### On relations between jacobians of certain modular curves, by Imin Chen

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.

Imin Chen <chen@math.mcgill.ca>