>1. Yichao Tian (Bonn, MCM)

>9:00-- 10:00

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>Title: The triple product Selmer group of an elliptic curve over a cubic real field

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>Abstract: Let F be a cubic totally real field, and E/F be a modular elliptic curve. We consider its triple product motive M attached to E, which is a 8-dimensional motive over Q. Assume that the functional equation of L(M,s) has sign -1. Under certain technical assumptions, one can construct a cohomology class in the p-adic Bloch-Kato Selmer group of M using  Hirzebruch-Zagier cycles. We prove that if this class is non-trivial, then the Bloch-Kato Selmer group of M has dimension 1. A key ingredient in the proof is a congruence formula for the cohomology class at certain unramified level raising primes, whose proof uses the fine geometry of the supersingular locus of  a Hilbert modular threefold at an inert prime. This is a joint work with Yifeng Liu.