Related papers: Mod $p$ Hilbert modular forms of parallel weight o…

Let p > 2 be prime. We prove the weight part of Serre's conjecture for rank two unitary groups for mod p representations in the unramified case (that is, the Buzzard-Diamond-Jarvis conjecture for unitary groups), by proving that any Serre…

Let \rho be a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that \rho has large image and admits a low weight crystalline modular deformation we show that any low weight…

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each $k\in\{12,16,18,20,22,26\}$, we give explicit rational primes $\l$…

For a rational prime $p \geq 3$ and an integer $n \geq 2$, we study the modularity of continuous 2-dimensional mod $p^n$ Galois representations of $\Gal(\bar{\Q}/\Q)$ whose residual representations are odd and absolutely irreducible. Under…

We establish Bernstein-centre type of results for the category of mod $p$ representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. We treat all the remaining open cases, which occur when $p$ is $2$ or $3$. Our arguments carry over for all primes…

Half-integral weight modular forms are naturally viewed as automorphic forms on the so-called metaplectic covering of $\operatorname{GL}_2(\mathbf{A}_{\mathbf{Q}})$ -- a central extension by the roots of unity $\mu_2$ in $\mathbf{Q}$. For…

Using the link between mod $p$ Galois representations of $\qu$ and mod $p$ modular forms established by Serre's Conjecture, we compute, for every prime $p\leq 1999$, a lower bound for the number of isomorphism classes of continuous Galois…

For each subset of primes in a totally real field above a rational prime $p$, there is the notion of partially classical Hilbert modular forms, where the empty set recovers the overconvergent forms and the full set of primes above $p$…

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote…

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$. Let $R_0$ be an unramified relative base ring over $W(k)\langle X_1^{\pm 1}, \ldots, X_d^{\pm 1}\rangle$, and…

Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space…

We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N…

Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single…

For every prime number p, we show the existence of a solvable number field L ramified only at {p and infinity whose p-Hilbert Class field tower is infinite.

Let $F$ be a totally real field in which $p$ is unramified and $B$ be a quaternion algebra which splits at at most one infinite place. Let $\overline{r}:\mathrm{Gal}(\overline{F}/F)\to \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular…

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular,…

Let $\ell \geq 5$ be a prime and let $N$ be a square-free integer prime to $\ell$. For each prime $p$ dividing $N$, let $a_p$ be either $1$ or $-1$. We give sufficient criteria for the existence of a newform $f$ of weight 2 for…

Under mild hypotheses, we prove that if F is a totally real field, k is the algebraic closure of the finite field with l elements and r : G_F --> GL_2(k) is irreducible and modular, then there is a finite solvable totally real extension…

Suppose that F/F+ is a CM extension of number fields in which the prime p splits completely and every other prime is unramified. Fix a place w|p of F. Suppose that rbar : Gal(F-bar/F) -> GL_3(Fp-bar) is a continuous irreducible Galois…

For a rational prime p, let k be a perfect field of characteristic p, K be a finite totally ramified extension of Frac(W(k)) of degree e and r be a non-negative integer satisfying r<p-1. In this article, we prove the upper numbering…