Exponential Sums Over Finite Fields and the Large Sieve

By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

Cyclotomic approximation lattices

In this paper, we will consider the Approximation Lattices for a p-adic number, as defined in a work of de Weger, and construct a generalization called the Cyclotomic Approximation Lattices. In the latter case, we consider approximation by a pair of cyclotomic integers instead of rational ones. This can be useful for studying p-adic continued fractions with cyclotomic integral part. The first section will introduce this work and provide motivations. The second one will give some background theorems on number rings. In the third section, we will recall the work of de Weger with a new proof for Theorem 3.6, the analogue of classical Lagrange's theorem for continued fractions. In the fourth one, we will then see the cyclotomic variant and its analogous properties.