Counting Multisections in Conic Bundles over a Curve defined over F_q

TL;DR

This paper develops a method to count irreducible branch covers of a conic bundle over a curve defined over a finite field, providing insights into algebraic numbers over function fields.

Contribution

It introduces a novel counting approach for branch covers in conic bundles over curves over finite fields, linking geometric and number-theoretic properties.

Findings

Count of irreducible branch covers grows with degree and height

Provides explicit formulas for the number of algebraic numbers over function fields

Establishes connections between geometric covers and algebraic number counts

Abstract

For a given conic bundle X over a curve C defined over F_q, we count irreducible branch covers of C in X of degree d and height e>>1. As a special case, we get the number of algebraic numbers of degree d and height e over the function field F_q (C).