Search

Let Γ be a subgroup of finite index of SL 2 ⁡ (퐙) . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ (2) , where we can consider convenient generalized Jacobians instead of J Γ . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves. [ABSTRACT FROM AUTHOR]

We define L-series of weakly holomorphic quasimodular forms and we derive functional equations of those L-series. We also prove a converse theorem for weakly holomorphic quasimodular forms. [ABSTRACT FROM AUTHOR]

Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ (n) is odd if and only if n is an odd square, we only need to consider τ ⁢ (p 2 ⁢ n) for primes p and natural numbers n ≥ 1 . This is a rather delicate question. In this direction, we show that for any ϵ > 0 and integer n ≥ 1 , the largest prime factor of τ ⁢ (p 2 ⁢ n) , denoted by P ⁢ (τ ⁢ (p 2 ⁢ n)) , satisfies P ⁢ (τ ⁢ (p 2 ⁢ n)) > (log ⁡ p) 1 8 ⁢ (log ⁡ log ⁡ p) 3 8 - ϵ for almost all primes p. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients. [ABSTRACT FROM AUTHOR]

We consider the Kudla–Millson theta series associated to a quadratic space of signature (N , N) . By combining a "see-saw" argument with the Siegel–Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree N is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon, Pozzi and Vonk to totally real fields. [ABSTRACT FROM AUTHOR]

Let k and n be positive even integers. For a Hecke eigenform h in the Kohnen plus subspace of weight k - n 2 + 1 2 for Γ 0 ⁢ (4) , let I n ⁢ (h) be the Duke–Imamoglu–Ikeda lift of h to the space of cusp forms of weight k for Sp n ⁡ (ℤ) . We then give an estimate of the Fourier coefficients of I n ⁢ (h) . It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form. [ABSTRACT FROM AUTHOR]

We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central L -values. We specialize our results to integer partitions, for which we investigate the landmark proofs of partition congruences by Atkin and by Ono. Based on a modulo ℓ analogue of the Maeda conjecture for certain partition generating functions, we conclude that their approach by Hecke operators acting diagonally modulo ℓ on modular forms is indeed close to optimal. This work is enabled by several structure results for Ramanujan-type congruences that we establish. In an extended example, we showcase how to employ them to also benefit experimental work. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for X 2 3 + ϵ < H < X 1 - ϵ there are constants B h such that ∑ X ≤ n ≤ 2 ⁢ X λ f ⁢ (n) 2 ⁢ λ f ⁢ (n + h) 2 - B h ⁢ X = O f , A , ϵ ⁢ (X ⁢ (log ⁡ X) - A ) for all but O f , A , ϵ ⁢ (H ⁢ (log ⁡ X) - 3 ⁢ A ) integers h ∈ [ 1 , H ] where { λ f ⁢ (n) } n ≥ 1 are normalized Hecke eigenvalues of a fixed holomorphic cusp form f. Our method is based on the Hardy–Littlewood circle method. We divide the minor arcs into two parts m 1 and m 2 . In order to treat m 2 , we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat m 1 , we apply Parseval's identity and Gallagher's lemma. [ABSTRACT FROM AUTHOR]

This article deals with various kinds of quantitative results about the comparison between the normalized Hecke eigenvalues of two distinct Siegel cuspidal Hecke eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts. We also prove some simultaneous sign change results for their eigenvalues. [ABSTRACT FROM AUTHOR]

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 , there exist infinitely many natural numbers 푛 such that τ ⁢ (p n) has at least 2 (1 - ϵ) ⁢ log ⁡ n log ⁡ log ⁡ n distinct prime factors for almost all primes 푝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above. [ABSTRACT FROM AUTHOR]

In this paper, we consider twisted Dirichlet series attached to quasimodular forms, study their analytic properties, and prove an analogue of Weil's converse theorem for quasimodular forms over congruence subgroups. We also give some applications of our results to a certain q-series and sign changes of the Fourier coefficients of quasimodular forms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, () for a fixed positive integer r, concerning both non-vanishing and non-negativity. [ABSTRACT FROM AUTHOR]

We study some arithmetics of Hermitian modular forms of degree two by applying the spectral theory on 3-dimensional hyperbolic space. This paper presents three main results: (1) a 3-dimensional analogue of Katok-Sarnak's correspondence, (2) an analytic proof of a Hermitian analogue of the Saito-Kurokawa lift by means of a converse theorem, (3) an explicit formula for the Fourier coefficients of a certain Hermitian Eisenstein series. [ABSTRACT FROM AUTHOR]

We characterize all cusp forms among the degree two Siegel modular forms by the growth of their Fourier coefficients. We also give a similar result for Jacobi forms over the group . [ABSTRACT FROM AUTHOR]

In this paper, we define the Euler phi function φ π( n) associated to the automorphic L-function L( s, π), where π is an automorphic irreductible unitary cuspidal representation of GL N(ℚ). Furthermore, we study the asymptotic properties for φ π( n) and prove some results on the summation function ∑ n≤ x φ π( n). [ABSTRACT FROM AUTHOR]