Let $p$ be a rational prime and $K$ a complete discrete valuation field with residue field $k$ of positive characteristic $p$. When $k$ is finite, generalizing the theory of Deligne [1], we construct in [10] and [11] a theory of local ${\epsilon }_0$-constants for representations, over a complete local ring with an algebraically closed residue field of characteristic $\ne p$, of the Weil group $W_K$ of $K$. In this paper, we generalize the results in [10] and [11] to the case where $k$ is an arbitrary perfect field.

We prove the compatibility of the local and global Langlands correspondences at places dividing $l$ for the $l$-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of ${GL}_n$ over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing $l$ and have Shin-regular weight.

The $p$-adic local Langlands correspondence for ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ attaches to any $2$-dimensional irreducible $p$-adic representation $V$ of $G_{{\mathbb{Q}}_p}$ an admissible unitary representation $\Pi \left(V\right)$ of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$. The unitary principal series of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ are those $\Pi \left(V\right)$ corresponding to trianguline representations. In this article, for $p\>2$, using the machinery of Colmez, we determine the space of locally analytic vectors $\Pi {\left(V\right)}_{\mathrm{an}}$ for all non-exceptional unitary principal series $\Pi \left(V\right)$ of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ by proving a conjecture of Emerton.

We prove formulas for the generating functions for $M_2$-rank differences for partitions without repeated odd parts. These formulas are in terms of modular forms and generalized Lambert series.

Let $G:=\mathrm{SO}{(n,1)}^{\circ }$ and $\Gamma (n-1)/2$ for $n=2,3$ and when $\delta >n-2$ for $n\ge 4$, we obtain an effective archimedean counting result for a discrete orbit of $\Gamma$ in a homogeneous space $H\setminus G$ where $H$ is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family $\{{ℬ}_T\subset H\setminus G\}$ of compact subsets, there exists $\eta >0$ such that $\#\left[e\right]\Gamma \cap {ℬ}_T=ℳ\left({ℬ}_T\right)+O\left(ℳ{\left({ℬ}_T\right)}^{1-\eta }\right)$ for an explicit measure $ℳ$ on $H\setminus G$ which depends on $\Gamma$. We also apply the affine sieve and describe the distribution of almost primes on orbits of $\Gamma$ in arithmetic settings....