Arithmetic Study Group: Level p paramodular congruences of Harder type
18 March 2014 13:00 in CM107
For a long time we have known about the existence of congruences between the Hecke eigenvalues of elliptic modular forms. Of course the most famous of these is the Ramanujan congruence for the tau function mod 691. Such congruences are important in describing, in some sense, the structure of Galois representations.
Around ten years ago, a well known paper by Harder exploited the cohomology of Siegel modular varieties in order to predict a far reaching generalization of Ramanujan's congruence. His conjecture describes a specific congruence between the Hecke eigenvalues of Siegel modular forms and elliptic modular forms (both of level 1).
In this talk I will briefly discuss Harder's conjecture along with a paramodular version (for prime levels). Then using conjectural work of Ibukiyama I show how we may translate into the realms of algebraic modular forms via something akin to the Eichler/Jacquet-Langlands correspondence. In this setting I provide a strategy for collecting evidence for the new conjecture, giving examples at previously unseen levels.
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