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p-adic adelic metrics and Quadratic Chabauty I

Amnon Besser, J. Steffen Müller, Padmavathi Srinivasan

TL;DR

This work develops a comprehensive framework for $p$-adic heights via $p$-adic Arakelov theory, establishing canonical adelic metrics on line bundles over number fields and canonical $p$-adic heights on abelian varieties. By relating these heights to Mazur–Tate and Coleman–Gross pairings, the authors unify real and $p$-adic height theories and provide a new, self-contained approach to Quadratic Chabauty that works even at primes of bad reduction. The main technical engine combines Vologodsky integration, higher $ar{ abla}$-operators, and curvature-controlled log functions to produce locally analytic height functions whose pull-backs yield quadratic forms on Jacobians. This allows a finite, computable set of $p$-adic conditions to bound rational points on curves, connecting with existing BD18 methods while providing a pure $p$-adic, non-fundamental-group-based perspective. The paper further clarifies links to Colmez, Nekovář, and Colmez-type Green functions, and offers a unified, purely $p$-adic theory of adelic metrics with potential computational advantages, including extensions to primes of bad reduction.

Abstract

We give a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of $p$-adic adelic metrics on line bundles. In particular, we describe a construction of canonical $p$-adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using $p$-adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes $p$ of bad reduction. One consequence of our work is that for any canonical height ($p$-adic or $\mathbb{R}$-valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime $q$ can be constructed using $q$-analytic methods.

p-adic adelic metrics and Quadratic Chabauty I

TL;DR

This work develops a comprehensive framework for -adic heights via -adic Arakelov theory, establishing canonical adelic metrics on line bundles over number fields and canonical -adic heights on abelian varieties. By relating these heights to Mazur–Tate and Coleman–Gross pairings, the authors unify real and -adic height theories and provide a new, self-contained approach to Quadratic Chabauty that works even at primes of bad reduction. The main technical engine combines Vologodsky integration, higher -operators, and curvature-controlled log functions to produce locally analytic height functions whose pull-backs yield quadratic forms on Jacobians. This allows a finite, computable set of -adic conditions to bound rational points on curves, connecting with existing BD18 methods while providing a pure -adic, non-fundamental-group-based perspective. The paper further clarifies links to Colmez, Nekovář, and Colmez-type Green functions, and offers a unified, purely -adic theory of adelic metrics with potential computational advantages, including extensions to primes of bad reduction.

Abstract

We give a new construction of -adic heights on varieties over number fields using -adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of -adic adelic metrics on line bundles. In particular, we describe a construction of canonical -adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using -adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes of bad reduction. One consequence of our work is that for any canonical height (-adic or -valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime can be constructed using -analytic methods.
Paper Structure (52 sections, 88 theorems, 113 equations)

This paper contains 52 sections, 88 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Notation
  4. Valuations and canonical local heights away from $p$
  5. $p$-adic Arakelov theory
  6. Vologodsky functions
  7. Log functions for line bundles with curvature forms
  8. The case of curves
  9. Iterated integrals as Vologodsky functions
  10. Single and double integrals on curves
  11. Log functions for curves
  12. Higher $\bar{\partial}$-operators on Vologodsky functions
  13. $k$-iterated Vologodsky functions and the operator $D_k$
  14. $k$-iterated Vologodsky integrals
  15. Canonical log functions on line bundles on abelian varieties
  16. ...and 37 more sections

Key Result

Theorem 1.1

Suppose that $\operatorname{rk} J(\mathbb{Q})=g$ and that the closure of $J(\mathbb{Q})$ has finite index in $J(\mathbb{Q}_p)$. Then the function is locally analytic and takes values on $C(\mathbb{Q})$ in $T$. Moreover, $T$ is finite and for every $t\in T$, there are only finitely many $x\in C(\mathbb{Q}_p)$ with $F(x)=t$.

Theorems & Definitions (176)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 166 more