  In this article, we set up a strategy to prove one divisibility towards the
main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint
modular Galois representations associated to Hida families. This conjecture
asserts the equality of the p-adic L-function interpoling the critical values
of the symmetric square of the modular forms in these families and the
characteristic ideal of the associated Selmer group. The idea is to introduce a
third characteristic ideal containing informations on the congruences between
cuspidal Siegel modular forms of genus 2 and the Klingen type Eisenstein series
and to prove the two divisibilities: The p-adic L-function divides the
Eisenstein ideal and that the Eisenstein ideal divides the characteristic ideal
of the Selmer group. In that paper we proved the latter divisibility.
