Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction

TL;DR

This paper investigates how Tamagawa numbers of hyperelliptic Jacobians change with base field and curve variations under semistable reduction, revealing unexpected constraints and providing explicit formulas for families of curves.

Contribution

It introduces explicit methods and formulae for Tamagawa numbers of hyperelliptic Jacobians, highlighting unique behaviors and constraints.

Findings

Strong constraints on Tamagawa number variation

Explicit formulas for infinite families of hyperelliptic curves

Reveals unexpected behaviors specific to hyperelliptic cases

Abstract

We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as one varies the base field or the curve, in the case of semistable reduction. We find that there are strong constraints on the behaviour that appears, some of which are unexpected and specific to hyperelliptic curves. Our methods are explicit and allow one to write down formulae for Tamagawa numbers of infinite families of hyperelliptic curves, of the kind used in proofs of the parity conjecture for Jacobians of curves of small genus.