Congruences between cusp forms and the geometry of Jacobians of modular curves (Q1318050)

Using minimal resolutions of the modular curves \(X_ 0(p^ rM)\) (where \(p \geq 5\) is a prime not dividing \(M)\), we first show that the map \(\gamma:J_ 0(pM)^ r\to J_ 0(p^ rM)\), obtained from the degeneracy maps, induces an isomorphism on the tori. This result is used to compute the kernel of the map \(\gamma:J_ 0(p)^ r \to J_ 0(p^ r)\) (with \(M=1)\). This latter result is then used to study the problem of establishing congruence relations between cusp forms of level \(p\) and \(p^ 2\). It is also used to determine the degree of an isogeny from the old subvariety of \(J_ 0(p^ 2)\) to the old quotient of \(J_ 0(p^ 2)\).