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

Journal article

In the early 90’s, Perrin-Riou [PR] introduced an important refinement of the Mazur-Swinnerton-Dyer p-adic L-function of an elliptic curve E over 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 %g and %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 %g ⊗ %h, in the style of the regulators that arise in [DLR1], and recovers Perrin-Riou’s original conjecture when g and h are Eisenstein series.

Authors: Darmon, H; Lauder, AG

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Annales Mathématiques du Québec
• Volume: 47
• Pages: 31–48
• Publication date: 2021-02-15
• Acceptance date: 2021-01-21
• DOI: 10.1007/s40316-021-00158-6
• ISSN: 2195-4755
• Language: English
• Keywords: Birch and Swinnerton-Dyer conjecture; Perrin-Riou conjectures; p-adic L-functions; FFR; elliptic curves