Jumps of Jacobians via orthogonal canonical forms

TL;DR

This paper introduces a canonical valuation on canonical forms of algebraic curves over discretely valued fields, showing it computes Edixhoven's jumps of Jacobians and enabling efficient calculations for certain curve classes.

Contribution

It establishes that the canonical valuation computes Jacobian jumps via orthogonal bases and provides a new, efficient method for calculating these jumps for specific curve types.

Findings

Canonical valuation $v_{can}$ computes Edixhoven's jumps of Jacobians.

Proof of the rationality of Jacobian jumps.

Efficient computation methods for $ riangle_v$-regular curves.

Abstract

Given a smooth, proper curve $C$ over a discretely valued field $k$, we equip the $k$-vector space $H^{0}(C,\omega_{C/k})$ with a canonical discrete valuation $v_{\mathrm{can}}$ which measures how canonical forms degenerate on regular integral models of $C$. More precisely, $v_{\mathrm{can}}$ maps a canonical form to the minimal value of its associated weight function, as introduced by Musta\c{t}\u{a}--Nicaise. Our main result states that $v_{\mathrm{can}}$ computes Edixhoven's jumps of the Jacobian of $C$ when evaluated in an orthogonal basis. As a byproduct, we deduce a short proof for the rationality of the jumps of Jacobians. We also show how $v_{\mathrm{can}}$ and the jumps can be computed efficiently for the class of $\Delta_v$-regular curves introduced by Dokchitser.