Algebraic Function Fields

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Math 612 { Algebraic Function Fields Julia Hartmann May 12, 2015 An algebraic function field is a finitely generated extension of transcendence degree 1 over some base field. This type of field extension occurs naturally in various branches of mathematics such as algebraic geometry, number theory, and the theory of compact Riemann surfaces. In this course, we study algebraic function fields from an algebraic point of view. Topics include: • Valuations on a function field: valuation rings, places, rational function fields, weak approximation • Divisors of a function field: divisor group, dimension of a divisor • Genus of a function field: definition, adeles, index of speciality • Riemann-Roch theorem: Weil differentials, divisor of a differential, duality theorem and Riemann-Roch theorem, canonical class, strong approximation • Fields of genus 0: divisor classes, rational function fields • Extensions of valuations: ramification index and relative degree, funda- mental equation, extensions and automorphisms • Completions: definition and existence of completions, complete discretely valued case, Hensel-Lemma, completions of function fields • The different: different exponent, Dedekind's different theorem • Hurwitz genus formula: divisors in extensions, lifting of differentials, genus formula, Lüroth'stheorem, constant extensions, inseparable extensions • Differentials of algebraic function fields: derviations, differentials, residues • The residue theorem: connection between differentials and Weil differentials, residue theorem • Congruence zeta function: finiteness theorems, zeta function, smallest positive divisor degree, functional equation, Riemann hypothesis for congruence function fields (Hasse-Weil) 1 • Applications to coding theory: gemetric Goppa codes, rational codes, de- coding, automorphisms, asymptotic bounds • Elliptic function fields and elliptic curves Additional topics will be added as time permits and based on students' interests. Prerequisites for the class are a solid background in algebra, in particular, field extensions and Galois theory. No prior knowledge of valuation theory will be assumed. This course is also suitable for undergraduate or master students who have taken 502/503. Grading will be based on regular homework assignments and a presentation or oral exam at the end of the semester. 2.

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