By the relative trace formula approach of Jacquet–Rallis, we prove the global Gan– Gross–Prasad conjecture for unitary groups under some local restrictions for the automor- phic representations.

Abstract We present a relative trace formula approach to the Gross–Zagier formula and its generalization to higher-dimensional unitary Shimura vari- eties. As a crucial ingredient, we formulate a conjectural arithmetic funda- mental lemma for unitary Rapoport–Zink spaces. We prove the conjecture when the Rapoport–Zink space is associated to a unitary group in two or three variables.

This paper proves Birch and Swinnerton-Dyer conjecture for Q-curves defined over real quadratic number field with everywhere good reductions.

This paper provides a new construction of a supply of Darmon-like points on elliptic curves E defined over a quadratic extension of totally real number fields.

Under mild assumptions, we prove the Iwasawa main conjecture for a cuspidal eigenform at a good prime p.