Counting quadratic points on Fano varieties

TL;DR

This paper develops a framework for counting rational points of bounded height on symmetric squares of weak Fano varieties, proving the Manin--Peyre conjecture for certain non-split quadric surfaces and introducing new techniques for lattice point counting and L-function analysis.

Contribution

It introduces a general method for point counting on symmetric squares of Fano varieties, relates it to the Manin--Peyre conjecture, and proves it for an infinite family of non-split quadric surfaces.

Findings

Established asymptotic formulas for point counts on specific surfaces.

Proved the Manin--Peyre conjecture for an infinite family of non-split quadric surfaces.

Developed new lattice point counting techniques over rings of integers.

Abstract

This paper initiates the systematic study of the number of points of bounded height on symmetric squares of weak Fano varieties. We provide a general framework for establishing the point count on $\text{Sym}^2 X$. In the specific case of surfaces, we relate this to the Manin--Peyre conjecture for $\text{Hilb}^2 X$, and prove the conjecture for an infinite family of non-split quadric surfaces. In order to achieve the predicted asymptotic, we show that a type II thin set of a new flavour must be removed. To establish our counting result for the specific family of surfaces, we generalise existing lattice point counting techniques to lattices defined over rings of integers. This reduces the dimension of the problem and yields improved error terms. Another key tool we develop is a collection of results for summing Euler products over quadratic extensions. We use this to show moments of $L$-functions at $s=1$ are constant on average in quadratic twist families.