Let p be a prime and L be a finite extension of ℚp. We study the ordinary parts of GL2(L)-representations arised in the mod p cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above p. The main tool of the proof is a theorem of Emerton \cite{Em3}.

In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and$k$-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

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

Let $${f \in \mathbb{Z}[x]}$$ , $${\deg f =3}$$ . Assume that$f$does not have repeated roots. Assume as well that, for every prime$q$, $${f(x)\not\equiv 0}$$ mod$q$^{2}has at least one solution in $${(\mathbb{Z}/q^2 \mathbb{Z})^*}$$ . Then, under these two necessary conditions, there are infinitely many primes$p$such that$f$($p$) is square-free.

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