Proof of an Exceptional Zero Conjecture for Elliptic Curves over Function Fields

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. 1. Historical introduction and announcement of results Historical background 1.1. 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 problem in [8] where he restated his original p-adic analogue of Stark’s conjecture in a form which makes good sense both over number fields and function fields. This conjecture was proved by D. Hayes for function fields in [10]. More recently the author formulated and proved an analogue of Beilinson’s theorem on the K2 of elliptic modular curves very similar to Hayes’ theorem in [16]. This result relates the value of a certain regulator, which takes values in the multiplicative group of a completion of the global function field in question, of some elements in the K2 of an elliptic curve defined over the function field to a Gross-type p-adic L-function attached to the elliptic curve. The proofs of both of these results show a striking similarity with their classical complex counterparts. In particular the rigid analytical regulator itself appearing in the second theorem is defined via non-archimedean integration. 2000 Mathematics Subject Classification. 11G18 (primary), 11G40 (secondary). Typeset by AMS-TEX 1