Shimura subgroups and degeneracy maps
Let \(M\) be a positive integer, and let \(X_0(M)\) and \(X_1(M)\) be the modular curves obtained by compactifying the quotients of the Poincaré upper half plane \({\mathcal H}\) by the modular groups \(\Gamma_0 (M)\) and \(\Gamma_1 (M)\), respectively. If \(J_0 (M)\) and \(J_1(M)\) denote the Jacobian varieties of \(X_0(M)\) and \(X_1(M)\), respectively, then the natural morphism \(X_1(M) \to X_0 (M)\) induces the homomorphism \(J_0(M) \to J_1(M)\) of Jacobian varieties whose kernel \(\Sigma(M)\) is known as the Shimura subgroup. Let \(M'\) be a positive divisor of \(M\), and let \(\varphi: J_0(M')^\tau \to J_0(M)\) be the map defined by all the degeneracy maps, where \(\tau\) is the number of positive divisors of \(M/M'\). In this paper, the author determines the kernel of \(\varphi\) for certain \(M\) and \(M'\), and relates the preimage of the Shimura subgroup \(\Sigma(M)\) under \(\varphi\) to the group \(\Sigma (M')^\tau\). He also studies the restriction of degeneracy maps to Shimura subgroups.
scientific article; zbMATH DE number 809106