Graduate courses

From 2006 consult AULA.

Algebraic Curves, Fall 2005

Description http://www.imf.au.dk/kurser/algcurves/Fall05/

The Arithmetic of Elliptic Curves, Spring 2004

Description http://www.imf.au.dk/courses/arithmellipticurves/

Algebraic Curves, Fall 2003

Description http://home.imf.au.dk/matjph/COX.ps

Dairy http://home.imf.au.dk/matjph/kurver03.html

The Arithmetic of Elliptic Curves. Spring 2001

Annoncement: Elliptic curves are intoduced from an algebraic geometric point of view. After a brief review of the basics of algebraic curves we proceed to the geometry of elliptic curves , the group law on elliptic curves and determination of the ring of endomorphisms. Elliptic curves over finite fields, over local and global fields will be the next topics.

In passing we will obtain the Hasse-Weil theorem on the number of rational points on an elliptic curve over a finite field and the Mordell-Weil theorem that the rational points on an elliptic curve over the rational numbers is a finitely generated Abelian group.

Literature: Joseph H. Silverman ``The Arithmetic of Elliptic Curves'', Springer, GTM, 1986

Prerequisites: Algebra 1 and basic theory of algebraic curves.

Lectures:tuesdays 9-11 in koll. G and wednesdays 11-13 in Aud. D3

Plan: ps, pdf

Noter af Jesper om APECS ps

Noter af Marc: Counting the number of isomorphism classes of elliptic curves with a given number of points over finite fields ps og programmer til beregning af H(Delta): pascalprogram DOS

Pensum ps

Afloesningsopgave ps

Bachelor-projekt Elliptiske kurver (F00).

Arbejdsprogram

� Steffen Bach Hansen (Rapporten som ps-fil)

� Kasper Kjolby (Rapporten som ps-fil)

� Peter Johannes Steffensen (Rapporten som ps-fil)

Elliptic Curves II (E99). Lectureplan

� Henrik Spalk: On global minimal Weierstrass equations

� Marc Skov Madsen: Global minimal Weierstrass forms for the Frey curves

� Jesper Petersen: Global minimal Weierstrass forms in MAPLE using Apecs.

� Fundamental domains for the Hecke groups of level 2, 4 and 8. Morten:

� Fundamentalomraade for Heckegruppen N=2 (postscript),

� Fundamentalomraade for Heckegruppen N=4 (postscript),

� Fundamentalomraade for Heckegruppen N=8 (postscript)

� Jesper Petersen: En sammenligning mellem Fourier-koefficienterne til en spidsform af niveau 11 og vaegt 2 og twists af visse elliptiske kurver (ps-fil)

� Morten Skarsholm Riasager: Generators for the Hecke groups Lidt om frembringere (ps-fil)

� C-program by Anders Nedergaard Jensen for calculating q-expansions C-source code

Elliptic Curves over Q and C (F99). Lectureplan

Henrik Spalk: Algebraisk Talteori og Mordells S�tning (ps) (April 1999) 13 sider, der knytter sig til A. Knapp: "Elliptic Curves"
Logi Ragnarsson: Elliptic Curves and Public-key Crypthography (ps) (Maj 1999) 5 sider.
Hans Christian Hulvej: Konstruktion af elliptisk kurver med foreskreven torsion og bestemmelse af torsionsgrupperne for specielle kurver. (Maj 1999). Noter 9 sider.
Anders H. Jensen: Bezout's theorem(ps) (Maj 1999) 4 sider.
Marc Skov : Anvendelse af elliptiske kurver til faktorisering af hele tal (Lenstra) (html)
Lene S�ndergaard: Isomorfiklasser af elliptiske kurver over endelige legemer.
Jesper Petersen: Algoritme til beregning af rangen af en elliptisk kurve. (ps) .

� Groebner Bases and Convex Polytopes (E98) (with Niels Lauritzen). Lectureplan(JPH) Lectureplan(Niels)

Jens: Beregning af minimalpolynomium (ps, pdf)
Niels: Gr�bner bases and matroids (ps, pdf)
Henrik: Dimension af en varietet (ps, pdf)
Anders and Jesper: Buchbergers algorithm and Hilbert functions (ps, pdf)
Per: Hilbert functions and the algorithm of Bayer-Stillman (ps, pdf)

� The Arithmetic of Elliptic Curves (F98). Lectureplan

Morten Skarsholm Risager: P-adic numbers. (ps)

Niels: Elliptiske kurver over C (ps)

Erik Woodward Knudsen: Supersingular elliptic curves and crypthography (ps)

� Algebraic Curves (E97). Lectureplan

Niels F. S�rensen: Siegel's s�tning om heltalspunkter p� en elliptisk kurve (ps)
Henrik Gadegaard Spalk: Primtalsfaktorisering ved elliptiske kurver (ps)

Allan Simonsen: Automatic Geometric Theorem Proving (ps)
Brian Rigelsen Jessen: Bevis for den svage Nullstellensatz (ps)

Morten Skarsholm Risager: Hasse-Weils s�tning om punkter p� elliptiske kurver over endelige legemer (ps)

Flemming Lorenzen: Kvadratiske transformationer (ps)

Claus Mazanti S�rensen : Kompleks multiplikation p� elliptiske kurver og Galois udvidelser over Q (ps)

� Invariant Theory (97).

Thomas Fangel: Invariant Theory of finite groups (pdf)

Torsten Rasmussen : Invariant teori for endelige reflektionsgrupper (pdf)