On Nekovar's heights, exceptional zeros and a conjecture of Mazur-Tate-Teitelbaum

Abstract

Let E/ℚ be an elliptic curve which has split multiplicative reduction at a prime p and whose analytic rank ran(E) equals one. The main goal of this article is to relate the second-order derivative of the Mazur-Tate-Teitelbaum p-adic L-function Lp(E,s) of E to Nekovář's height pairing evaluated on natural elements arising from the Beilinson-Kato elements. Along the way, we extend a Rubin-style formula of Nekovář to apply in the presence of exceptional zeros. Our height formula allows us, among other things, to compare the order of vanishing of Lp(E,s) at s=1 with its (complex) analytic rank ran(E) assuming the non-triviality of the height pairing. This has consequences toward a conjecture of Mazur, Tate, and Teitelbaum.