PROGRAM OF THE COURSE: INTRODUCTION TO ALGEBRAIC CURVES 2011-2012 (WS) GIULIO PERUGINELLI

Algebraic curves over a field k. Introduction: geometric and algebraic point of view. Field of rational functions of a curve. Notion of places. Valuation ring, DVR, residue field of a valuation ring, degree of a place. Discrete valuations over a field F . Correspondance between valuations and places. Corespondance between geometric points of a curve and places of its rational function field. Zero and poles of a place. Existence of places on a function field (after Chevalley). Example: rational function field k(x) in one variable over a field k. Independence of valuations. Divisors and their degree. Divisor class group (free abelian group on the set of divisors). Vector space L(D) associated to a divisor of a function field and its dimension l(D). Bounds on l(D). Genus g = g(F/k) of a function field F/k. Riemann-Roch theorem for algebraic curves. Index of speciality of a divisor. Adele and Weil differential: properties. Canonical divisor of a function field F/k. Consequences of Riemann-Roch Theorem. Strong Approximation Theorem for distinct places (independent valuations). Theorem of Luroth. Parametrization of an algebraic curve of genus zero. Elliptic function fields and group law on the set of places. Codes: definition and invariants: Lenght, dimension and minimum distance. Bounds and relations among these quantities. Examples: Parity check code, Repetition code, Geometric Goppa code (use of alg.curves over a finite field to construct codes; application of Riemann-Roch theorem). Error correcting code. Riemann-zeta function of an algebraic curves over a finite field. Properties and relations. Counting points (places) on function fields over a finite field, class number of a function field (cardinality of the group of divisor classes of degree zero). Euler’s formula. Function field equation for the Riemann zeta function. Hasse-Weil’s theorem. Hasse-Weil’s bound.

Riferimenti bibliografici [Chev] Chevalley, Claude. Introduction to the theory of algebraic functions of one variable. Mathematical Surveys, No. VI. American Mathematical Society, Providence, R.I. 1963 xi+188 pp. [Ros] Rosen, Michael. Number theory in function fields. Graduate Texts in Mathematics, 210. SpringerVerlag, New York, 2002. xii+358 pp. [Stich] Stichtenoth, Henning. Algebraic function fields and codes. Second edition. Graduate Texts in Mathematics, 254. Springer-Verlag, Berlin, 2009. xiv+355 pp.

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