Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups

TL;DR

This paper determines the Weil abscissa for certain procyclic groups formed from primes in specific residue classes, using quadratic form representations and a theorem of Iwaniec to analyze the zeta function.

Contribution

It establishes the Weil abscissa for procyclic groups associated with primes in particular residue classes, linking quadratic forms to the analysis of the Weil representation zeta function.

Findings

Weil abscissa equals 2 for primes p ≡ 1 mod 3, 1 mod 4, and 1 or 3 mod 8.

Integers with prime factors in these sets can be represented by specific binary quadratic forms.

A theorem of Iwaniec is used to construct a minorant for the Weil zeta function.

Abstract

Here we show that the Weil abscissa of the procyclic groups $\prod_{p \in S} \mathbb{Z}_p$ equals $2$ for three sets $S$: (i) the set of primes $p \equiv 1 \bmod 3$, (ii) the set of primes $p \equiv 1 \bmod 4$ and (iii) the set of primes $p \equiv 1,3 \bmod 8$. Our argument is based on the observation that integers all of whose prime factors lie in $S$ can be represented by a suitable binary quadratic form, which allows us to use a theorem of Iwaniec to exhibit a minorant for the Weil representation zeta function.