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Trigonal quotients of modular curves \(X_{0}(N)\). (English) Zbl 1115.11031

The paper completes the list of trigonal Atkin-Lehner quotients of modular curves \(X_0(N)\). The authors have determined some parts of the list in their previous works [Acta Arith. 88, No. 2, 129–140 (1999; Zbl 0947.11018), Proc. Japan Acad., Ser. A 75, No. 9, 172–175 (1999; Zbl 0963.11031), Proc. Japan Acad., Ser. A 76, No. 6, 83–86 (2000; Zbl 0973.11064)].
In this paper they consider the remaining case of those quotients \(X_0(N)/W\), with \(W\) a proper subgroup of the Atkin-Lehner group \(W(N)\) with \(2<| W| <| W(N)|\). Some geometric lemma and arithmetic criteria allow the authors to build tables with all the pairs \(N, W\) such that \(X_0(N)/W\) is trigonal of genus 3 and 4 and to provide plane models of the trigonal curves \(X_0(N)/W\) of genus greater than four.

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
11G05 Elliptic curves over global fields
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References:

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