Kohnen and Zagier studied two types of cusp forms [Formula: see text] on [Formula: see text] defined by binary quadratic forms. They showed the cusp forms [Formula: see text] have rational even periods, while [Formula: see text] have rational odd periods. In this paper, we examine analogous cusp forms [Formula: see text] on [Formula: see text] for higher levels [Formula: see text]. Using locally harmonic Maass forms on [Formula: see text], we explicitly compute the even (resp. odd) period polynomials of [Formula: see text] (resp. [Formula: see text]). Consequently, we establish that the space of cusp forms of weight [Formula: see text] on [Formula: see text] has rational structures, attributed to the periods of these cusp forms.

We construct a differential operator on sheaves of p-adic modular forms defined over the locus of p-rank ≥ 1 of the Siegel threefold, by applying a revisited version of the approach that S. Howe recently introduced in [7] to construct the theta operator in the elliptic case.

Let [Formula: see text] denote the Riemann zeta function; let [Formula: see text] and [Formula: see text] denote the Möbius and Liouville functions respectively, while [Formula: see text] and [Formula: see text] respectively denote their corresponding summatory functions. We consider the correlations [Formula: see text] and [Formula: see text] where [Formula: see text] is arbitrary and [Formula: see text] is suitably chosen. Under the Riemann hypothesis and simplicity of the nontrivial zeros [Formula: see text] of [Formula: see text] we show that [Formula: see text] and [Formula: see text] as [Formula: see text] where [Formula: see text]. These results combined with numerical observations suggest that there is anticorrelation between [Formula: see text] and [Formula: see text] as well as between [Formula: see text] and [Formula: see text], where the correlation is computed using a logarithmic average. This would imply effective upper bounds on [Formula: see text].

This paper applies the modular approach to obtain effectively computable bounds for Fermat-type equations over number fields, while also discussing the differences and obstructions that arise when considering such equations over totally real versus totally complex number fields. We use these techniques to study the generalized Fermat equation [Formula: see text] over quadratic fields [Formula: see text] of class number one. Extending the results of Freitas&Siksek and Turcas, we show that when [Formula: see text], there is an effective and explicit bound, depending on the field [Formula: see text], such that the latter equation does not have certain types of special solutions. We also discuss, for [Formula: see text], the solutions of a variant of the above equation. Our results over imaginary quadratic fields are conjectural. Serre’s modularity conjecture and an analogue of Eichler-Shimura over totally complex fields are assumed.

In this paper we prove rationality results for critical values for [Formula: see text]-functions attached to representations in the residual spectrum of [Formula: see text]. We use the Jacquet-Langlands correspondence to describe their partial [Formula: see text]-functions via cuspidal automorphic representations of the group [Formula: see text] over a quaternion algebra. Using ideas inspired by results of Grobner and Raghuram we are then able to compute the critical values as a Shalika period up to a rational multiple.

For a non-zero algebraic number [Formula: see text] of degree [Formula: see text], let [Formula: see text] denote its logarithmic Weil height. It is known that when [Formula: see text] is small, and [Formula: see text] is large, the conjugates of [Formula: see text] are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a [Formula: see text]-adic analogue of this result by obtaining lower bounds for [Formula: see text] in terms of the number of its conjugates that lie in a finite extension of [Formula: see text], for some prime [Formula: see text]. As a consequence, we prove Lehmer’s conjecture for all [Formula: see text] such that [Formula: see text] many of its conjugates lie in a finite extension of [Formula: see text].

A subset [Formula: see text] is called [Formula: see text]-almost square universal if every sufficiently large positive integer can be written as a sum of at most [Formula: see text] squares of integers from [Formula: see text]. In this article, we study the minimal number [Formula: see text] with this property, where [Formula: see text] denotes the residue class of [Formula: see text] modulo [Formula: see text], with [Formula: see text] and [Formula: see text]. We further prove that [Formula: see text] is [Formula: see text]-square universal for some [Formula: see text] if and only if [Formula: see text], and determine the minimal such number [Formula: see text] in these cases.

In this paper, we examine a unique identity—Entry 27 from Ramanujan’s Notebook IV. By leveraging the symmetry properties of integral quadratic forms, we provide natural representations of both sides of the identity in terms of three distinct classes of binary quadratic forms with discriminant −252. The combination of integral quadratic forms and exact covering systems leads to a collection of surprisingly elegant identities.

In this work, we establish several results on distinguishing Siegel cusp forms of degree two. In particular, a Hecke eigenform of level one can be determined by its second Hecke eigenvalue under a certain assumption. Moreover, we can distinguish two Hecke eigenforms of level one by using L-functions.

We use the Petrow-Young [11] subconvexity bound for Dirichlet [Formula: see text]-functions to show that [Formula: see text] has exponent of distribution [Formula: see text] when we allow an average over [Formula: see text] mod [Formula: see text], thereby giving an equidistribution result for [Formula: see text] which goes past the [Formula: see text] barrier for the first time.