An Algorithm for Modular Elliptic Curves over Real Quadratic Fields
| Autor: | Lassina Dembélé |
|---|---|
| Rok vydání: | 2008 |
| Předmět: |
Discrete mathematics
Mathematics::Number Theory General Mathematics 11Gxx Sato–Tate conjecture Twists of curves Modular curve Supersingular elliptic curve Elliptic curve 11-xx Oda conjecture Modular elliptic curve elliptic curves with everywhere good reduction elliptic curves Counting points on elliptic curves Schoof's algorithm Algorithm Hilbert modular forms Mathematics |
| Zdroj: | Experiment. Math. 17, iss. 4 (2008), 427-438 |
| ISSN: | 1944-950X 1058-6458 |
| DOI: | 10.1080/10586458.2008.10128875 |
| Popis: | Let $F$ be a real quadratic field with narrow class number one, and $f$ a Hilbert newform of weight $2$ and level $\mathfrak{n}$ with rational Fourier coefficients, where $\mathfrak{n}$ is an integral ideal of $F$. By the Eichler--Shimura construction, which is still a conjecture in many cases when $[F:\Q]>1$, there exists an elliptic curve $E_f$ over $F$ attached to $f$. In this paper, we develop an algorithm that computes the (candidate) elliptic curve $E_f$ under the assumption that the Eichler--Shimura conjecture is true. We give several illustrative examples that explain among other things how to compute modular elliptic curves with everywhere good reduction. Over real quadratic fields, such curves do not admit any parameterization by Shimura curves, and so the Eichler--Shimura construction is still conjectural in this case. |
| Databáze: | OpenAIRE |
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