College of Liberal Arts and Sciences

# Department of Mathematics

## Algebra Seminar

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Title: $p$-adic L-functions for symplectic groups and Klingen Eisenstein ideal
Speaker: Zheng Liu (IAS / McGill University)
Time: Wednesday, April 25, 2018 at 11:15 am
Place: MONT 313Abstract: The Iwasawa--Greenberg Main Conjecture predicts that the characteristic ideal of the Selmer group of $p$-adic deformation of a motive is generated by the corresponding $p$-adic L-function. One strategy for proving one divisibility is to construct cohomology classes through the congruences between Eisenstein series and cusp forms, which was used by Ribet in proving the converse to Herbrand's theorem, and further developed by Mazur--Wiles, Wiles, Skinner--Urban, etc, for proving the main conjectures for GL(1) and GL(2). A key step in this strategy is to study certain Eisenstein series including constructing $p$-adic families of Klingen Eisenstein series, relating the constant terms to $p$-adic L-functions and analyzing the non-degenerate Fourier coefficients. I will first illustrate the idea of Eisenstein congruences in the GL(1) case, and then explain some results for symplectic groups.

Organizer: Liang Xiao