Exceptional zero formulae and a conjecture of Perrin-Riou

Abstract

Let $$A/\mathbf{Q}$$ 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 $$A(\mathbf{Q})$$ , 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 weight-two newform associated with A, let $$f_{\infty }$$ be the Hida family of f, and let $$L_{p}(f_{\infty },k,s)$$ be the Mazur–Kitagawa two-variable p-adic L-function attached to $$f_{\infty }$$ . We prove a p-adic Gross–Zagier formula, expressing the quadratic term of the Taylor expansion of $$L_{p}(f_{\infty },k,s)$$ at $$(k,s)=(2,1)$$ as a non-zero rational multiple of the extended height-weight of a Heegner point in $$A(\mathbf{Q})$$ .