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Bounds on the number of torsion points of given order on curves embedded in their jacobians

John Boxall

TL;DR

The paper develops a framework to bound and compute the torsion points of order dividing $N$ lying on a smooth curve $X$ embedded in its Jacobian $J$, using a wronskian determinant $\Delta_N$ built from Hasse–Schmidt derivatives applied to a basis of functions with prescribed poles at a base point. A central construction is the $\mathsf{D}_N\times N$ matrix $\mathcal{M}_N$ whose rank controls the presence of torsion points outside ramification, with $\Delta_N$ encoding when the pushforward $(N-g+1)_*X$ lands in the negative theta-translate $W^-_{g-1}$. The authors derive lower bounds for ramification valuations, establish zero-order bounds for minors of $\mathcal{M}_N$, and obtain explicit cardinality bounds for $X[N]$, especially for $N\ge 2g-1$, including sharp results in hyperelliptic and tame-ramification cases, along with a range of illustrative examples. Their method yields a practical, computable criteria for $\Delta_N\neq0$ and hence effective bounds on torsion, with linear-in-$g$ behavior in many regimes and connections to the Manin–Mumford context in characteristic zero. The work both generalizes and complements prior results (notably Bo23 and Pareschi–Parreau) by providing a unified approach via function-field wronskians and Hasse–Schmidt calculus that applies across characteristics and offers explicit computational pathways.

Abstract

Working over an algebraically closed field of arbitrary characteristic we study, for integers $N\geq 2$ and $g\geq 2$, the set of points of order dividing $N$ lying on an irreducible smooth proper curve of genus $g$ embedded in its jacobian using a fixed base point. We discuss bounds on its cardinality and describe an efficient method for computing the set. Our method uses wronskians similar to those used in the study of Weierstrass points and the strength of our bounds is related to whether or not a certain multiple of the curve is contained in the negative of the theta divisor. Several examples are discussed. This generalizes our previous work [https://doi.org/10.1216/rmj.2023.53.357] dealing with the case of hyperelliptic curves embedded in their jacobian using a Weierstrass point as base point.

Bounds on the number of torsion points of given order on curves embedded in their jacobians

TL;DR

The paper develops a framework to bound and compute the torsion points of order dividing lying on a smooth curve embedded in its Jacobian , using a wronskian determinant built from Hasse–Schmidt derivatives applied to a basis of functions with prescribed poles at a base point. A central construction is the matrix whose rank controls the presence of torsion points outside ramification, with encoding when the pushforward lands in the negative theta-translate . The authors derive lower bounds for ramification valuations, establish zero-order bounds for minors of , and obtain explicit cardinality bounds for , especially for , including sharp results in hyperelliptic and tame-ramification cases, along with a range of illustrative examples. Their method yields a practical, computable criteria for and hence effective bounds on torsion, with linear-in- behavior in many regimes and connections to the Manin–Mumford context in characteristic zero. The work both generalizes and complements prior results (notably Bo23 and Pareschi–Parreau) by providing a unified approach via function-field wronskians and Hasse–Schmidt calculus that applies across characteristics and offers explicit computational pathways.

Abstract

Working over an algebraically closed field of arbitrary characteristic we study, for integers and , the set of points of order dividing lying on an irreducible smooth proper curve of genus embedded in its jacobian using a fixed base point. We discuss bounds on its cardinality and describe an efficient method for computing the set. Our method uses wronskians similar to those used in the study of Weierstrass points and the strength of our bounds is related to whether or not a certain multiple of the curve is contained in the negative of the theta divisor. Several examples are discussed. This generalizes our previous work [https://doi.org/10.1216/rmj.2023.53.357] dealing with the case of hyperelliptic curves embedded in their jacobian using a Weierstrass point as base point.
Paper Structure (20 sections, 24 theorems, 82 equations)

This paper contains 20 sections, 24 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Lower bounds on valuations at ramified points
  3. The matrix $\mathcal{M}_N$
  4. Bounds on zero divisor degrees
  5. Orders of zeros at torsion points
  6. Non-vanishing of $\Delta_N$
  7. Applications and examples
  8. Torsion of small order
  9. Simplifying the bound
  10. Proof of Theorem \ref{['theo:intromain']}
  11. The worst case
  12. The case of tame ramification
  13. Hyperelliptic curves with base point a Weierstrass point
  14. The case $p\neq 2$
  15. The case $p=2$
  16. ...and 5 more sections

Key Result

Proposition 1.1

We have $\mathop{\mathrm{\sharp}}\nolimits{X[2]}\leq 2g+2$ and, if $N\geq 3$, then except when $p$ is purely inseparable for $X$ and $N-1$ is a power of $p$, in which case unless $(p,N)=(2,3)$.

Theorems & Definitions (48)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 38 more