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.

Voloch and Walker used liftings of elliptic curves to create error-correcting codes, for which lifting of points with smaller degrees for their coordinate functions give better parameters. Thus, one might be interested in constructing liftings with minimal possible degrees. Here we study liftings of twisted Edwards curve with minimal degrees. We prove that up to the second coordinate these minimal degrees are given by the canonical lifting and elliptic Teichmüller lift, give upper and lower bounds for the degrees of Edwards lifts, and prove that the lower bound can be achieved for the third coordinate.

In this paper, we evaluate Fine type integrals of level 30. For an example, [Formula: see text] In the process, we discover certain new Eisenstein series identities of level 30.

There exist results proving that within certain families of totally real number fields, the minimal rank of a universal quadratic lattice over such a field can be arbitrarily large. Kala introduced a technique of extending such results to larger fields — e.g., from quadratic fields to fields of arbitrary even degree — under some conditions. We present improvements to this technique by investigating the structure of subfields within the smallest common superfield of two number fields, using Galois theory to translate this into a group-theoretic problem. In particular, we show that if totally real number fields with minimal rank of a universal lattice [Formula: see text] exist in degree d, then they also exist in degree [Formula: see text] for all [Formula: see text].

In this paper, we investigate the average formula for the coefficients of general L-functions without assuming the generalized Ramanujan conjecture. As an application, we consider the higher moments of Fourier coefficients of Hecke–Maass cusp forms. Our results also improve the previous results.

The Pólya group [Formula: see text] of an algebraic number field [Formula: see text] is the subgroup of the ideal class group [Formula: see text] generated by the ideal classes of the products of prime ideals of the same norm. If [Formula: see text] is trivial, then the number field [Formula: see text] is said to be a Pólya field. In the first part of this paper, we extend a recent result of ours and prove the existence of infinitely many number fields having Pólya groups as large as possible and those number fields being the compositum of a simplest cubic field and a quintic field. Using a result of Zylinski, we also prove that those fields are of index [Formula: see text]. Next, we produce three new families [Formula: see text], [Formula: see text] and [Formula: see text] of totally real bi-quadratic Pólya fields [Formula: see text] involving prime numbers [Formula: see text] and [Formula: see text] that satisfy certain quadratic residue conditions among themselves. It is worthwhile to note that in each of these fields, exactly five primes ramify in [Formula: see text] and this is the maximum possible number of ramified primes in a bi-quadratic Pólya field over [Formula: see text].

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan’s Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong q-analogues by virtue of their reduction to the classical beta evaluations as [Formula: see text] and explicit evaluations at CM points for [Formula: see text]. We also determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

Let [Formula: see text] be the set of primes and [Formula: see text] the set of positive integers. Let also [Formula: see text] be positive real numbers and [Formula: see text] the set of odd integers which can be represented as [Formula: see text] where [Formula: see text] and [Formula: see text]. Recently, Chen and Xu proved that the set [Formula: see text] has positive lower asymptotic density, provided that [Formula: see text] and at least one of [Formula: see text] is an integer. Their result reduces to the famous theorem of Romanoff by taking [Formula: see text] In this paper, we remove the unnecessary condition: ‘at least one of [Formula: see text] is an integer’.

Let E be a non-CM elliptic curve defined over [Formula: see text]. There is an isogeny-torsion graph associated to E and there is also a Galois representation [Formula: see text] associated to E for every prime [Formula: see text] In this paper, we explore the relation between these two objects when [Formula: see text]. More precisely, we give a classification of 3-adic Galois images associated to vertices of the isogeny-torsion graph of [Formula: see text]

Drawing inspiration from the work of Nathanson and Yamada, we prove an effective and explicit version of Chen’s theorem. By contrast, existing proofs of Chen’s theorem are ineffective due to their use of the Siegel–Walfisz theorem. Our main result is that every even integer larger than [Formula: see text] can be written as the sum of a prime and the product of at most two primes. We also prove that all even integers [Formula: see text] can be written as the sum of a prime and the product of at most [Formula: see text] primes. The main idea will be to follow a proof of Chen’s theorem due to Nathanson, being more careful with the treatment of potential Siegel zeros in order to obtain an effective and explicit result. In following this framework we also prove an explicit version of the linear sieve, which substantially improves upon the previous best one by Nathanson.