Giacomo Cherubini: Coprime-universal quadratic forms

Speaker
Giacomo Cherubini, Istituto Nazionale di Alta Matematica

Abstract
Given a prime p>3, we prove that there is an explicit set S_p of positive integers –whose cardinality does not exceed 31 and whose elements do not exceed 290– such that a positive definite integral quadratic form is coprime-universal with respect to p (i.e. it represents all positive integers coprime to p) if and only if it represents all the elements in S_p. This generalizes works of Bhargava and Hanke (p=1, i.e., no coprimality conditions), Rouse (p=2), and De Benedetto and Rouse (p=3). The proof is based on algebraic and analytic methods, plus a large computational part. Joint work with Matteo Bordignon. When p=5,23,29,31, our result is conditional on GRH, which is used to prove that certain ternary forms are coprime-universal, generalizing results of Ono and Soundararajan on Ramanujan’s ternary form, Lemke Oliver (regular forms) and Rouse (coprime-universal forms when p=2). During the talk, I will describe the strategy of proof, spending time on the conditional result to explain how to use the Shimura correspondence and Waldspurger’s theorem to reduce the desired claims to a finite computation on modular forms of weight 3/2 and weight 2. The assumption of GRH allows us to reduce the computation to integers up to 10^10 (vs 10^85 without GRH). The computation is then performed using Pari-GP and Magma.