Related papers: Spectral expansions of overconvergent modular func…

Let $p$ be a prime, $F$ be a totally real field in which $p$ is unramified and $\rho: \mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a totally odd, irreducible, continuous representation. The geometric…

We develop an analytic theory of cusps for Scholze's $p$-adic modular curves at infinite level in terms of perfectoid parameter spaces for Tate curves. As an application, we describe a canonical tilting isomorphism between an anticanonical…

The moduli dependence of $(2,2)$ superstring compactifications based on Calabi--Yau hypersurfaces in weighted projective space has so far only been investigated for Fermat-type polynomial constraints. These correspond to Landau-Ginzburg…

We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+\epsilon})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the…

Let $\lambda(m)$ be the $m$th coefficient of a modular form $f(z)=\sum_{m\geq 1} \lambda(m)q^m$ of weight $k\geq 4$, let $p^n$ be a prime power, and let $\varepsilon>0$ be a small number. An approximate of the Atkin-Serre conjecture on the…

We prove the weight elimination direction of the Serre weight conjectures as formulated by Herzig for forms of $U(n)$ which are compact at infinity and split at places dividing $p$ in generic situations. That is, we show that all modular…

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function $\bar{p}(n)$ and two smallest parts functions: $\bar{\operatorname{spt1}}(n)$ for overpartitions and $\operatorname{M2spt}(n)$ for partitions…

The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small…

The modular properties of the simple vertex operator superalgebra associated to the affine Kac-Moody superalgebra $\widehat{\mathfrak{osp}} \left( 1 \middle\vert 2 \right)$ at level $-\frac{5}{4}$ are investigated. After classifying the…

We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a…

In this article, we prove an omega-result for the Hecke eigenvalues $\lambda_F(n)$ of Maass forms $F$ which are Hecke eigenforms in the space of Siegel modular forms of weight $k$, genus two for the Siegel modular group $Sp_2(\Z)$. In…

A seminal open question of Pisier and Mendel--Naor asks whether every degree-regular graph which satisfies the classical discrete Poincar\'e inequality for scalar functions, also satisfies an analogous inequality for functions taking values…

In this paper, we prove a conjecture of Andrews and Bachraoui relating a generating function arising from two-color partitions (with odd smallest part and restrictions on the even parts) to a Hecke-type double sum. Our proof is based on…

We show (under some hypothesis in small dimensions) that the analytic degree of the divisor of a modular form on the orthogonal group O(2,p) is determined by its weight. Moreover, we prove that certain integrals, occurring in Arakelov…

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…

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n…

Inspired by Borcherds' questions, Guerzhoy constructed a new type of Hecke operators $\mathcal{T}(p)$, called the multiplicative Hecke operators, which acts on the space of meromorphic modular forms on the full modular group ${\rm SL}(\Z)$.…

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\Gamma$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\mathrm{Gal}\left(\overline{K}/K\right)$ in…

We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modularity condition is…

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…