Potential good reduction of hyperelliptic curves

TL;DR

This paper investigates the minimal number of primes outside which infinitely many hyperelliptic curves over a number field have potentially good reduction, providing new bounds and insights into their reduction properties.

Contribution

It introduces the invariant c_K(g) for hyperelliptic curves and establishes a lower bound relating it to prime counts, advancing understanding of reduction behavior.

Findings

c_K(g) > π_{K, odd}(2g) + 1

Provides conditional and unconditional upper bounds for c_K(g)

Enhances knowledge of reduction properties of hyperelliptic curves

Abstract

Let $K$ be a number field, and $g \geq 2$ a positive integer. We define $c_K(g)$ as the smallest integer $n$ such that there exist infinitely many $\overline{K}$-isomorphism classes of genus $g$ hyperelliptic curves $C/K$ with all Weierstrass points in $K$ having potentially good reduction outside $n$ primes in $K$. We show that $c_K(g) > \pi_{K, \textrm{odd}}(2g) + 1$, where $\pi_{K, \textrm{odd}}(n)$ denotes the number of odd primes in $K$ with norm no greater than $n$, as well as present a summary of various conditional and unconditional results on upper bounds for $c_K(g)$.