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Let X be an arithmetic hyperbolic surface, {\psi} a Hecke-Maass form, and {\gamma} a geodesic segment on X. We obtain a power saving over the local bound of Burq-G\'erard-Tzvetkov for the L^2 norm of ...
We evaluate a determinant of generalized Fibonacci numbers, thus providing a common generalization of several determinant evaluation results that have previously appeared in the literature, all of the...
We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We u...
We resolve a function field version of two conjectures concerning the variance of the number of primes in short intervals (Goldston and Montgomery) and in arithmetic progressions (Hooley). A crucial i...
We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke op...
Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equatio...
Using Lefschetz numbers of certain involutions, we provide lower bounds for the cuspidal cohomology of principal congruence subgroups of Bianchi groups. The asymptotic lower bounds that follow from ou...
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successful...
Let E be an elliptic curve over Q, and let F be a finite abelian extension of Q. Using Beilinson's theorem on a suitable modular curve, we prove a weak version of Zagier's conjecture for L(E/F,2), whe...
We investigate the remainder in the asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space, which remain unchanged along some linear subspace and exp...
The Linnik phenomenon concerning the repelling effect of Siegel's zeros is extended to L-functions in a family wider than hitherto considered.
We study the spatial distribution of the representation of a large integer as a sum of three squares, on the small and critical scale as well as their electrostatic energy. The main results announced ...
Recently Adams and Vogan made a conjecture about the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. We prove this conjecture for the temp...
For $i, j > 0, i + j = 1$, the set of badly approximable vectors with weight $(i, j)$ is defined by $Bad(i, j) = \{(x, y) \in \R^2 : \exists c > 0 \forall q\in\N, \;\; \max\{q||qx||^{1/i}, q||qy||^{1/...
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence o...

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