  In this paper, we recall an alternative proof of Merel's conjecture which
asserts that a certain explicit correspondence gives the isogeny relation
between the Jacobians associated to the normalizer of split and non-split
Cartan subgroups. This alternative proof does not require extensive
representation theory and can be formulated in terms of certain finite
geometries modulo $\ell$.
  Secondly, we generalize these arguments to exhibit an explicit correspondence
which gives the isogeny relation between the Jacobians associated to split and
non-split Cartan subgroups. An interesting feature is that the required
explicit correspondence is considerably more complicated but can expressed as a
certain linear combination of double coset operators whose coefficients we are
able to make explicit.
