In this paper we prove the Iwasawa main conjecture for the Rankin-Selberg product of a general modular form and a CM form of higher weight. This result has been used by others to prove the converse of the theorem of Gross-Zagier and Kolyvagin, and the BSD formula for elliptic curves of analytic rank one.

We study the modularity of the genus zero open Gromov-Witten potentials and its generating matrix factorizations for elliptic orbifolds. These objects constructed by Lagrangian Floer theory are a priori well-defined only around the large volume limit. It follows from modularity that they can be analytically continued over the global Kähler moduli space.

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.

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