Proof of an exceptional zero conjecture for elliptic curves over function fields

Abstract

Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refinement of the Birch-Swinnerton-Dyer conjecture if the analytic rank of the elliptic curve is zero. Mathematics Subject Classification (2000)11G18(primary) · 11G40 (secondary) Historical introduction and announcement of results (Historical background)In the more recent history of number theory one of the major themes is to formulate and prove so-called p-adic analogues of class number formulas. The first result of this sort was perhaps Leopoldt's class number formula. In general these theorems and conjectures relate p-adic L-functions, whose characteristic property is that they interpolate suitable normalized special values of classical complex L-functions to p-adic analytic invariants of global number theoretical objects, as the p-adic logarithm of global units in number fields or p-adic heights of global points in elliptic curves defined over number fields.The strong analogy between number fields and function fields of one variable over finite fields has been known already to mathematicians in the classical period of the 19th century. This analogy suggests that the aforementioned results and conjectures should have appropriate analogues over function fields as well, although their precise form is not at all clear if someone looks at classical results such as Leopoldt's class number formula. It was first B. Gross who made progress on this A. Pál

Selmer groups for elliptic curves in \mathbb{Z}_l^d-extensions of function fields of characteristic p

“…As for the analytic side, the best candidate we know of has been provided by Pál. In [16] We also remark that Ochiai and Trihan [15] are able to prove that their Selmer dual is always torsion (a necessary condition to have a non-zero characteristic ideal). So one expects the analog to be true for our Sel E (F) ∨ l as well.…”