Related papers: Murmurations of modular forms in the weight aspect

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular…

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and…

We report on the computation of torsion in certain homology theories of congruence subgroups of SL(4,Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories…

We study the action of the derived Hecke algebra on the space of weight one forms. By analogy with the topological case, we formulate a conjecture relating this to a certain Stark unit. We verify the truth of the conjecture numerically, for…

We introduce and study modular truncations of the Ackermann function viewed as self-maps on finite rings. These maps form a hierarchy of rapidly increasing compositional complexity indexed by recursion depth. We investigate their structural…

We discover a non-trivial relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$, where $N > 1$ is…

We associate weight complexes of (homological) motives, and hence Euler characteristics in the Grothendieck group of motives, to arithmetic varieties and Deligne-Mumford stacks; this extends the results in the paper "Descent, Motives and…

We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that…

We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod $p$ Hilbert modular forms, by making use of modularity lifting theorems and computations in $p$-adic Hodge theory.

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…

A description is given of all primitive differential series mod p of order 1 which are eigenvectors of all the Hecke operators and which are differential Fourier expansions of differential modular forms of arbitrary order and given weight;…

In this paper, we prove a conjecture of Broadhurst and Zudilin \cite{BZ17} concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds…

Let $q:=e^{2 \pi iz}$, where $z \in \mathbb{H}$. For an even integer $k$, let $f(z):=q^h\prod_{m=1}^{\infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $\Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th…

We give a presentation of the cohomology ring of spatial polygon spaces $M(r)$ with fixed side lengths $r \in \mathbb R^n_+$. These spaces can be described as the symplectic reduction of the Grassmaniann of 2-planes in $\mathbb C^n$ by the…

In this paper we consider certain families of arithmetic subgroups of SO^0(p,q) and SL_3(R), respectively. We study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules. We show that for natural…

We construct Eigenvarieties for PEL Shimura varieties which interpolate cuspidal, finite slope automorphic forms for PEL Shimura varieties appearing as global sections of (coherent) automorphic sheaves, under the hypothesis that the primes…

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f…

The enumeration $d_k(n)$ of $k$-elongated plane partition diamonds has emerged as a generalization of the classical integer partition function $p(n)$. Congruences for $d_k(n)$ modulo certain powers of primes have been proven via elementary…

In this article, we will generalize an explicit formula proved by Quer for the Brauer class of the endomorphism algebra of abelian varieties associated to modular forms of weight 2 to the case of Hilbert modular forms of parallel weight 2,…

We determine the structure of the first resonance variety of the cohomology ring of the group of automorphisms of a finitely generated free group which act by conjugation on a given basis.