Stark points on elliptic curves via Perrin-Riou's philosophy
| Autor: | Henri Darmon, Alan G. B. Lauder |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Cusp (singularity)
Conjecture Mathematics::Number Theory General Mathematics 010102 general mathematics Formal group 01 natural sciences Cusp form Combinatorics Elliptic curve symbols.namesake Number theory 0103 physical sciences Eisenstein series De Rham cohomology symbols 010307 mathematical physics 0101 mathematics Mathematics |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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