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Effective Siegel’s theorem for modular curves. (English) Zbl 1225.11088

Let \(\Gamma\) be a congruence subgroup of \(\mathrm{SL}_2({\mathbb Z})\) and \(X_{\Gamma}\) be the corresponding modular curve. We denote by \(j\) the modular invariant function. We prove that if the level of \(\Gamma\) is a prime power distinct from 25, then either \(X_{\Gamma}\) is of genus 0 and has at most two poles or for any number field \(K\) such that the couple \((X,j)\) is defined over \(K\) and for any finitely generated subring \(R\) of \(K\), the set \(X(R,j) = \{P\in X(K)/\;j(P)\in R\}\) is effectively determined in terms of \(X\), \(j\), \(K\) and \(R\). Similarly, if the level of \(\Gamma\) does not divides the number \(2^{20}\cdot 3^7\cdot 5^3\cdot 7^2\cdot 11\cdot 13\), then the set \(X(R,j)\) is effectively determined in terms of \(X\), \(j\), \(K\) and \(R\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G18 Arithmetic aspects of modular and Shimura varieties
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