Arithmetic of hyperelliptic curves over local fields

Abstract

We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a "cluster picture" associated to the curve, that describes the p-adic distances between the roots of f(x), and show that this elementary combinatorial object encodes the curve's Galois representation, conductor, whether the curve is semistable, and if so, the special fibre of its minimal regular model, the discriminant of its minimal Weierstrass equation and other invariants.

93 pages

References in corpus (1)

• Models of Hyperelliptic Curves with Tame Potentially Semistable Reduction

Cited by in corpus (8)

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• Jumps of Jacobians via orthogonal canonical forms
• Local constancy of reduction type and related invariants for curves in $p$-adic families
• Computing Euler factors of genus 2 curves at odd primes of almost good reduction
• Semistable reduction of plane quartics at $p=3$