Exceptional zero formulae and a conjecture of Perrin-Riou

Let A/Q be an elliptic curve with split multiplicative reduction at a prime p.We prove (an analogue of) a conjecture of Perrin-Riou, relating p-adic Beilinson–Kato elements to Heegner points in A(Q), and a large part of the rank-one case of the Mazur–Tate–Teitelbaum exceptional zero conjecture for the cyclotomic p-adic L-function of A. More generally, let f be the weighttwo newform associated with A, let f∞ be the Hida family of f , and let L p( f∞, k, s) be theMazur–Kitagawa two-variable p-adic L-function attached to f∞.We prove a p-adicGross–Zagier formula, expressing the quadratic term of theTaylor expansion of L p( f∞, k, s) at (k, s) = (2, 1) as a non-zero rational multiple of the extended height-weight of a Heegner point in A(Q).