Plectic points and Hida-Rankin p-adic L-functions

Abstract

Plectic points were introduced by Fornea and Gehrmann as certain tensor products of local pointson elliptic curves over arbitrary number fields $F$. In rank $r\leq [F:\mathbb{Q}]$-situations, they conjecturally come from p-adic regulators of basis of the Mordell-Weil group defined over dihedral extensions of $F$. In this article we define two variable anticyclotomic $p$-adic L-functions attached to a family of overconvergent modular symbols defined over $F$ and a quadratic extension $K/F$. Their restriction to the weight space provide Hida-Rankin $p$-adic L-functions. If such a family passes through an overconvergent modular symbol attached to a modular elliptic curve $E/F$, we obtain a $p$-adic Gross-Zagier formula that computes higher derivatives of such Hida-Rankin $p$-adic L-functions in terms of plectic points. This result generalizes that of Bertolini and Darmon, which has been key to demonstrating the rationality of Darmon points.