On Relations between Jacobians of Certain Modular Curves

We confirm a conjecture of L. Merel (H. Darmon and L. Merel, J. Reine Angew. Math.490 (1997), 81–100) describing a certain relation between the jacobians of various quotients of X(p) in terms of specific correspondences. The method of proof involves reducing this conjecture to a question about certain Z[ GL2(Fp)]-module homomorphisms, which is in turn answered by exhibiting some peculiar relations in a double coset algebra.