Pesenti–Szpiro inequality for optimal elliptic curves

Abstract

We study Pesenti–Szpiro inequality in the case of elliptic curves over F q ( t ) which occur as subvarieties of Jacobian varieties of Drinfeld modular curves. In general, we obtain an upper-bound on the degrees of minimal discriminants of such elliptic curves in terms of the degrees of their conductors and q. In the special case when the level is prime, we bound the degrees of discriminants only in terms of the degrees of conductors. As a preliminary step in the proof of this latter result we generalize a construction (due to Gekeler and Reversat) of 1-dimensional optimal quotients of Drinfeld Jacobians.