Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Neil P. Dummigan
Affiliation:

University of Sheffield
Date:

Tue, 2020-01-21 10:15 - 12:00 I will start from the situation of a cuspidal Hecke eigenform f of real quadratic character, congruent to its complex conjugate modulo a prime P ramified in the coefficient field.

Following Hida, P then divides an appropriately normalised near-central critical value of the adjoint L-function of f. In fact the same is true of the other critical values. The Bloch-Kato conjecture then predicts non-zero elements of P torsion in Selmer groups, which can be constructed using experimentally-observed congruences involving non-parallel weight, level one Hilbert modular forms. Using p-adic deformation in Hida's nearly ordinary family, these appear to be equivalent to congruences between certain parallel-weight (but level p, non-trivial character) Hilbert modular cusp forms and Eisenstein series.

**Links:**

[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/3444

[3] http://www.mpim-bonn.mpg.de/node/9809