Reducibility of Galois representations and critical L-values: another mod 691 congruence

I will start with Ramanujan's famous mod 691 congruence, after a
brief discussion of rational torsion on elliptic curves, and the
Birch-Swinnerton-Dyer conjecture. Then I will look at an experimental mod
691 congruence of Bergstroem, Faber and van der Geer, involving the Hecke
eigenvalues of a genus 3 Siegel cusp form. This, like Ramanujan's congruence
and Harder's conjecture, is an example of an ``Eisenstein'' congruence. I
will then explain the coincidence of the moduli, visiting the Bloch-Kato
conjecture and Selmer groups along the way."