Giacomo Cherubini: Coprime-universal quadratic forms

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.