We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms that are uniform as the lattice moves through “congruence” subgroups. We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, $f(c,d)=c^2+d^2$, but restrict $(c,d)$ to the orbit $O=(0,1)\Gamma$, where $\Gamma$ is an infinite-index, nonelementary, finitely generated subgroup of $\mathrm{SL}(2,\mathbb{Z})$. Assume that the Reimann surface $\Gamma \backslash \mathbb{H}$ has a cusp at infinity. We show that the set of values $f(O)$ contains infinitely many integers having at most $R$ prime factors for any $R>4/(\delta -\theta )$, where $\theta >1/2$ is the spectral gap and $\delta <1$ is the Hausdorff dimension of the limit set of $\Gamma$. If $\delta >149/150$, then we can take $\theta =5/6$, giving $R=25$. The limit of this method is $R=9$ for $\delta -\theta >4/9$. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture

We formulate a Serre-type conjecture for $n$-dimensional Galois representations that are tamely ramified at $p$. The weights are predicted using a representation-theoretic recipe. For $n=3$, some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for $n=4$ by using automorphic inductions of Hecke characters

In this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$. In particular, we obtain Weyl's law with an estimation on the remainder term. This extends some of the main results of Duistermaat, Kolk, and Varadarajan ([DKV1]) to this setting

In this article, we introduce new enumerative invariants of curves on Calabi-Yau $3$-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on those moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions