Curves and Function Fields


Presented topics

   (17.2.) 0.Motivation. Where to find function field for a curve given by a polynomial? 1.Algebras over a field. K-algebras as vector spaces. Duble factorizing lemma, modularity law.

   (18.2.) Structure of dual spaces over field extensions. Linear independence in a field of rational functions.

   (24.2.) Multiplication of subrings and ideals, a base of a field extension forms a base of the extension of rational-function fields. Description and examples of an algebraic function field. The field of konstants of an algebraic function field is of a finite degree.

   (25.2.) 2. Valuation rings. Local rings. Nakayama's lemma, local domains with a principal maximal ideal, a noetherian local domain with a principal maximal ideal is uniserial. Existence of valuation rings of a field.

   (2.3.) Noetherian valuation rings are maximal. Prime ideals in K[x,y]. Discrete valuations and valuations of prime elements. Discrete valuation rings.

   (3.3.) Discrete valuation rings are preciselly noetherian local rings with the principal maximal ideal. Normalized discrete valuations and uniformizing elements. Valuation rings of an algebraic function field. Description of normalized discrete valuations of an algebraic function field. Places and their valuation rings.

   (9.3.) Normalized discrete valuations of algebraic fubction fields determined by places. 3. Weierstrass equiation polynomials. The group of affine mappings and affine automorphisms.

   (10.3.) Translations and unitriangular affine automorphisms map Weierstrass equiation polynomials (WEP) to WEP. A short WEP, description of affine automorphisms inducing equivalence of WEP.

   (16.3.) Smoothness and singular points. Correspondence of tangents and linear parts. Smoothness is an invariant of K-equivalence. Short WEP y2-f(x) over fields of the characteristic different from 2 are smooth if and only if f(x) is separable.

   (17.3.) 4. Coordinate rings. Description of maximal ideals of polynomial rings over a field. Prime ideals of the ring K[x,y] and planar curves. Coordinate rings and function fields of curves.

   (23.3.) Algebraic function fields are preciselly function fields of curves. Each WEP is absolutely ireducible, Algebraic function fields given by absolutely ireducible polynomials have the trivial field of constants.

   (24.3.) Algebraic function fields given by an equation. 5. Places. m-weighted polynomials and m-socles, "shifting" to the variable x.

   (30.3.) Existence and uniqueness of normalized discrete valuations with positive value on a and b for algebraic function fields given by equations y + m(a,b)=0 for polynomials m of multiplicity > 1.

   (31.3.) Every algebraic function field could be given by equation of type y + m(a,b)=0 for polynomials m of multiplicity > 1. Existence and uniqueness of normalized discrete valuations determined by smooth points, detecting of tangenths using these normalized discrete valuations.

   (6.4.) Computing of valuations given by lines going through rational smooth point. Maximal ideals of local subrings Rg of an AFF determined by points of the curve.

   (7.4.) Local rings Rg for singular points g are not valuation. The maximal ideal Pg is a place for "positive" smooth points g.

   (14.4.) Description of "positive" places of degree 1. The Weak Approximation Theorem.

   (20.4.) Consequences of the Weak Approximation Theorem: There are infinitely many places, there exists a basis of an arbitrary valuation ring of AFF modulo the corresponding place which is contained in a power of finitely many distinct places, an upper bound on valuations and degrees of places containing a single element.

   (21.4.) There are only finitely many places containing a fixed element. Places of degree one over WEP. 6. Divisors. Divisors are formal sums of places. Principal divisors, degree of divisors, decsription of zero principal divisors. A class group and Riemann-Roch spaces.

   (27.4.) Correspondence of dimension of Riemann-Roch spaces and degree od divisor. Degree of positive and negative part of a principal divisor.

   (28.4.) Riemann theorem and the notion of genus. Index of speciality and adeles, computing index of speciality by applying adeles.

   (4.5.) The strong Approximation Theorem. 7. Weil differntials. Spaces of Weil differntials, canonical divisors.

   (5.5.) Dimension of Riemann-Roch spaces can be computed as dimension of a corresponding space of Weil differntials. Riemann-Roch theorem, the main consequence of Riemann-Roch theorem allows computation of dimension of a Riemann-Roch space without knowladge of canonical divisor, An AFF of genus 0.

   (11.5.) 8. The associative law. Elliptic function fields are exactly those given by a smooth WEP.

   (12.5.) The transfer of Picard group to points of a smooth WEP curve, computation using properties of places.

   (18.5.) Computing in the group on an elliptic curve. 9. Projective curves. Homegeneous polynomials and their smoothnes.

   (19.5.) Description of the concept of AFF's using projective curves.