Richard Griffon

Isogenies of elliptic curves over function fields

I will report on joint work with Fabien Pazuki, in which we study elliptic curves over function fields and the isogenies between them. Specifically, we prove analogues in the function field setting of two famous theorems about isogenous elliptic curves over number fields. The function field versions of these theorems, though having a similar flavour to their number field counterparts, display some striking differences.
The first of these results completely describes the variation of the Weil height of the j-invariant of elliptic curves within an isogeny class. In particular, we show that the modular height remains constant under an isogeny of degree prime to the characteristic.
Our second main theorem is an “isogeny estimate” in the spirit of theorems by Masser–Wüstholz and by Gaudron–Rémond. Unavoidable inseparability issues aside, we prove a uniform isogeny bound in this setting.
After stating our results and giving quick sketches of their proof, I will, time permitting, mention a few Diophantine applications.