An additive problem in the Fourier coefficients of cusp forms

 We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the binary additive divisor sum which has been studied extensively. As an application we derive, extending work of Duke, Friedlander and Iwaniec, a subconvex estimate on the critical line for L-functions associated to character twists of these cusp forms. Received: 2 October 2001 / Revised version: 9 September 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): Primary 11F30, 11F37; Secondary 11M41.     

The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II

We prove a general subconvex bound in the level aspect for Rankin–Selberg L-functions associated with two primitive holomorphic or Maass cusp forms over Q. We use this bound to establish the equidistribution of incomplete Galois orbits of Heegner points on Shimura curves associated with indefinite quaternion algebras over Q. Mathematics Subject Classification (2000) 11F66, 11F67, 11M41     

Bounds for modular L-functions in the level aspect

Let f be a primitive (holomorphic or Maaß) cusp form of level q and non-trivial nebentypus. Then for the associated L-function satisfies , where the implied constant depends polynomially on s and the Archimedean parameters of f (weight or Laplacian eigenvalue).     

Popular distances in 3-space

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn^1/3 log log n m(n) n^3/5 β(n), where c> 0 is a constant and β(n) is an extremely slowly growing function, related to the inverse of the Ackermann function.     

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke–Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda $ . It is shown that $\left||f \right||_\infty \ll _{\lambda ,\epsilon } N^{-\frac{1}{6}+\epsilon } \left||f \right||_2$ for any $\epsilon >0$ .     

Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem

Let K be a totally real number field, π an irreducible cuspidal representation of ${{\rm GL}_{2}(K){\backslash}{\rm GL}_{2}(\mathbb{A}K)}$ with unitary central character, and χ a Hecke character of conductor ${\mathfrak{q}}$ . Then ${L(1/2, \pi\oplus\chi) \ll (\mathcal{N}\mathfrak{q})^{\frac{1}{2}-\frac{1}{8}(1-2\theta)+\epsilon}}$ , where 0 ≤ θ ≤ 1/2 is any exponent towards the Ramanujan–Petersson conjecture (θ =  1/9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a generalized Kuznetsov formula.     

Covering lattice points by subspaces

We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body C , symmetric about the origin 0. This enables us to prove the following statement, which settles a problem of G. Halász. The maximum number of n-wise linearly independent lattice points in the n-dimensional ball r B n of radius r around 0 is O(rn/(n-1)). This bound cannot be improved. We also show that the order of magnitude of the number of diferent (n - 1)-dimensional subspaces induced by the lattice points in r&Bgr;n is rn/(n-1).     

A Problem on Zero Subsums in Abelian Groups

Let a₁, a₂, ... be elements of an abelian group such that a_mhas order larger than m^m. Then the multiset of the a_m's can be partitioned into two parts which are free of zero subsums. The result was motivated by a question of Pál Erds.