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

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