Flach system on quaternionic Hilbert-Blumenthal surfaces and distinguished periods
Title: Flach system on quaternionic Hilbert-Blumenthal surfaces and distinguished periods
Speaker: Haining Wang (Fudan University)
Time: 10:00-11:00 Beijing time, May 8, 2023
Zoom ID: 4552601552 Passcode: YMSC
Location: online seminar (this week only)
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.