Related papers: Computing Classical Modular Forms for Arbitrary Co…
Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…
We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $\Gamma_0(N)$ for $N$ satisfying some congruence conditions and for…
In this paper we prove a conjectured modular equation of Farkas and Kra, which involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjectured identity…
We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL(2,Z) in terms of theta series. We apply this general setup to obtain closed and easily computable…
The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of…
We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL$(\bold Z)$ in terms of theta series. We apply this general setup to obtain closed and easily computable…
We study congruences between cuspidal modular forms and Eisenstein series at levels which are square-free integers and for equal even weights. This generalizes our previous results from Naskr\k{e}cki [17] for prime levels and provides…
Let $\Gamma<\mathrm{SL}_2(\mathbb{Z})$ be a non-elementary finitely generated subgroup and let $\Gamma(q)$ be its congruence subgroup of level $q$ for each $q\in \mathbb{N}$. We obtain an asymptotic formula for the matrix coefficients of…
We study from an algebraic point of view the question of extending an action of a group \(\Gamma\) on a commutative domain \(R\) to a formal pseudodifferential operator ring \(B=R(\!(x\,;\,d)\!)\) with coefficients in \(R\), as well as to…
The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $\Gamma$ we construct an arithmetic subgroup…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ over the ranges $0<k<p$ and $0<k<p/r$, where $p$ is a prime larger than the positive integer $r$. Here $x$ is an indeterminate, thus…
We describe an algorithm that we used to compute the q-expansions of all weight 2 cusp forms of prime level at most 2,000,000 and dimension at most 6. We also present an algorithm that we used to verify that there was only one cusp form of…
It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to…
We begin the process of classifying all supersymmetric theories with quantum modified moduli. We determine all theories based on a single SU or Sp gauge group with quantum modified moduli. By flowing among theories we have calculated the…
We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2…
Our recent extension of Arnold's classification includes all singularities of corank up to two equivalent to a germ with a non-degenerate Newton boundary, thus broadening the classification's scope significantly by a class which is…
We investigate rational $G$-modules $M$ for a linear algebraic group $G$ over an algebraically closed field $k$ of characteristic $p > 0$ using filtrations by sub-coalgebras of the coordinate algebra $k[G]$ of $G$. Even in the special case…
In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for $\mathrm{GL}_2$, this…
We consider the Clifford algebra and the Clifford group associated with any quadratic module, degenerate or not, over an arbitrary commutative ring with 1. We determine some of the important subalgebras of the Clifford algebra under some…