Abstract

We prove a formula, which, given a principally polarized abelian variety $(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of $A$ with the Néron–Tate height of a symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $(A,\lambda )$.

Highlights

  • Let (A, λ) be a principally polarized abelian variety of positive dimension over the field of algebraic numbers Q

  • The Neron–Tate height of higher-dimensional cycles was first constructed by Philippon [Phi91] and soon afterwards reobtained using different methods by, among others, Gubler [Gub94], Bost et al [Bos96b, BGS94] and Zhang [Zha95]

  • The Neron–Tate height hL(Θ) is non-negative and is an invariant of the pair (A, λ). Another natural invariant of (A, λ) is the stable Faltings height hF (A) of A introduced by Faltings in [Fal83] as a key tool in his proof of the Mordell conjecture

Read more

Summary

Introduction

Let (A, λ) be a principally polarized abelian variety of positive dimension over the field of algebraic numbers Q. The Neron–Tate height hL(Θ) is non-negative and is an invariant of the pair (A, λ). In [Aut, Question], it is asked whether an extension of (1.3) might hold for arbitrary principally polarized abelian varieties over Qof the following shape. For each v ∈ M (k)0, there should exist a natural local invariant αv ∈ Q 0 of (A, λ) at v such that the equality αv log N v +. In [dJ18, Theorem 1.6], the first-named author exhibited natural αv ∈ Q 0, and established (1.4), for all Jacobians and for arbitrary products of these In both [Aut, dJ18], the local non-archimedean invariants αv are expressed in terms of the combinatorics of the dual graph of the underlying semistable curve at v

Main result
Tropical moments
Lower bounds for the stable Faltings height
Elliptic curves
Jacobians
The function field case
Semistable group schemes
Berkovich analytification
Metrics and Green’s functions
Canonical metrics
Neron functions
Raynaud extensions
Non-archimedean uniformization of abelian varieties
Non-archimedean theta functions
Tropical Riemann theta function
Translations of line bundles
Canonical metrics and theta functions
10. Semistable models and canonical metric
10.2 Direct images
11. Proof of Theorem B
12.1 Faltings metric and L2-metric
12.2 Canonical metrics
13. Stable Faltings height and key formula
13.1 Arakelov degree
13.2 Stable Faltings height
14. Neron–Tate heights
14.3 Adelic intersections
14.6 Connection with the Neron model
15. Proof of Theorem A
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call