New explicit formulas for Faltings’ delta-invariant

Abstract

Abstract. In this paper we give new explicit formulas for Faltings' δ-invariant in terms of integrals of theta functions, and we deduce an explicit lower bound for δ only in terms of the genus and an explicit upper bound for the Arakelov-Green function in terms of δ. Furthermore, we give a canonical extension of δ and the Zhang-Kawazumi invariant ϕ to the moduli space of indecomposable principally polarised complex abelian varieties.

Néron-Tate heights of cycles on jacobians

“…Then the identityδ F (C) − 4g log(2π) = −12 κ 0 g + 24 I(J, λ) + 2 ϕ(C) holds.Proof. This is the main result of[46],but slightly rewritten. Let θ be the Riemann theta-function of (J, λ), seen as a global section of a suitable symmetric ample line bundle L representing λ. Endow L with its standard admissible metric giving θ the norm θ defined on [23, p. 401].…”

“…By our choice of α ∈ Div 1 X we can write (2g − 2)α =ω +μ whereμ is a suitable admissible line bundle on Spec k. Then, applying Theorem 6.1 with D = (g − 1)α we find As it turns out, in [46] precisely the right amount of combinatorics is carried out in order to determine the universal constant b ′ . More precisely, combining [46, Specializing to a curve of genus g over some number field k such that the associated jacobian has everywhere good reduction (such curves exist by [36, Théorème 0.7, Exemple 0.9]), and using that ω,ω is always positive, we deduce from Theorem 5.3 that a = 1 12 .…”

“…Apart from [52], our proof of Theorem 1.1 is heavily inspired by a method developed recently by R. Wilms in the context of the moduli space of curves [46]. In particular, our proof provides an algorithm to compute a, b and c from m and g. Upon hearing about our proof, D. Holmes implemented the algorithm in SAGE, and kindly made it publicly available via https://arxiv.org/abs/1610.01932.…”

“…moreover we have I(J, λ) > 0 by Jensen's inequality. A fundamental identity relating ϕ(C) with the Faltings delta-invariant δ F (C) from[23, p. 402] and the invariant I(J, λ) has recently been found by R. Wilms in his paper[46].Theorem 5.1. (R. Wilms) Let δ F (C) be the Faltings delta-invariant of the compact connected Riemann surface C. Let κ 0 = log(π √ 2).…”

On CM points away from the Torelli locus

“…The proof of this theorem is immediate using the following Noether formula for arithmetic surfaces together with some recent improvement by Wilms, explained to us by Prof. Ariyan Javanpeykar. The improvement from Wilms [38] provides a link between the δ-invariant of Faltings and the ϕ-invariant of Kawazumi-Zhang, cf. [40].…”