Curves and Function Fields
Presented topics
- week:
(15.02.)
0.Motivation. Curves over a field of positive characterisic: translation of geometry to language of algebra.
1.Algebras over a field. K-algebras as vector spaces [L, 1.1].
(16.02.) Duble factorizing lemma, modularity law. Structure of dual spaces over field extensions
[L, 1.2-4].
- week:
(22.02.) Structure of L-space on the space Hom(V,K) [L, 1.5]. 2.Algebraic function fields. Linear independence over a polynomial ring. Multiplication of subrings and ideals, a basis of a field extension forms a basis of the extension of rational-function fields [L, 2.1-4].
(23.02.) Description and examples of an algebraic function field, the field of konstants of an algebraic function field is of a finite degree [L, 2.5-7].
Practicals. Constructions of algebraic function fields and computing their fields of konstants.
- week:
(1.3.) Charakterization of AFF [L, 2.8]. 3. Valuation rings. Local rings, Nakayama's lemma, noetherian local domain with a principal maximal ideal is uniserial [L, 3.1-3].
(2.3.) Valuation and uniserial rings [L, 3.4].
Practicals. Examples of valuation rings, localization,
structure properties of valuation rings.
- week:
(8.3.) Valuation rings of a field are exactly maximal subrings. Noetherian valuation rings are maximal
[L, 3.5-7].
4. Discrete valuation rings. Construction end examples of (normalized) discrete valuations [L, 4.1].
Practicals. Noetherian valuation rings.
(9.3.) Discrete valuation rings are preciselly noetherian local domains with the principal maximal ideal [L, 4.2]. Normalized discrete valuations and uniformizing elements [L, 4.3-6]. Valuation rings of an algebraic function field, description of normalized discrete valuations of an algebraic function field [L, 4.7].
- week:
(9./11.3.) Places of algebraic function fields. Unicity of VR of AFF and NDV determined by a place [L, 4.8-9].
Practicals. Discrete valuation rings.
5. Weierstrass equiation . The group of affine mappings and affine automorphisms.
Translations and unitriangular affine automorphisms map Weierstrass equiation polynomials (WEP) to WEP. A short WEP, description of affine automorphisms inducing equivalence of WEP [L, 5.1-4].
record of the on-line lecture on 9 March
- week:
(22.3.) 6. Singularities. Smoothness and singular points. Correspondence of tangents and linear parts [L, 6.1-2] Practicals: Transformation of WEP.
(23.3.) 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 [L, 6.3-5]. Practicals: Determining smoothness of WEP.
7. Coordinate rings. Description of maximal ideals of polynomial rings over a field [L, 7.1].
- week:
(29.3.) Prime ideals of the ring K[x,y] and planar curves. Coordinate rings and function fields of planar curves [L, 7.2-5].
Practicals: How do maximal ideals containing WEP look like?
(30.3.) Algebraic function fields and function fields of curves. Algebraic function fields given by an equation. [L, 7.6-8].
8. Absolutely ireducible polynomials.
Each WEP is absolutely ireducible, algebraic function fields given by absolutely ireducible polynomials have the trivial field of constants. [L, 8.1-5].
- week:
(5.4.) 9. Places determined by a pair.
Description of an AFF by polynomial f with positive multiplicity and the tangent y at the point (0,0)
Multiplicity, m-weighted multiplicity, and m-socles [L, 9.1].
(6.4.) Weighted multiplicity of an element of K[u,v].
Existence and uniqueness of normalized discrete valuations with positive value on u,v [L, 9.2-5].
Practicals: Computing of m-weighted multiplicity.
- week:
(12.4.) Computing of valuations given by lines going through rational smooth point [L, 9.6-8].
Practicals: Computing of discrete valuations.
(13.4.) 10. Localization in a coordinate ring.
Maximal ideals of local subrings Rg of an AFF determined by points of the curve, local rings Rg for singular points g are not VR.
Subalgebra K[u,v] and places of an AFF [L, 10.1-4].
- week:
(19.4.) Description of places of degree 1, places of degree 1 over WEP which are not given by localization in a rational point [L, 10.5-8].
Practicals: Structure of places of degree 1.
(20.4.) 11. The Weak Approximation Theorem. Weak Approximation Theorem and its consequences: 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 of [L:K(s)] using valuations and degrees of places containing s. [L, 11.1-6].
- week:
(26.4.) The place "in infinity" of a WEP, characterization of places of degree one over WEP [L, 11.7-9].
Practicals: Finding places over WEP of degree 1 and higher.
12. Divisors. Principal divisors, degree of a divisor description of zero principal divisors [L, 12.1].
(27.4.) Degree of a divisor as a group homomorphism, description of zero principal divisors,
Riemann-Roch spaces. Correspondence of dimension of Riemann-Roch spaces and degree od divisor [L, 12.2].
- week:
(3.5.) The degree of positive and negative part of a principal divisor. Correspondence degree of a principal divisor and the dimension of the corresponding Riemann-Roch space [L, 12.3-9]. Practicals: Computing of principal divisors.
(4.5.) Riemann theorem and the notion of genus [L, 12.10-11]. 13. Adeles and Weil differentials. Index of speciality and adeles, computing index of speciality by applying adeles.
Correspondence of Weil differentials and indexes of speciality [L, 13.1-2].
- week:
(10.5.) The L-space of Weil differentials is of dimension 1,
l(W-A)=i(A) for any canonical divisor W [L, 13.3-5].
14. Riemann-Roch Theorem. Riemann-Roch Theorem and its consequences [L, 14.1-3].
- week:
(17.5.) AFFs of genus 0 a 1, AFFs of genus 0 with a place of degree 1 are isomorphic to K(x) [L, 14.4-8].
15. Elliptic function field and elliptic curves. Elliptic function fields are exactly those given by a smooth WEP [L, 15.1-3].
(18.5.) The transfer of Picard group to points of a smooth WEP curve, description of intersections with lines as principal divisors,
computation on the curve. [L, 15.4-8].
[L] - List of claims.