Related papers: A note on the minimal level of realization for a m…

We show that the systems of Hecke eigenvalues occurring in the spaces of Siegel modular forms (mod p) of fixed dimension g, fixed level N, and varying weight, are the same as the systems occurring in the spaces of Siegel cusp forms with the…

We show that, under suitable assumptions, the systems of Hecke eigenvalues arising from (mod p) modular forms of PEL-type associated to an algebraic group G of type A or C coincide with the Hecke eigensystems arising from (mod p) algebraic…

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…

Let $F/\mathbb{Q}$ be any totally real number field and $\frak{N}$ an ideal of its ring of integers of norm $N$ and define, for every even $n$, the $[F:\mathbb{Q}]$-dimensional multiweight $\textbf{n}=(n,...,n)$. We prove that for a non CM…

We establish the existence of many holomorphic Hecke eigenforms $f$ of large weight $k$ for the full modular group, for which the least positive integer $n_f$ such that $\lambda_f(n_f)<0$ satisfies $n_f \ge (\log k)^{1-o(1)}.$ This is…

Let $ F$ be an imaginary quadratic field and $\mathcal{O}$ its ring of integers. Let $ \mathfrak{n} \subset \mathcal{O} $ be a non-zero ideal and let $ p> 5$ be a rational inert prime in $F$ and coprime with $\mathfrak{n}$. Let $ V$ be an…

Let K be an imaginary quadratic field with class number one and ring of integers O. We prove that mod l, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Gamma of SL(2,O) can be realized in…

Let $F$ be an arbitrary totally real field. Under weak conditions we prove the existence of certain Eisenstein congruences between parallel weight $k \geq 3$ Hilbert eigenforms of level $\mathfrak{mp}$ and Hilbert Eisenstein series of level…

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of…

Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $\Gamma_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consider the problem of deciding whether $f_1$ and $f_2$ have…

We fix $\ell$ a prime and let $M$ be an integer such that $\ell\not|M$; let $f\in S_2(\Gamma_1(M\ell^2))$ be a newform supercuspidal of fixed type related to the nebentypus, at $\ell$ and special at a finite set of primes. Let $\TT^\psi$ be…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…

Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $\mathfrak p$ be a prime ideal of the Hecke field of $f$. We denote by $SP_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k \equiv 0 \mod 2, \ j \equiv 0…

We derive an explicit upper bound for the number of systems of Hecke eigenvalues coming from Siegel modular forms (mod p) of dimension g and level N relatively prime to p. In the special case of elliptic modular forms (g=1), our result…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of…

Let $k \ge 2$ be an even integer, $ \ell \ge \max\{5, k-1\} $ be a prime, and $N$ be a squarefree positive integer. It is known that if the $\rm{mod}\,\ell$ Galois representation $\overline{\rho}_f$ associated with a newform $f$ of weight…

Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let…

In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod $p$ modular forms (of fixed level $\Gamma(N)$ coprime to $p$, and any weight $k$) are the same as those arising from functions $\Omega(N) \to \bar{\mathbb…

Let $\{\lambda_f(n)\}_{n \geq 1}$ be the normalized Hecke eigenvalues of a given holomorphic cusp form $f$ of even weight $k$. We show under the assumption of the existence of Littlewood's type zero free region for $L(s, f, \chi)$, where…