Related papers: Theta Cycles of Modular Forms Modulo $p^2$

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We…

We prove a higher weight general Gross--Zagier formula for CM cycles on Kuga--Sato varieties over modular curves of arbitrary levels. To formulate and prove this result, we prove several results on the modularity of CM cycles, in the sense…

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $\alpha$ is irrational and $\beta$ is real then for any fixed $1<c<12/11$ there exist infinitely…

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…

We give a complete picture of when the tensor product of an induced module and a Weyl module is a tilting module for the algebraic group $SL_2$ over an algebraically closed field of characteristic $p$. Whilst the result is recursive by…

It is proved that the theta series of an even lattice whose level is a power of a prime $\ell$ is congruent modulo $\ell$ to an elliptic modular form of level~1. The proof uses arithmetic and algebraic properties of lattices rather than…

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are…

We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with…

The paper develops the result of second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular…

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p, N) \in \{(3, 1), (5, 1), (7, 1), (3, 4)\}$. In these…

In this paper, we investigate traces of cycle integrals of certain meromorphic modular forms. By relating them to regularised theta lifts we provide explicit formulae for them in terms of coefficients of harmonic Maass forms.

Let $p$ be a prime, and let $k,n,m,n_0$ and $m_0$ be nonnegative integers such that $k\ge 1$, and $_0$ and $m_0$ are both less than $p$. K. Davis and W. Webb established that for a prime $p\ge 5$ the following variation of Lucas' Theorem…

We use deformation theory to study the big Hecke algebra acting on mod-2 modular forms of prime level $N$ and all weights, especially its local component at the trivial representation. For $N = 3, 5$, we prove that the maximal reduced…

Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields.

We show that almost all permutations have some power that is a cycle of prime length. The proof includes a theorem giving a strong upper bound on the proportion of elements of the symmetric group having no cycles with length in a given set.

In a previous paper we attached to classical complex newforms $f$ of weight $2$ certain $\delta_p$-modular forms $f^{\sharp}$ of order $2$ and weight $0$; the forms $f^{\sharp}$ can be viewed as "dual" to $f$ and played a key role in some…

We consider the periods of the linear congruential and the power generators modulo $n$ and, for fixed choices of initial parameters, give lower bounds that hold for ``most'' $n$ when $n$ ranges over three different sets: the set of primes,…

Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves $E/\mathbb{Q}$. We…

Problem 8.1 in Astaiza et. al. asks about the relationship between the cycle decomposition of a permutation $\sigma$ and that of its symmetric tensor power $\sigma ^{\odot k}$. In this paper, we investigate this question and give formulas…

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta…