1,015 research outputs found
Arithmetic level raising on triple product of Shimura curves and Gross-Schoen diagonal cycles II: Bipartite Euler system
In this article, we study the Gross-Schoen diagonal cycle on the triple
product of Shimura curves at a place of good reduction. We prove the unramified
arithmetic level raising theorem for triple product of Shimura curves and we
deduce from it the second reciprocity law which relates the image of the
diagonal cycle under the Abel-Jacobi map to certain period integral of
Gross-Kudla type. Along with the first reciprocity law we proved in a previous
work, we show that the Gross-Schoen diagonal cycles form a Bipartite Euler
system for the symmetric cube motive of modular forms. As an application we
provide some evidence for the rank case of the Bloch-Kato conjecture for
the symmetric cube motive of a modular form
Flach system on Quaternionic Hilbert--Blumenthal surfaces and distinguished periods
We study arithmetic properties of certain quaternionic periods of Hilbert
modular forms arising from base change of elliptic modular forms. These periods
which we call the distinguished periods are closely related to the notion of
distinguished representation that appear in work of
Harder--Langlands--Rapoport, Lai, Flicker--Hakim on the Tate conjectures for
the Hilbert--Blumenthal surfaces and their quaternionic analogues. In
particular, we prove an integrality result on the ratio of the distinguished
period and the quaternionic Peterson norm associated to the modular form. 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
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 Peterson norms of the modular forms
- …