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

Martin Jann; Steffen Kionke

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arXiv · 2602.09797

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

Martin Jann·Steffen Kionke

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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.

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Martin Jann, Steffen Kionke. 2026-02-10. Primes represented by quadratic forms and the Weil abscissa of abelian profinite groups. https://arxiv.org/abs/2602.09797

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