Mathematical Research Letters

Volume 22 (2015)

Number 5

Asymptotics of the Néron height pairing

Pages: 1337 – 1371

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n5.a5

Authors

David Holmes (Mathematical Institute, University of Leiden, The Netherlands)

Robin de Jong (Mathematical Institute, University of Leiden, The Netherlands)

Abstract

The aim of this paper is twofold. First, we study the asymptotics of the Néron height pairing between degree-zero divisors on a family of degenerating compact Riemann surfaces parametrized by an algebraic curve. We show that if the monodromy is unipotent the leading term of the asymptotic formula is controlled by the local non-archimedean Néron height pairing on the generic fiber of the family. Second, we prove a conjecture of R. Hain to the effect that the ‘height jumping divisor’ related to the normal function ${(2g - 2) x} - K$ on the moduli space $\mathcal{M}_{g,1}$ of $1$-pointed curves of genus $g \geq 2$ is effective. Both results follow from a study of the degeneration of the canonical metric on the Poincaré bundle on a family of principally polarized abelian varieties.

Keywords

Green’s function, height jumping, Lear extension, Néron pairing, normal function, reduction graph

2010 Mathematics Subject Classification

Primary 14G40. Secondary 14D06, 14D07, 14H15.

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