Plane Algebraic Curves; p-adic Numbers (4V+2U, 9.0 LP)
|SWS||Type||Course Form||CP (Effort)||Presence-Time / Self-Study|
|-||K||Lecture with exercise classes (V/U)|
|4||V||Lecture||6.0 CP||56 h||124 h|
|2||U||Exercise class (in small groups)||3.0 CP||28 h||62 h|
|(4V+2U)||9.0 CP||84 h||186 h|
Possible Study achievement
- Verification of study performance: proof of successful participation in the exercise classes (ungraded)
- Details of the examination (type, duration, criteria) will be announced at the beginning of the course.
- Affine and projective spaces, in particular, the projective line and the projective plane,
- Plane algebraic curves over the complex numbers,
- Smooth and singular points,
- Bézout's Theorem for projective plane curves,
- Topological genus of a curve and the genus formula,
- Rational maps between plane curves and the Riemann-Hurwitz formula.
In addition, a selection of the following topics is covered:
- Polar and Hessian curves,
- Dual curves and Plücker’s formulas,
- Linear systems and divisors on smooth curves,
- Real projective curves,
- Puiseux-parameterizations of plane curve singularities,
- Invariants of plane curve singularities,
- Elliptic curves,
- Further aspects of plane algebraic curves.
- construction of p-adic numbers,
- p-adic integers, units,
- p-adic topology,
- Hensel's lemma,
- algebraic closure,
- Newton polygon,
- inertia and ramification groups
Competencies / intended learning achievements
In the part "p-adic Numbers", the students have studied extensions of a number system through p-adic numbers which are fundamental for number theory. In particular, they have studied their main properties and some simple applications.
- G. Fischer: Ebene algebraische Kurven,
- E. Brieskorn, H. Knörrer: Plane Algebraic Curves,
- E. Kunz: Introduction to Plane Algebraic Curves,
- F. Kirwan: Complex Algebraic Curves,
- R. Miranda: Algebraic Curves and Riemann Surfaces.
- I.N. Stewart, D.O. Tall: Algebraic Number Theory,
- M. Trifković: Algebraic Theory of Quadratic Numbers.