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- Fernando Rodriguez-Villegas, David Boyd, Sean Keel, David J Saltman, John Tate, Riad Mohamad Masri +11 others
- 2005

Acknowledgments To begin, I want to thank my advisor, Fernando Rodriguez-Villegas, for helpful discussions and encouragement during the last three years. I owe much to my friend and collaborator Jim Kelliher for his patience while listening to me explain my ideas, and Misha Vishik for his constant encouragement. I benefited from the advice and suggestions… (More)

Let π 1 , π 2 be cuspidal automorphic representations of PGL 2 (R) of conductor 1 and Hecke eigenvalues λπ 1,2 (n), and let h > 0 be an integer. For any smooth compactly supported weight functions W 1,2 : R × → C and any Y > 0 a spectral decomposition of the shifted convolution sum

Let K be a totally real number field, π an irreducible cuspidal representation of GL 2 (K)\ GL 2 (A K) with unitary central character, and χ a Hecke character of conductor q. Then L(1/2, π ⊗ χ) ≪ (N q) 1 2 − 1 8 (1−2θ)+ε , 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… (More)

Let π be a regular algebraic cuspidal automorphic representation of GL 2 over an imaginary quadratic number field K, and let ℓ be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible ℓ-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv)… (More)

Let f be a primitive (holomorphic or Maaß) cusp form of level q and nontrivial neben-typus. Then for Re s = 1 2 the associated L-function satisfies L(f, s) q 1 4 − 1 1889 , where the implied constant depends polynomially on s and the archimedean parameters of f (weight or Laplacian eigenvalue). Résumé. Soit f une forme modulaire cuspidale primitive… (More)

- Harald A. Helfgott, zhù màn mèlan, Gergely Harcos
- 2003

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 rB n of… (More)

- Gergely Harcos
- 2008

In the analytic theory of automorphic L-functions one often encounters sums of the form D f (a, b; h) = am±bn=h s(m)t(n)f (am, bn) where a, b, h are positive integers, s(m) and t(n) are the normalized Fourier coefficients of holomorphic or Maass cusp forms coming from automorphic representations of GL(2) over Q and f is some nice weight function on (0, ∞) ×… (More)

- Gergely Harcos
- 2005

This dissertation contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation π of GL m over a number field. The approximation involves a smooth truncation of the Dirichlet series L(s, π) and L(s, ˜ π) after… (More)

Let a 1 , a 2 ,. .. be elements of an abelian group such that a m has 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 Paul Erd˝ os.