Hida families and rational points on elliptic curves

Let E be an elliptic curve over the rationals, and let p be a split multiplicative prime for E. Assume that E has at least two primes of multiplicative reduction. This article studies the restriction of the Mazur-Kitagawa two-variable p-adic L-function attached to E to the central critical line, when the sign of the p-adic functional equation is +1. In this case, it relates the second derivative of the above mentioned p-adic L-function at the point (2,1) to the square of the formal group logarithm of a rational point on E. This is an instance of the p-adic Birch and Swinnerton-Dyer conjecture for E, and represents a counterpart of the Greenberg-Stevens exceptional zero formula in the rank one setting.