Algorithms on Elliptic Curves
Průběh přednášky
(19.2.) Lecture: Motivation and structure of the course. 1.Curves and functional fields. Terminology, notation and basic facts about coordinate rings and functional fields of affine curves [D, C.1 and C.3].
Practicals: affine curves in R2, irreducible components, use of the signature of the real quadratic form.
(26.2.) Practicals: Functional fields of lines, construction of algebraic closure of a finite field.
Lecture: Curves in projective space, correctness of the definition of the functional field of an irreducible projective curve. Concepts of smooth and singular curves. Normalized discrete valuations on functional fields. [D, C.2-C.5].
(5.3.) Practicals: Extension of an affine curve into projective space. Normalized discrete valuation of the function field K(x).
Lecture: Places of the function field and the corresponding normalized discrete valuation [D, C.5], equivalent description of the group structure on an elliptic curve [D, W.3]. 2. Weierstrass curves. Geometric motivation of the notion of genus of a curve and genus of a curve over a general field. Weierstrass polynomials (WEP) and Weierstrass curves. The function field (and therefore group) of an elliptic curve can be expressed in terms of a Weierstrass curve. [D, W.1-4]
(12.3.) Practicals: K-equivalence of affine WEP. Computing of short WEP.
Lecture: Affine transformation of WEP. Singularities of WEP pf the form y2-f(x) [D, W.1-4]
[D] - Lecture notes of Aleše Drápal