A Spectral Correspondence for Maaß Waveforms

Let be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order in an indefinite quaternion division algebra over . Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maa? newforms for the Hecke congruence group , when d is the discriminant of the maximal order . We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for and the newforms of level d, respectively. Submitted: July 1998, final version: January 1999.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

A note on Hecke eigenvalues of Hermitian Siegel Eisenstein series

In this note we compute Hecke eigenvalues of Siegel Eisenstein series on the unitary group \(\operatorname{U}(2,2)\) induced from a character. We do this for Hecke operators at primes not dividing the level of the Eisenstein series.     

Distribution of Hecke eigenvalues

We give two results concerning the distribution of Hecke eigenvalues of . The first result asserts that on certain average the Sato-Tate conjecture holds. The second result deals with the Gaussian central limit theorem for Hecke eigenvalues.

     

On bounding Hecke-Siegel eigenvalues

We use the action of the Hecke operators (1≤j≤n) on the Fourier coefficients of Siegel modular forms to bound the eigenvalues of these Hecke operators. This extends work of Duke-Howe-Li and of Kohnen, who provided bounds on the eigenvalues of the operator T(p).     

Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 ? k. The operator ξ_2-k (resp. D ^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are “dual” under ξ_2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D ^k-1.     

Asymptotic densities of Maass newforms

We define the counting function for Maass newforms of Hecke congruence groups and calculate the three main terms of this counting function. We then give necessary and sufficient conditions for this expansion to have the same shape as if it were counting eigenvalues related to cocompact surfaces. We relate the result to classical instances of the Jacquet-Langlands correspondence.     

On the traces of Hecke operators

From Eichler's results on traces of Hecke operators and making essential use of the theory of Atkin-Lehner on newforms, we derive in a simple manner an equality between the traces of Hecke operators acting on different spaces of cusp forms.     

Symmetric square L-values and dihedral congruences for cusp forms

Let be a prime, and k=(p+1)/2. In this paper we prove that two things happen if and only if the class number . One is the non-integrality at p of a certain trace of normalised critical values of symmetric square L-functions, of cuspidal Hecke eigenforms of level one and weight k. The other is the existence of such a form g whose Hecke eigenvalues satisfy “dihedral” congruences modulo a divisor of p (e.g. p=23, k=12, g=Δ). We use the Bloch-Kato conjecture to link these two phenomena, using the Galois interpretation of the congruences to produce global torsion elements which contribute to the denominator of the conjectural formula for an L-value. When , the trace turns out always to be a p-adic unit.     

On the Control Theorem for the Symplectic Group

We build a theory of -adic Siegel modular forms related to the Klingen parabolic subgroup of GSp(4). These correspond to families of cohomology classes of increasing levels whose Hecke eigenvalues enjoy strong congruence properties. In the spirit of Hida's theory, a control theorem to relate the family to finite-level members is proved for almost all primes p; in particular we show that the error term appearing in degree one cohomology is killed by the ordinary idempotent.     

Self-replication and Borwein-like algorithms

We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing this, we see explicit computational algorithms that generate Hecke eigenvalues for such forms.     

On the Distribution of Values of <Emphasis Type="Italic">L</Emphasis>(1, <Emphasis Type="Italic">f</Emphasis>)

Let S_k(N)⁺ be the set of newforms of weight k for Γ₀(N), and let L (s, f), f ∈ S_k(N)⁺, be the Hecke L-function of the form f. It is proved that for every integer m ≥ 1, k = 2, and N = p → ∞,

$\mathop \sum \limits_{f \in S_2 (N)^ + } L^m (1,f) = \frac{1}{{12}}B_m N + O(N^{1 - \alpha } ),$

where B_m is a constant defined in the paper, and α = α(m) > 0 is a certain constant. This result implies the existence of the distribution function of the sequence

$\{ L(1,f),f \in S_2 (N)^ + \} ,\quad N = p \to \infty ,$

and also yields an explicit expression for the corresponding characteristic function. Bibliography: 11 titles.

     

Shimura lifts of half-integral weight modular forms arising from theta functions

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.      

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.     

Triple correlations of Fourier coefficients of cusp forms

We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms \(\sum _{H\le h\le 2H}W\left( \frac{h}{H}\right) \sum _{X\le n\le 2X}\lambda _{1}(n-h)\lambda _{2}(n)\lambda _{3}(n+h)\), which is nontrivial provided that \(H\ge X^{2/3+\varepsilon }\). The result can be viewed as a cuspidal analogue of a recent result of Blomer on triple correlations of divisor functions.     

On a convolution series attached to a Siegel Hecke cusp form of degree 2

We prove that the “naive” convolution Dirichlet series $D_{2}(s)$ attached to a degree 2 Siegel Hecke cusp form $F$ , has a pole at $s=1$ . As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor-zeta function attached to $F$ is $s = 1$ .     

Formal Hecke algebras and algebraic oriented cohomology theories

In the present paper, we generalize the construction of the nil Hecke ring of Kostant–Kumar to the context of an arbitrary formal group law, in particular, to an arbitrary algebraic oriented cohomology theory of Levine–Morel and Panin–Smirnov (e.g., to Chow groups, Grothendieck’s \(K_0\) , connective \(K\) -theory, elliptic cohomology, and algebraic cobordism). The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings, respectively. We also introduce a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra, respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.     

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 (Harder?s 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.     

Modules for Yokonuma-type Hecke Algebras

This paper describes the module categories for a family of generic Hecke algebras, called Yokonuma-type Hecke algebras. Yokonuma-type Hecke algebras specialize both to the group algebras of the complex reflection groups G(r,1,n) and to the convolution algebras of (B \(^{\prime }\),B \(^{\prime }\))-double cosets in the group algebras of finite general linear groups, for certain subgroups B \(^{\prime }\) consisting of upper triangular matrices. In particular, complete sets of inequivalent, irreducible modules for semisimple specializations of Yokonuma-type Hecke algebras are constructed.