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On the explicit models of Shimura’s elliptic curves. (English) Zbl 0608.14023

The author studies the abelian varieties \(B_ N\), rational over the field \({\mathbb{Q}}(\sqrt{N})\), which G. Shimura [J. Math. Soc. Japan 25, 523-544 (1973; Zbl 0266.14017)] has considered in relation with Nebentypus cusp forms of level N, in three cases (vis. \(N=29, 37, 41)\) where they have dimension \(1\). He computes their j-invariants by estimating their periods with sufficient accuracy, and deduces from known examples of elliptic curves over \({\mathbb{Q}}(\sqrt{N})\), with good reduction everywhere, explicit Weierstraß equations for the \(B_ N's\).
Reviewer: D.Bertrand

MSC:

14H25 Arithmetic ground fields for curves
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols

Citations:

Zbl 0266.14017
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