Symplectic extension graphs and Ribet's lemma

In his pioneering paper, Ribet uses Eisenstein congruences to prove the converse to Herbrand's theorem. A key ingredient in his proof is the construction of nonsplit extensions between certain characters, commonly called Ribet's lemma.  In the talk, I will go over Ribet's lemma, its generalization due to Bellaiche, and our extension of Bellaiche's result. Given an irreducible representation, Bellaiche shows the associated extension graph (the edges correspond to nonsplit extensions of residual factors realized by the representation) is strongly connected using the theory of Bruhat--Tits buildings. I will define a symplectic variant of Bellaiche's graph and prove the existence of certain paths in it by the use of symplectic buildings.