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We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations are uniform in the volume and spectral aspects. 

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power $p^n$of a fixed odd prime $p$, a pure depth-aspect analogue of theorems of Kowalski–Sawin and Ricotta–Royer–Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as $n\to <\#comment/>\mathrm{\infty <\#comment/>}$to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every $p$converges in law as $p\to <\#comment/>\mathrm{\infty <\#comment/>}$, and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the $p$-adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments. 

For a fairly general family of L L -functions, we survey the known consequences of the existence of asymptotic formulas with power-saving error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q q , and prove that it satisfies such formulas. We derive arithmetic consequences: a positive proportion of central values L ( f ⊗ χ , 1 / 2 ) L(f\otimes \chi ,1/2) are non-zero, and indeed bounded from below; there exist many characters χ \chi for which the central L L -value is very large; the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols. 

Abstract We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.