Department of Mathematics2024-11-0920161073-792810.1093/imrn/rnv2052-s2.0-84966355968http://dx.doi.org/10.1093/imrn/rnv205https://hdl.handle.net/20.500.14288/7358Let 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.MathematicsJournal Articlehttps://www.scopus.com/inward/record.uri?eid=2-s2.0-84966355968&doi=10.1093%2fimrn%2frnv205&partnerID=40&md5=d88d792b12c3a9c3ec6a19c3eb95c2751182