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.

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.

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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For any natural number$m$(>1) let$P$($m$) denote the greatest prime divisor of$m$. By the problem of Erdős in the title of the present paper we mean the general version of his problem, that is, his conjecture from 1965 that $$\frac{P(2^n-1)}{n} \to \infty \quad {\rm as}\, n \to \infty$$ (see Erdős [10]) and its far-reaching generalization to Lucas and Lehmer numbers. In the present paper the author delivers three refinements upon Yu [40] required by C. L. Stewart for solving completely the problem of Erdős (see Stewart [25]). The author gives also some remarks on the solution of this problem, aiming to be more streamlined with respect to the$p$-adic theory of logarithmic forms.