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Involutory elliptic curves over \(\mathbb{F}_q(T)\). (English) Zbl 0930.11040

Let \(\mathbb{F}_q\) be a field with \(q\) elements, \(n\in\mathbb{F}_q[T]-\{0\}\) and \(G\) be a subgroup of the Atkin-Lehner involutions of the Drinfeld modular curve \(X_0(n)\). The author determines all \(n\) and \(G\) such that the quotient curve \(G\setminus X_0(n)\) is rational or elliptic. This problem is related, as suggested in the title, to the classical (and solved) problem on involutory elliptic curves. Interesting calculations about the genus of \(G\setminus X_0(n)\) are done in §2.

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11G05 Elliptic curves over global fields
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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References:

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