A formula of Perrin-Riou and characteristic power series of signed Selmer groups

Abstract

We prove a conjecture of Kundu--Ray, following from the $p$-adic Birch--Swinnerton-Dyer conjecture for supersingular primes by Bernardi--Perrin-Riou and Kato's Main Conjecture, predicting an expression for the leading term (up to a $p$-adic unit) of a characteristic power series of Kobayashi's signed Selmer groups attached to elliptic curves $E/\mathbb{Q}$ with supersingular reduction at a prime $p>2$ with $a_p=0$. The proof is deduced from a similar formula due to Perrin-Riou for a generator of her module of arithmetic $p$-adic $L$-functions with values in the Dieudonn\'{e} module of $E$.