N\'eron-Tate heights of cycles on jacobians

TL;DR

This paper presents a method to compute Néron-Tate heights of cycles on Jacobians, providing explicit formulas and applications to bounds and height relations in arithmetic geometry.

Contribution

It introduces a novel approach for calculating Néron-Tate heights of cycles on Jacobians, including explicit formulas for key cycles and applications to height bounds and conjectures.

Findings

Explicit formulas for Néron-Tate heights of specific cycles

A new lower bound for the essential minimum of Abel-Jacobi images

Proof of a height relation formula proposed by Autissier

Abstract

We develop a method to calculate the N\'eron-Tate height of tautological integral cycles on jacobians of curves defined over number fields. As examples we obtain closed expressions for the N\'eron-Tate height of the difference surface, the Abel-Jacobi images of the square of the curve, and of any symmetric theta divisor. As applications we obtain a new effective positive lower bound for the essential minimum of any Abel-Jacobi image of the curve and a proof, in the case of jacobians, of a formula proposed by Autissier relating the Faltings height of a principally polarized abelian variety with the N\'eron-Tate height of a symmetric theta divisor.