CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES

Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

On Hecke eigenvalues of cusp forms in almost all short intervals

Let [Formula: see text] be a function such that [Formula: see text] as [Formula: see text]. Let [Formula: see text] be the [Formula: see text]th Hecke eigenvalue of a fixed holomorphic cusp form [Formula: see text] for [Formula: see text]. We show that for any real-valued function [Formula: see text] such that [Formula: see text], mean values of [Formula: see text] over intervals [Formula: see text] are bounded by [Formula: see text] for all but [Formula: see text] many integers [Formula: see text], in which [Formula: see text] is the average value of [Formula: see text] over primes. We generalize this for [Formula: see text] for [Formula: see text].

Some Density Results on Sets of Primes for Hecke Eigenvalues

Let f and g be two distinct holomorphic cusp forms for S L 2 ℤ , and we write λ f n and λ g n for their corresponding Hecke eigenvalues. Firstly, we study the behavior of the signs of the sequences λ f p λ f p j for any even positive integer j . Moreover, we obtain the analytic density for the set of primes where the product λ f p i λ f p j is strictly less than λ g p i λ g p j . Finally, we investigate the distribution of linear combinations of λ f p j and λ g p j in a given interval. These results generalize previous ones.

A note on arithmetic behavior of Hecke eigenvalues of Siegel cusp forms of degree two

Let [Formula: see text] and [Formula: see text] be Siegel cusp forms for the group [Formula: see text] with weights [Formula: see text], [Formula: see text], respectively. Suppose that neither [Formula: see text] nor [Formula: see text] is a Saito–Kurokawa lift. Further suppose that [Formula: see text] and [Formula: see text] are Hecke eigenforms lying in distinct eigenspaces. In this paper, we investigate simultaneous arithmetic behavior and related problems of Hecke eigenvalues of these Hecke eigenforms, some of which improve upon results of Gun et al.

A Hecke-equivariant decomposition of spaces of Drinfeld cusp forms via representation theory, and an investigation of its subfactors

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum’s isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory, it relates certain intertwining maps to hyperderivatives of Bosser-Pellarin, and it begins an investigation into dimension formulas for the subquotients mentioned above. We end with some numerical data for $$A={\mathbb {F}}_3[t]$$ A = F 3 [ t ] that displays a new obstruction to an analog of a Maeda conjecture by discovering a conjecturally infinite supply of $${\mathbb {F}}_3(t)$$ F 3 ( t ) -rational eigenforms with combinatorially given (conjectural) Hecke eigenvalues at the prime t.

Divisor Problems Related to Hecke Eigenvalues in Three Dimensions

In this paper, we consider divisor problems related to Hecke eigenvalues in three dimensions. We establish upper bounds and asymptotic formulas for these problems on average.