Related papers: On the average least negative Hecke eigenvalue

We revisit the $k$-Hessian eigenvalue problem on a smooth, bounded, $(k-1)$-convex domain in $\mathbb R^n$. First, we obtain a spectral characterization of the $k$-Hessian eigenvalue as the infimum of the first eigenvalues of linear…

For each nonprincipal Dirichlet character $\chi$, let $n_\chi$ be the least $n$ with $\chi(n) \notin \{0,1\}$. We show that as the average of $n_\chi$ over all nonprincipal characters $\chi$ modulo $q$ is $\ell(q) + o(1)$, where $\ell(q)$…

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems,…

We present examples of Maass forms on Hecke congruence groups, giving low eigenvalues on $\Gamma_0(p)$ for small prime $p$, and the first 1000 eigenvalues for $\Gamma_0(11)$. We also present calculations of the $L$-functions associated to…

Shifted convolution sums play a prominent r\^ole in analytic number theory. We investigate pointwise bounds, mean-square bounds, and average bounds for shifted convolution sums for Hecke eigenforms.

Let $f$ be a holomorphic Hecke cusp form of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $(\lambda_f(n))_{n\geq 1}$ denote its sequence of normalised Hecke eigenvalues. We compute the first and second moments of the sums…

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$.…

Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is…

We prove that, for fixed tame level (N,p) = 1, there are only finitely many Hecke eigenforms f of level Gamma_1(N) and even weight with a_p(f) = 0 which are not CM.

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)}…

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard…

In this paper, we study the first moment of central values of Hecke $L$-functions associated with quartic characters.

Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and…

For a prime level $N$, we compute the index of the anemic Hecke algebra of weight 2 level $\Gamma_0(N)$ forms inside the full Hecke algebra. We prove that the index is fully described by the presence of weight 1 Katz forms of the same…

We determine the shape which minimizes, among domains with given measure, the first eigenvalue of the anisotropic laplacian perturbed by an integral of the unknown function. Using also some properties related to the associated \lq\lq…

A well known result of Breulmann states that Hecke eigenvalues of Saito-Kurokawa lifts are positive. In this article, we show that the Hecke eigenvalues of an Ikeda lift at primes are positive. Further, we derive lower and upper bounds of…

We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindel\"of hypothesis. We obtain asymptotic…

In this paper, we formulate conjectures on the joint distribution of several Hecke eigenforms. We prove an asymptotic formula of the joint mass of two Hecke eigenforms under the generalized Riemann Hypothesis (GRH) and the generalized…

We construct the basic representation of the double affine Hecke algebra at critical level $q=1$ associated to an irreducible reduced affine root system $R$ with a reduced gradient root system. For $R$ of untwisted type such a…

Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new…