Let K be a number field containing the n-th roots of unity for some n 3. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s, χ) with χ an n-th order Hecke character: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.

Abstract. We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

Using the theory of metaplectic forms, we study the asymptotic behavior of cubic exponential sums over the ring of Eisenstein integers. In the first part of the paper, some non-trivial estimates on average over arithmetic progressions are obtained. In the second part of the paper, we prove that the sign of cubic exponential sums changes infinitely often, as the modulus runs over almost prime integers.

The main objective of this article is to study the asymptotic behavior of Salié sums over arithmetic progressions. We deduce from our asymptotic formula that Salié sums possess a bias towards being positive. The method we use is based on the Kuznetsov formula for modular forms of half-integral weight. Moreover, in order to develop an explicit formula, we are led to determine an explicit orthogonal basis of the space of modular forms of half-integral weight.

No abstract