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Modular units and cuspidal divisor class groups of \(X_1(N)\). (English) Zbl 1208.11076

Let \(\mathcal{F}_1^{\infty}(N)\) denote the group of modular units on the curve \(X_1(N)\) that have divisors supported on the cusps lying over \({\infty}\) of the curve \(X_0(N)\). In this article, for each integer greater than four, the author constructs an explicit basis of \(\mathcal{F}_1^{\infty}(N)\) in terms of the Siegel functions. This enables one to compute the group structure of the rational torsion subgroup \(\mathcal{C}_1^{\infty}(N)\) generated by the divisor classes \([(P)-(\infty)]\) where the points \(P\) run through the cusps lying over \(\infty\). In addition, he makes a conjecture on the structure of the \(p\)-primary part of \(\mathcal{C}_1^{\infty}(p^n)\) for a regular prime \(p\).

MSC:

11G16 Elliptic and modular units
11F03 Modular and automorphic functions
11G18 Arithmetic aspects of modular and Shimura varieties
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