  The topic of this paper concerns a certain relation between the jacobians of
various quotients of the modular curve $X(p)$, which relates the jacobian of
the quotient of $X(p)$ by the normaliser of a non-split Cartan subgroup of
$GL_2(F_p)$ to the jacobians of more standard modular curves. In this paper, we
confirm a conjecture of Merel found in a paper of Darmon-Merel, "Winding
quotients and some variants of Fermat's Last Theorem", Crelle, v. 490, p.
81-100, 1997, which describes this relation in terms of explicit
correspondences. The method used is to reduce the conjecture to showing a
certain $Z[GL_2(F_p)]$-module homomorphism is an isomorphism. This is
accomplished by using some peculiar relations between double coset operators to
find a expression for the eigenvalues of this homomorphism in terms of Legendre
character sums and Soto-Andrade sums. A ramification argument then shows that
these eigenvalues are non-zero.
