Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case

摘要

Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensionsof imaginary quadratic fields has been studied by several authors, startingfrom the works of Bertolini-Darmon and Iovita-Spiess, under the crucialassumption that the prime $p$ is unramified in $K$. We start in this articlethe systematic study of anticyclotomic $p$-adic $L$-functions when $p$ isramified in $K$. In particular, when $f$ is a weight $2$ modular form attachedto an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and$p$ is ramified in $K$, we show an analogue of the exceptional zeroesphenomenon investigated by Bertolini-Darmon in the setting when $p$ is inert in$K$. More precisely, we consider situations in which the $p$-adic $L$-function$L_p(E/K)$ of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ doesnot vanish identically but, by sign reasons, has a zero at certain characters$\chi$ of the Hilbert class field of $K$. In this case we show that the valueat $\chi$ of the first derivative of $L_p(E/K)$ is equal to the formal grouplogarithm of the specialization at $p$ of a global point on the elliptic curve(actually, this global point is a twisted sum of Heegner points). Thisgeneralizes similar results of Bertolini-Darmon, available when $p$ is inert in$K$ and $\chi$ is the trivial character.