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Jumps of Jacobians via orthogonal canonical forms

Enis Kaya, Michaël Maex, Art Waeterschoot

TL;DR

This work develops a canonical valuation $v_{\text{can}}$ on the space of canonical forms $V=H^0(C,\omega_{C/k})$ for a smooth proper curve $C$ over a discretely valued field, tying degeneration of holomorphic forms to arithmetic invariants. The authors prove that, when evaluated in an orthogonal basis, $v_{\text{can}}$ recovers Edixhoven's jumps of the Jacobian $\mathrm{Jac}(C)$, and that these jumps are rational with the stabilisation index $e(C)$ governing the image of $v_{\text{can}}$. The approach hinges on constructing an $R$-basis of $\Omega(C)$ from an orthogonal $v_{\text{can}}$-basis and analyzing its behavior under tame base change, leading to an explicit computation of jumps. For $\Delta_v$-regular curves ( Dok21 ), the method yields efficient, explicit formulas: the jumps are the decimal parts of the valuations attached to interior Newton polygon points, enabling practical calculation via Baker’s theorem. Overall, the paper provides a new, computable bridge between weight functions, canonical valuations, and the rational jumps of Jacobians, with concrete algorithms for a broad class of curves.

Abstract

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 $v_{\mathrm{can}}$ 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 Mustaţă--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 $Δ_v$-regular curves introduced by Dokchitser.

Jumps of Jacobians via orthogonal canonical forms

TL;DR

This work develops a canonical valuation on the space of canonical forms for a smooth proper curve over a discretely valued field, tying degeneration of holomorphic forms to arithmetic invariants. The authors prove that, when evaluated in an orthogonal basis, recovers Edixhoven's jumps of the Jacobian , and that these jumps are rational with the stabilisation index governing the image of . The approach hinges on constructing an -basis of from an orthogonal -basis and analyzing its behavior under tame base change, leading to an explicit computation of jumps. For -regular curves ( Dok21 ), the method yields efficient, explicit formulas: the jumps are the decimal parts of the valuations attached to interior Newton polygon points, enabling practical calculation via Baker’s theorem. Overall, the paper provides a new, computable bridge between weight functions, canonical valuations, and the rational jumps of Jacobians, with concrete algorithms for a broad class of curves.

Abstract

Given a smooth, proper curve over a discretely valued field , we equip the -vector space with a canonical discrete valuation which measures how canonical forms degenerate on regular integral models of . More precisely, maps a canonical form to the minimal value of its associated weight function, as introduced by Mustaţă--Nicaise. Our main result states that computes Edixhoven's jumps of the Jacobian of 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 and the jumps can be computed efficiently for the class of -regular curves introduced by Dokchitser.
Paper Structure (18 sections, 14 theorems, 44 equations, 1 figure)

This paper contains 18 sections, 14 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setup and motivation.
  3. Canonical valuation.
  4. Properties of $v_\mathrm{can}$.
  5. Main result.
  6. Orthogonal canonical forms.
  7. A brief outline of the proof.
  8. Overview of the paper.
  9. Preliminaries
  10. Notation and assumptions.
  11. Jumps of Abelian varieties.
  12. Original definition of jumps.
  13. Weight functions.
  14. A canonical valuation on $H^0(C,\omega_{C/ k})$.
  15. Proof of Theorem \ref{['thm:MainThmcopy']}
  16. ...and 3 more sections

Key Result

Theorem 1.7

Let $k$ be a Henselian discretely valued field with algebraically closed residue field $\tilde{k}$ and let $C$ be a smooth, proper and geometrically connected $k$-curve of genus $g\ge 1$ admitting a divisor of degree $1$. Then for all $j\in [0,1)$, the integer $\dim_{\tilde{k}}\mathrm{Gr}^{-j}V$ equ

Figures (1)

  • Figure 1.15: Left: the subdivided Newton polygon $\Delta$ for the given affine equation of $C$, decorated with a $v$-function whose values on lattice points are indicated. Right: special fiber of the minimal snc model $\mathscr{C}$ of $C$.

Theorems & Definitions (42)

  • Remark 1.5
  • Theorem 1.7
  • Corollary 1.8
  • Remark 1.9
  • Proposition 1.12
  • Example 1.14
  • Remark 1.16
  • Definition 2.5: $(k'/k)$-jumps
  • Definition 2.6: $k$-jumps
  • Lemma 2.10
  • ...and 32 more