JUMPS OF JACOBIANS VIA ORTHOGONAL CANONICAL FORMS

ENIS KAYA, MICHAËL MAEX AND ART WAETERSCHOOT 

Given a smooth, proper curve C over a discretely valued field k, we equip the k-vector space H^0(C,ω_C/k) with a canonical discrete valuation v_can which measures how canonical forms degenerate on regular integral models of C. More precisely, v_can maps a canonical form to the minimal value of its associated weight function, as introduced by Mustaţă–Nicaise. Our main result states that v_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_can and the jumps can be computed efficiently for the class of ∆_v-regular curves.