Une minoration du minimum essentiel sur les varietes abeliennes

TL;DR

This paper generalizes a lower bound for the essential minimum on abelian varieties, relying on a conjecture about primes, and introduces a new approach to simplify the proof process.

Contribution

It extends previous bounds to higher codimension on abelian varieties and simplifies the proof by modifying the transcendence strategy.

Findings

Established a new lower bound for the essential minimum in general codimension

Simplified the zero lemma and combinatorics in the proof

Developed a descent procedure adapted to abelian varieties

Abstract

We extend to the general codimension a lower bound for the essential minimum on abelian varieties found in a previous work, under a conjecture about ordinary primes in abelian varieties. This lower bound is the best expected, ``up to an epsilon'', in the degree of the subvariety. Following a strategy of Amoroso, we change the transcendance phasis in order to simplify the zero lemma and its combinatorics. The last argument is a descent procedure on varieties, which is far more intricate in the abelian setting since there is not, in general, a lifting of the Frobenius in characteristic zero.