Jumps of Jacobians via orthogonal canonical forms

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