Coprime-Universal Quadratic Forms

TL;DR

This paper characterizes positive-definite integral quadratic forms that represent all integers coprime to a prime p, extending the 290-Theorem and related results, with some results conditional on GRH.

Contribution

It generalizes the concept of universality to coprime-universality for primes p, extending previous theorems and providing conditional proofs for specific primes.

Findings

Characterization of coprime-universal quadratic forms for primes p>3.

Extension of the 290-Theorem to coprime-universality.

Conditional proofs for primes 5, 23, 29, 31 under GRH.

Abstract

Given a prime $p>3$, we characterize positive-definite integral quadratic forms that are coprime-universal for $p$, i.e. representing all positive integers coprime to $p$. This generalizes the $290$-Theorem by Bhargava and Hanke and extends later works by Rouse ($p=2$) and De Benedetto and Rouse ($p=3$). When $p=5,23,29,31$, our results are conditional on the coprime-universality of specific ternary forms. We prove this assumption under GRH (for Dirichlet and modular $L$-functions), following a strategy introduced by Ono and Soundararajan, together with some more elementary techniques borrowed from Kaplansky and Bhargava. Finally, we discuss briefly the problem of representing all integers in an arithmetic progression.