function fields

Let F be the function field of an elliptic curve X over \({\mathbb{F}_q}\). In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over F. We determine these formulas in the language of the graph of an Hecke operator, for which we use its interpretation in terms of \({\mathbb{P}^1}\)-bundles on X. This allows a purely geometric approach, which involves, amongst others, a classification of the \({\mathbb{P}^1}\)-bundles on X. We apply the computed formulas to calculate the dimension of the space of unramified cusp forms and the support of a cusp form. We show that a cuspidal Hecke eigenform does not vanish in the trivial \({\mathbb{P}^1}\)-bundle. Further, we determine the space of unramified F′-toroidal automorphic forms where F′ is the quadratic constant field extension of F. It does not contain non-trivial cusp forms. An investigation of zeros of certain Hecke L-series leads to the conclusion that the space of unramified toroidal automorphic forms is spanned by the Eisenstein series E( · , s) where s + 1/2 is a zero of the zeta function of X—with one possible exception in the case that q is even and the class number h equals q + 1.

Products of nearly holomorphic eigenforms

We prove that the product of two nearly holomorphic Hecke eigenforms is again a Hecke eigenform for only finitely many choices of factors.

On the AF-algebra of a Hecke eigenform

AbstractAn AF-algebra is assigned to each cusp form f of weight 2; we study properties of this operator algebra when f is a Hecke eigenform.

L-functions of [formula omitted

van Geemen and van Straten [B. van Geemen, D. van Straten, The cuspform of weight 3 on Γ 2 ( 2 , 4 , 8 ) , Math. Comp. 61 (1993) 849–872] showed that the space of Siegel modular cusp forms of degree 2 of weight 3 with respect to the so-called Igusa group Γ 2 ( 2 , 4 , 8 ) is generated by 6-tuple products of Igusa theta constants, and each of them are Hecke eigenforms. They conjectured that some of these products generate Saito–Kurokawa representations, weak endoscopic lifts, or D-critical representations. In this paper, we prove these conjectures. Additionally, we obtain holomorphic Hermitian modular eigenforms of GU ( 2 , 2 ) of weight 4 from these representations.

Computations with classical and p-adic modular forms

AbstractWe present p-adic algorithms for computing Hecke polynomials and Hecke eigenforms associated to spaces of classical modular forms, using the theory of overconvergent modular forms. The algorithms have a running time which grows linearly with the logarithm of the weight and are well suited to investigating the dimension variation of certain p-adically defined spaces of classical modular forms.

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin–Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ0(n)⊂GL2(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b1. We give elements in the Hecke annihilator of b1. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.

A Variant of Lehmer’s Conjecture, II: The CM-case

Abstract Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

Some Siegel modular standard L-values, and Shafarevich–Tate groups

We explain how the Bloch–Kato conjecture leads us to the following conclusion: a large prime dividing a critical value of the L-function of a classical Hecke eigenform f of level 1, should often also divide certain ratios of critical values for the standard L-function of a related genus two (and in general vector-valued) Hecke eigenform F. The relation between f and F (Harderʼs conjecture in the vector-valued case) is a congruence involving Hecke eigenvalues, modulo the large prime. In the scalar-valued case we prove the divisibility, subject to weak conditions. In two instances in the vector-valued case, we confirm the divisibility using elaborate computations involving special differential operators. These computations do not depend for their validity on any unproved conjecture.

On the number of partitions with distinct even parts

Let p e d ( n ) be the number of partitions of n wherein even parts are distinct (and odd parts are unrestricted). We obtain many congruences for p e d ( n ) mod 2 and mod 4 by the theory of Hecke eigenforms.

Anticyclotomic p-adic L-function of Central Critical Rankin-Selberg L-value

Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and \pi the unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character \chi of M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and \chi over the idele class group of M to the central critical value L(1/2,{\pi}\otimes{\chi}). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Katz where he did that for Eisenstein series.

Simultaneous nonvanishing of automorphic L-functions at the central point

Let g be a holomorphic Hecke eigenform and {uj} an orthonormal basis of even Hecke–Maass forms for \({\textup{SL}(2,\mathbb{Z})}\). Denote L(s, g × uj) and L(s, uj) the corresponding L-functions. In this paper, we give an asymptotic formula for the average of \({L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)}\), from which we derive that there are infinitely many uj’s such that \({L(\frac{1}{2},g\times u_j)L(\frac{1}{2},u_j)\neq0}\).

One level density of low-lying zeros of families of L-functions

AbstractIn this paper, we prove some one level density results for the low-lying zeros of families of L-functions. More specifically, the families under consideration are that of L-functions of holomorphic Hecke eigenforms of level 1 and weight k twisted with quadratic Dirichlet characters and that of cubic and quartic Dirichlet L-functions.

CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

The Manin constant of elliptic curves over function fields

We study the p-adic valuation of the values of normalised Hecke eigenforms attached to non-isotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic p. We derive upper bounds on the smallest attained valuation in terms of the minimal discriminant under a certain assumption on the function field and provide examples to show that our estimates are optimal. As an application of our results we also prove the analogue of the degree conjecture unconditionally for strong Weil curves with square-free conductor defined over function fields satisfying the assumption mentioned above.

Spectral Expansions of Overconvergent Modular Functions

The main result of this article is an instance of the conjecture made by Gouvea and Mazur in [11], which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. For N = 1, p = 2 and k = 0 we show that this follows from the combinatorial approach initiated by Emerton [9] and Smithline [16], using the classical LU decomposition and results of Buzzard–Calegari [1]; this implies the conjecture for all [Formula]. Similar results follow for p = 3 and p = 5 with the assumption of a plausible conjecture, which would also imply formulae for the slopes analogous to those of [1]. We also show that (for general p and N) the space of weight 0 overconvergent forms carries a natural inner product with respect to which the Hecke action is self-adjoint. When N = 1 and p ∈ {2, 3, 5, 7, 13}, combining this with the combinatorial methods allows easy computations of the q-expansions of small slope overconvergent eigenfunctions; as an application we calculate the q-expansions of the first 20 eigenfunctions for p = 5, extending the data given in [11].

A new bound K2/3+ for Rankin-Selberg L-functions for Hecke congruence subgroups

Let f be a holomorphic Hecke eigenform for Γ0(N ) of weight k, or a Maass eigenform for Γ0(N ) with Laplace eigenvalue 1/4 + k. Let g be a fixed holomorphic or Maass cusp form for Γ0(N ). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f ⊗ g) is proved in the k-aspect: L(1/2+ it, f ⊗ g) N ,g,t,e k, while a convexity bound is only k. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to −1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.