Generalised divisor sums of binary forms over number fields

TL;DR

This paper generalizes the estimation of divisor sums over binary form values to arbitrary number fields, incorporating Jacobi symbols with varying arguments, and provides asymptotic estimates crucial for progress on Manin's conjecture.

Contribution

It introduces a novel approach to divisor sums over binary forms over number fields, extending previous results from the rational case and enabling applications to Manin's conjecture.

Findings

Provides asymptotic estimates for divisor sums over binary forms in number fields.

Establishes lower bounds for these sums with applications to algebraic geometry.

First such asymptotic evaluation over arbitrary number fields.

Abstract

Estimating averages of Dirichlet convolutions $1 \ast \chi$, for some real Dirichlet character $\chi$ of fixed modulus, over the sparse set of values of binary forms defined over $\mathbb{Z}$ has been the focus of extensive investigations in recent years, with spectacular applications to Manin's conjecture for Ch\^atelet surfaces. We introduce a far-reaching generalization of this problem, in particular replacing $\chi$ by Jacobi symbols with both arguments having varying size, possibly tending to infinity. The main results of this paper provide asymptotic estimates and lower bounds of the expected order of magnitude for the corresponding averages. All of this is performed over arbitrary number fields by adapting a technique of Daniel specific to $1\ast 1$. This is the first time that divisor sums over values of binary forms are asymptotically evaluated over any number field other than $\mathbb{Q}$. Our work is a key step in the proof, given in subsequent work, of the lower bound predicted by Manin's conjecture for all del Pezzo surfaces over all number fields, under mild assumptions on the Picard number.