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

Iwasawa theory of modular forms over anticyclotomic Zp-extensions of imaginary quadratic fields K has been studied by several authors, starting from the works of Bertolini–Darmon and Iovita–Spiess, under the crucial assumption that the prime p is unramified in K. We start in this article the systematic study of anticyclotomic p-adic L-functions when p is ramified in K. In particular, when f is a weight 2 modular form attached to an elliptic curve E/Q having multiplicative reduction at p, and p is ramified in K, we show an analogue of the exceptional zeroes phenomenon 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 Lp(E/K) of E over the anticyclotomic Zp-extension of K does not vanish identically but, by sign reasons, has a zero at certain characters χ of the Hilbert class field of K. In this case we show that the value at χ of the first derivative of Lp(E/K) is equal to the formal group logarithm of the specialization at p of a global point on the elliptic curve (actually, this global point is a twisted sum of Heegner points). This generalizes similar results of Bertolini–Darmon, available when p is inert in K and χ is the trivial character.