On Nekovář's Heights, Exceptional Zeros and a Conjecture of Mazur–Tate–Teitelbaum

Let E/\ Q be an elliptic curve which has split multiplicative reduction at a prime p and whose analytic rank r\ ₀₍ (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 Nekova´r's height pairing evaluated on natural elements arising from the Beilinson–Kato elements. Along the way, we extend a Rubin-style formula of Nekova´r 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 r\ ₀₍ (E) assuming the non-triviality of the height pairing. This has consequences toward a conjecture of Mazur, Tate, and Teitelbaum.