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Seminar on Algebraic Function Fields SemesterWiSe 2024 / 25 Lecturer Klaus Künnemann, Riccardo Pengo Type of course (Veranstaltungsart) Seminar German title Seminar on Algebraic Function Fields Contents Let k be a field. An algebraic function field in one variable over k is a finitely generated field extension K/k of transcendence degree one such that k is algebraically closed in K. Algebraic functions fields in one variable appear at many places in mathematics. In the theory of holomorphic functions they appear as fields of meromorphic function on a compact Riemann surface. In algebraic geometry they appear as fields of rational functions on a projective algebraic curve. In algebraic number theory, algebraic function fields in one variable over finite fields appear as analogs of algebraic number fields. The seminar will not assume any knowledge of Riemann surfaces, algebraic geometry or algebraic number theory. It is however recommended to attend this seminar in parallel to the course on algebraic number theory. We introduce the transcendence degree, define points and divisors for algebraic function fields and ask about the dimension of the k-vector space of algebraic functions with a given pole and zero behavior. This question is answered by the famous Theorem of Riemann-Roch. We will give applications of Riemann-Roch and classify function fields of small genus. If time permits we will also discuss function fields over finite fields and prove the Riemann hypothesis in the function field case. Literature Pete L. Clark: Algebaric Curves: An Algebraic Approach, Course Notes 2020. --- David M. Goldschmidt: Algebraic Functions and Projective Curves, Springer GTM 2003. --- Henning Stichtenoth: Algebraic Function Fields and Codes, Springer GTM 2009 Recommended previous knowledge Algebra and some commutative algebra Time/Date Monday 16h15 - 17h45 Location M102 Registration
BSem, MSem ECTS 4,5 ECTS |