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

Abstract

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