Bounding sup-norms of cusp forms of large level

Let f be an L ²-normalized weight zero Hecke-Maaß cusp form of square-free level N, character χ and Laplacian eigenvalue λ≥1/4. It is shown that ‖f‖_∞? _λ N ^?1/37, from which the hybrid bound ‖f‖_∞?λ ^1/4(N λ)^?δ (for some δ>0) is derived. The first bound holds also for f=y ^k/2 F where F is a holomorphic cusp form of weight k with the implied constant now depending on k.     

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).     

Period Integrals and Rankin–Selberg L-functions on GL(n)

We compute the second moment of a certain family of Rankin–Selberg L-functions L(f ×?g, 1/2) where f and g are Hecke–Maass cusp forms on GL(n). Our bound is as strong as the Lindel?f hypothesis on average, and recovers individually the convexity bound. This result is new even in the classical case n?=?2.     

Applications of the Kuznetsov formula on GL(3)

We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl’s law, exceptional eigenvalues, a large sieve and L-functions are given.     

The number of integer points on Vinogradov’s quadric

An asymptotic formula is obtained for the number of integer solutions of bounded height on Vinogradov’s quadric. Two leading terms are determined, and a strong estimate for the error term is given.     

Prime paucity for sums of two squares

Almost all solutions of the diophantine equations , with all x_j prime numbers, are diagonal ones.This is established here in quantitative form.     

THE AVERAGE VALUE OF DIVISOR SUMS IN ARITHMETIC PROGRESSIONS

Let (n) denote the Fourier coefficients of cusp forms or thenumber of divisors of n. Estimates of the type are shown, uniformly in q X. The methods canbe extended to other arithmetic functions, for example, thenumber of representations of n as a sum of two squares or k-freenumbers. As an application, sums of the type _{n X}(n) (n) forany q-periodic function can be estimated non-trivially.     

Galois cohomology of completed link groups

In this paper we compute the Galois cohomology of the pro- completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in whose linking number diagram is irreducible modulo (e.g. none of the linking numbers is divisible by ).

The result is that (with -coefficients) the Galois cohomology is naturally isomorphic to the -cohomology of the discrete link group.

The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself.

     

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.     

Thin bases of order h

A subset is called a basis of order h if every positive integer can be represented as a sum of h members of A. Thin bases of order h will be constructed in this paper, for each h?2, where the value of is smaller than that of thin bases known so far. In the most important case h=2 it is shown that for the considered class of bases (which generalizes an ansatz of Stöhr) the result is best possible up to an ε>0.