A numerical approach toward the \(p\)-adic Beilinson conjecture for elliptic curves over \(\mathbb{Q}\). (English) Zbl 07684029
Summary: Restricting ourselves to elliptic curves over \(\mathbb{Q}\), we reformulate the \(p\)-adic Beilinson conjecture due to Perrin-Riou, which is customized to our computational approach. We then develop a new algorithm for numerical verifications of the \(p\)-adic Beilinson conjecture, which is based on the theory of rigid cohomology and \(F\)-isocrystals.
MSC:
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11S40 | Zeta functions and \(L\)-functions |
| 14F30 | \(p\)-adic cohomology, crystalline cohomology |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F85 | \(p\)-adic theory, local fields |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
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