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Coleman-Gross Heights and $p$-adic Néron Functions on Jacobians of Genus $2$ Curves

Francesca Bianchi, Enis Kaya, J. Steffen Müller

TL;DR

The paper develops a framework for $p$-adic Néron functions on abelian varieties and proves that global $p$-adic heights, including Mazur--Tate heights, decompose into sums of local $p$-adic Néron functions when consistent auxiliary data are used. Focusing on the Jacobian of genus $2$ curves, it establishes a direct local comparison between Coleman--Gross and Mazur--Tate heights, extending Balakrishnan–Besser’s elliptic results to genus $2$ and treating both unramified and ramified primes. In the ramified case, the authors relate $p$-adic Néron functions to Colmez Green functions and $v$-adic theta/sigma data, giving explicit descriptions via Norman’s theta function and Blakestad’s zeta functions. They also prove analyticity results for the canonical Mazur--Tate splitting in semistable ordinary genus $2$ and provide explicit descriptions of the canonical subspace $W_v$, linking to unit-root theory in good reduction and offering computational pathways for the canonical height in practice. Altogether, the work unifies several $p$-adic height frameworks in genus $2$ and furnishes concrete tools for quadratic Chabauty-type computations and explicit height decompositions.

Abstract

We develop a theory of $p$-adic Néron functions on abelian varieties, depending on various auxiliary choices, and show that the global $p$-adic height functions constructed by Mazur and Tate can be decomposed into a sum of $p$-adic Néron functions if the same auxiliary choices are made. We also consider a decomposition of the $p$-adic height constructed by Coleman and Gross for good reduction, and extended to arbitrary reduction by Colmez and Besser, into a sum of certain local height functions for Jacobians of odd degree genus~$2$ curves. We show that this local height function is equal to the $p$-adic Néron function with the same auxiliary choices, regardless of the reduction type of the curve. This extends work of Balakrishnan and Besser for elliptic curves. When the curve has semistable reduction and the reduction of the Jacobian is ordinary, we also describe the $p$-adic Néron function that arises from the canonical Mazur--Tate splitting explicitly in terms of a generalisation of the $p$-adic sigma function constructed by Blakestad.

Coleman-Gross Heights and $p$-adic Néron Functions on Jacobians of Genus $2$ Curves

TL;DR

The paper develops a framework for -adic Néron functions on abelian varieties and proves that global -adic heights, including Mazur--Tate heights, decompose into sums of local -adic Néron functions when consistent auxiliary data are used. Focusing on the Jacobian of genus curves, it establishes a direct local comparison between Coleman--Gross and Mazur--Tate heights, extending Balakrishnan–Besser’s elliptic results to genus and treating both unramified and ramified primes. In the ramified case, the authors relate -adic Néron functions to Colmez Green functions and -adic theta/sigma data, giving explicit descriptions via Norman’s theta function and Blakestad’s zeta functions. They also prove analyticity results for the canonical Mazur--Tate splitting in semistable ordinary genus and provide explicit descriptions of the canonical subspace , linking to unit-root theory in good reduction and offering computational pathways for the canonical height in practice. Altogether, the work unifies several -adic height frameworks in genus and furnishes concrete tools for quadratic Chabauty-type computations and explicit height decompositions.

Abstract

We develop a theory of -adic Néron functions on abelian varieties, depending on various auxiliary choices, and show that the global -adic height functions constructed by Mazur and Tate can be decomposed into a sum of -adic Néron functions if the same auxiliary choices are made. We also consider a decomposition of the -adic height constructed by Coleman and Gross for good reduction, and extended to arbitrary reduction by Colmez and Besser, into a sum of certain local height functions for Jacobians of odd degree genus~ curves. We show that this local height function is equal to the -adic Néron function with the same auxiliary choices, regardless of the reduction type of the curve. This extends work of Balakrishnan and Besser for elliptic curves. When the curve has semistable reduction and the reduction of the Jacobian is ordinary, we also describe the -adic Néron function that arises from the canonical Mazur--Tate splitting explicitly in terms of a generalisation of the -adic sigma function constructed by Blakestad.
Paper Structure (43 sections, 54 theorems, 175 equations)

This paper contains 43 sections, 54 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Elliptic curves
  3. Previous comparison results
  4. Notation
  5. Vologodsky functions and integrals
  6. Vologodsky functions
  7. Vologodsky integrals
  8. The $\bar{\partial}$-operator
  9. Coleman--Gross heights
  10. Local Coleman--Gross height pairings for disjoint divisors
  11. Local Coleman--Gross height pairings for disjoint divisors and $\chi_v$ unramified
  12. Local Coleman--Gross height pairings for disjoint divisors and $\chi_v$ ramified
  13. Properties of local Coleman--Gross height pairings for disjoint support
  14. Local Coleman--Gross height pairings for arbitrary divisors
  15. Local Coleman--Gross height pairings for arbitrary zero-cycles and $\chi_v$ unramified
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $J/F$ be the Jacobian of an odd degree genus $2$ curve $C/F$. Fix a continuous idèle class character $\chi\colon \mathbb{A}_F^\times/F^\times \to \mathbb{Q}_p$ and, for each $v$ such that $\chi_v$ is ramified, a complementary subspace $W_v$ that is isotropic with respect to the cup product pairi

Theorems & Definitions (137)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 127 more