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401-3113-68L  Exponential Sums over Finite Fields

SemesterAutumn Semester 2018
LecturersE. Kowalski
Periodicitynon-recurring course
Language of instructionEnglish

AbstractExponential sums over finite fields arise in many problems of number theory. We will discuss the elementary aspects of the theory (centered on the Riemann Hypothesis for curves, following Stepanov's method) and survey the formalism arising from Deligne's general form of the Riemann Hypothesis over finite fields. We will then discuss various applications, especially in analytic number theory.
ObjectiveThe goal is to understand both the basic results on exponential sums in one variable, and the general formalism of Deligne and Katz that underlies estimates for much more general types of exponential sums, including the "trace functions" over finite fields.
ContentExamples of elementary exponential sums
The Riemann Hypothesis for curves and its applications
Definition of trace functions over finite fields
The formalism of the Riemann Hypothesis of Deligne
Selected applications
Lecture notesLectures notes from various sources will be provided
LiteratureKowalski, "Exponential sums over finite fields, I: elementary methods:
Iwaniec-Kowalski, "Analytic number theory", chapter 11
Fouvry, Kowalski and Michel, "Trace functions over finite fields and their applications"