N\'eron models and the height jump divisor

TL;DR

This paper introduces an algebraic analogue of the height jump divisor for Jacobians of curves, providing explicit formulas, proving effectivity conjectures, and offering new proofs of classical theorems on height variation in abelian varieties.

Contribution

It defines an algebraic analogue of the height jump divisor, derives explicit combinatorial formulas, and proves a conjecture on its effectivity, advancing understanding of height variations in families of abelian varieties.

Findings

Proved the effectivity of the height jump divisor.

Derived explicit combinatorial formulas for height jump.

Provided a new proof of the variation of heights in families of abelian varieties.

Abstract

We define an algebraic analogue, in the case of jacobians of curves, of the height jump divisor introduced recently by R. Hain. We give explicit combinatorial formulae for the height jump for families of semistable curves using labelled reduction graphs. With these techniques we prove a conjecture of Hain on the effectivity of the height jump, and also give a new proof of a theorem of Tate, Silverman and Green on the variation of heights in families of abelian varieties.