Minoration de la hauteur canonique pour les modules de Drinfeld \`a multiplications complexes

TL;DR

This paper establishes an effective lower bound for the canonical height of Drinfeld modules with complex multiplication over certain Galois extensions, extending analogous results from number fields to function fields.

Contribution

It provides the first explicit lower bound for the canonical height of Drinfeld modules with complex multiplication in the function field setting.

Findings

Effective lower bound for canonical height established

Results extend number field analogs to function fields

Applicable to Galois extensions with bounded local degrees

Abstract

Lower Bound for the Canonical Height for Drinfeld Modules with Complex Multiplication. Let K be a fi nite extension of Fq(T), let L=K be a Galois extension with Galois group G and let E be the sub eld of L fixed by the center of G. Assume that there exists a finite place v of K such that the local degrees of E=K above v are bounded. Let $\phi$ be a Drinfeld module with complex multiplication. We give an e fective lower bound for the canonical height of $\phi$ on L outside the torsion points of $\phi$ . In the number field case, this problem was solved by F. Amoroso, S. David and U. Zannier.