2-dimensional simple factors of \(J_ 0(N)\)
Let \(S_2 (\Gamma_0 (N))\) be the space of cusp forms of weight 2 for the modular group \(\Gamma_0 (N)\), with \(N\) a positive integer. Given a newform \(f \in S_2 ( \Gamma_0 (N))\) Shimura constructed a simple abelian variety \({\mathcal A}_f\) which is a quotient of the Jacobian \(J_0 (N)\) of the modular curve \(X_0 (N)\) associated with \(\Gamma_0 (N)\). The paper under review presents an algorithm for computing the period matrix of the abelian variety \({\mathcal A}_f\). The type of the induced polarization of \({\mathcal A}_f\) is given by certain intersection numbers in the cuspidal cohomology \(H^1_p (\Gamma_0 (N), \mathbb{Z})\) of \(\Gamma_0 (N)\). If \({\mathcal A}_f\) is a principally polarized abelian surface, then \({\mathcal A}_f\) is the Jacobian of a smooth curve \(C\) of genus \(g\). \S 3 of the paper exhibits an algorithm for computing an equation for \(C\) in this case. Moreover some examples are given.
principally polarization abelian surface
scientific article; zbMATH DE number 784070
scientific article