Conjectures uniformes sur les variétés abéliennes
Résumé
We consider several conjectures on abelian varieties over number fields whose common feature is a bound that depends only on the dimension of the variety and the degree of the number field. For example, Coleman’s conjecture predicts that only a finite number of rings can occur as endomorphism rings once these two parameters are fixed. We show that this conjecture implies the existence of a small polarization as well as a uniform isogeny conjecture (without Faltings height) which in turn implies the uniform torsion conjecture. We then discuss several variants of the Lang–Silverman conjecture on heights and implications between them. In particular, we show how a rather weak version is, under Coleman’s conjecture, equivalent to much more precise versions. We build on Bertrand’s work to give a bound explicit in terms of the polarization.