Stark points on elliptic curves via Perrin-Riou’s philosophy

Abstract

In the early 90’s, Perrin-Riou (Ann Inst Fourier 43(4):945–995, 1993) introduced an important refinement of the Mazur–Swinnerton-Dyer p-adic L-function of an elliptic curve E over $$\mathbb {Q}$$ , taking values in its p-adic de Rham cohomology. She then formulated a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for this p-adic L-function, in which the formal group logarithms of global points on E make an intriguing appearance. The present work extends Perrin-Riou’s construction to the setting of a Garret–Rankin triple product (f, g, h), where f is a cusp form of weight two attached to E and g and h are classical weight one cusp forms with inverse nebentype characters, corresponding to odd two-dimensional Artin representations $$\varrho _g$$ and $$\varrho _h$$ respectively. The resulting p-adic Birch and Swinnerton-Dyer conjecture involves the p-adic logarithms of global points on E defined over the field cut out by $$\varrho _g\otimes \varrho _h$$ , in the style of the regulators that arise in Darmon et al. (Forum Math 3(e8):95, 2015), and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series.