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 upperbound 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.

Analogue of the degree conjecture over function fields

“…D] (modulo the assumptions we made). Using the argument in the proof of [17,Thm.6.4], one can show that deg ℘ is bounded from below by α · N (n E )/(deg n) 2 , where α is a constant depending only on q, g and deg ∞. Thus, log(deg ℘) deg n E and the bound in the theorem is essentially the best possible.…”

“…We only indicate the main steps which go into the proof of this theorem and refer to [17,Thm. A.4] for the details.…”

Untitled

“…Although this hypothesis is frequently made (sometimes implicitly) in the literature (see for example [Papikian 2005;Rück and Tipp 2000]), it is actually false. One of the aims of this paper is to exhibit many cases when m(E) is not zero.…”

Strong Weil curves over Fq(T) with small conductor

“…More precisely, they are obtained from finitely many curves by repeated application of the Frobenius isogeny. Papikian [Pa1,Pa2] has shown that in certain situations the strong Weil curve is not the Frobenius of another curve over F q (T ), but from examples it is known that this is not a general phenomenon.…”