FOURIER COEFFICIENTS OF GLpNq AUTOMORPHIC FORMS IN ARITHMETIC PROGRESSIONS

Fourier Coefficients of Gl(n) Automorphic Forms in Arithmetic Progressions

Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: – The first negative coefficient of the sequence {af(tn)}n∈N, – The number of coefficients af(tn ) of same signs, – Non-vanishing of coefficients af(tn ) in short intervals and in arithmetic progressions, where af(n) is the n-th Fourier coefficient of f and t is a square...

On the Decay of the Fourier Transform and Three Term Arithmetic Progressions

In this paper we prove a basic theorem which says that if f : Fpn → [0, 1] has few ‘large’ Fourier coefficients, and if the sum of the norms squared of the ‘small’ Fourier coefficients is ‘small’ (i.e. the L2 mass of f̂ is not concetrated on small Fourier coefficients), then Σn,df(n)f(n+ d)f(n + 2d) is ‘large’. If f is the indicator function for some set S, then this would be saying that the set...

Adelic Fourier - Whittaker Coefficients and the Casselman - Shalika formula

In their paper Metaplectic Forms, D. A. Kazhdan and S. J. Patterson developed a generalization of automorphic forms that are defined on metaplectic groups. These groups are non-trivial covering groups of usual algebraic groups, and the forms defined on them are representations that respect the covering. As in the case for automorphic forms, these representations fall into a principle series, in...

Applications of a Pre–trace Formula to Estimates on Maass Cusp Forms

By using spectral expansions in global automorphic Levi–Sobolev spaces, we estimate an average of the first Fourier coefficients of Maass cusp forms for SL2(Z), producing a soft estimate on the first numerical Fourier coefficients of Maass cusp forms, which is an example of a general technique for estimates on compact periods via application of a pre–trace formula. Incidentally, this shows that...