Trace and Norm L-functions
Antonio Rojas Leon
Departamento de Álgebra, Universidad de Sevilla
Spain
Exponential sums de…ned over the rational points of an algebraic variety
over a …nite …eld have been extensively studied in the literature, and the ladic cohomology theory provides a very convenient tool to obtain estimates
and equidistribution results for them. But sometimes we need to study sums
which are de…ned over more general sets, like the set of points on a commutative algebraic group with a …xed norm (e.g. Soto-Andrare type sums). These
sums can be studied using Weil descent, but this gives sub-optimal estimates in
most cases. In this talk we will introduce some L-functions that are associated
with these exponential sums (in the same way that classical L-functions encode
the properties of classical exponential sums), show that, as their classical counterparts, they are rational, and the improved estimates that one can deduce
from them. As a consequence, we will give some improvements of the HasseWeil estimates for exponential sums which are invariant under a large group of
translations or homoteties.
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