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
Summary
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
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