Related papers: Vanishing Fourier Coefficients of Hecke Eigenforms

In 2008, Soundararajan showed that there exists a normalized Hecke eigenform $f$ of weight $k$ and level one such that $$ L(1/2, f ) ~\geq~ \exp\Bigg( (1 + o(1)) \sqrt{\frac{2\log k}{\log\log k} }\Bigg) $$ for sufficiently large $k \equiv 0…

In this paper, we prove effective quantitative decorrelation of values of two Hecke eigenforms as the weight goes to infinity. As consequences, we get an effective version of equidistribution of mass and zeros of certain linear combinations…

We consider Diophantine inequalities of the kind |f(x)| \le m, where F(X) \in Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m\ge 1. We say…

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

We study the $1$- or $2$-level density of families of $L$-functions for Hecke--Maass forms over an imaginary quadratic field $F$. For test functions whose Fourier transform is supported in $\left(-\frac 32, \frac 32\right)$, we prove that…

We study classes of atomic models At_T of a countable, complete first-order theory T . We prove that if At_T is not pcl-small, i.e., there is an atomic model N that realizes uncountably many types over pcl(a) for some finite tuple a from N,…

Let F_0 be a non-archimedean local field of odd residual characteristic and let G be the unramified unitary group U(2,2) defined over F_0. In this paper, we give a classification of the irreducible smooth representations of G of…

For an odd prime $p$, let $\phi$ denote the quadratic character of the multiplicative group ${\mathbb F}_p^\times$, where ${\mathbb F}_p$ is the finite field of $p$ elements. In this paper, we will obtain evaluations of the hypergeometric…

We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular…

We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar…

Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of…

For $m \geq 1$, let $N \geq 1$ be coprime to $m$, $k \geq 2$, and $\chi$ be a Dirichlet character modulo $N$ with $\chi(-1)=(-1)^k$. Then let $T_m^{\text{new}}(N,k,\chi)$ denote the restriction of the $m$-th Hecke operator to the space…

We prove that for any m > 1 given any m-tuple of Hecke eigenforms $f_i$ of level 1 whose weights satisfy the usual regularity condition there is a self-dual cuspidal automorphic form $\pi$ of $\GL_{2^m}(\Q)$ corresponding to their tensor…

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group $SL_2(\mathbb{Z})$.…

Given a pair of distinct non-CM normalized eigenforms having integer Fourier coefficients $a_1 (n)$ and $a_2(n)$, we count positive integers $n$ with $(a_1(n), a_2(n))=1$ and make a conjecture about the density of the set of primes $p$ for…

In this paper we consider the question of when the set of Hecke eigenvalues of a cusp form on $GL_n(A_F)$ is contained in the set of Hecke eigenvalues of a cusp form on $GL_m(A_F)$ for $n \leq m$.This question is closely related to a…

Let $f(z)=q+\sum_{n\geq 2}a(n)q^n$ be a weight $k$ normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in \cite{AH} for $k=2$ by ruling out or locating all…

In our recent papers the centralizer construction was applied to the series of classical Lie algebras to produce the quantum algebras called (twisted) Yangians. Here we extend this construction to the series of the symmetric groups S(n). We…

We show that given a simple abelian variety $A$ and a normal variety $V$ defined over a finitely generated field $K$ of characteristic zero, the set of non-constant morphisms $V \to A$ satisfying certain tangency conditions imposed by a…

A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=\frac{k}{2}.$ It was shown by Kohnen…