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Geodesic restrictions of eigenfunctions on arithmetic surfaces
Geodesic restrictions of eigenfunctions arithmetic surfaces Number Theory
2012/4/23
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 ...
A determinant of generalized Fibonacci numbers
generalized Fibonacci numbers Cassini’s identity determinant evaluation
2012/4/18
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...
Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case
Modularity lifting results parallel weight one Artin conjecture tamely ramified case Number Theory
2012/4/18
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...
The variance of the number of prime polynomials in short intervals and in residue classes
variance the number of prime polynomials short intervals residue classes Number Theory
2012/4/23
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...
On the Petersson scalar product of arbitrary modular forms
Petersson scalar product arbitrary modular forms Number Theory
2012/4/18
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...
Heegner points and Jochnowitz congruences on Shimura curves
Heegner points Shimura curves L-functions Jochnowitz congruences
2012/4/18
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...
On the Dimension of Cohomology of Bianchi Groups
Dimension Cohomology Bianchi Groups Number Theory
2012/4/18
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...
Counting rational points over number fields on a singular cubic surface
rational points over number fields singular cubic surface Number Theory
2012/4/18
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...
On Zagier's conjecture for base extensions of elliptic curves
Zagier's conjecture base extensions of elliptic curves Number Theory
2012/4/18
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...
The number of integer points in a family of anisotropically expanding domains
number of integer points family of anisotropically expanding domains Number Theory
2012/4/17
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.
Local statistics of lattice points on the sphere
Local statistics lattice points sphere Number Theory
2012/4/17
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 ...
Genericity and contragredience in the local Langlands correspondence
Genericity contragredience local Langlands correspondence Representation Theory Number Theory
2012/4/17
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...
Badly approximable vectors on a vertical Cantor set
Badly approximable vectors vertical Cantor set Number Theory
2012/4/17
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/...
Ramanujan type congruences for modular forms of several variables
Congruences for modular forms Cusp forms Number Theory
2012/4/17
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...