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Flach system on quaternionic Hilbert-Blumenthal surfaces and distinguished periods

组织者

刁晗生 , 胡悦科 , 埃马纽埃尔·勒库图里耶 , 凯撒·鲁普

演讲者

Haining Wang

时间

2023年05月08日 10:00 至 11:00

地点

Online

摘要

In this talk, we will report some integrality result on the ratio of the quaternionic distinguished period associated to a Hilbert modular form and the quaternionic Petersson norm associated to a modular form. These distinguished periods are closely related to the notion of distinguished representations that play a prominent role in the proof of the Tate conjecture for Hilbert modular surfaces by Langlands-Rapoport-Harder. Our method is based on an Euler system argument initiated by Flach by producing elements in the motivic cohomologies of the quaternionic Hilbert–Blumenthal surfaces with control of their ramification behaviours. We show that these distinguished periods give natural bounds for certain subspaces of the Selmer groups of these quaternionic Hilbert–Blumenthal surfaces. The lengths of these subspaces can be determined by using the Taylor–Wiles method and can be related to the quaternionic Petersson norms of the modular forms.