Prepare your way for current research topics in COMPUTATIONAL NUMBER THEORY such as advanced methods for primality testing and factoring integers, cryptosystems, computational aspects in algebraic number theory, the computation of L-functions, and local methods and their various applications.
Winter semester 2020/2021
- Class: Number Theoretical Algorithms
In this class we will study methods for testing primality, factoring integers and computing discrete logarithms in finite fields. Further, we will introduce the continued fraction method for rational approximation. This can be applied to some Diophantine equations. (Literature: Cohen: A Course in Computational Algebraic Number Theory, Forster: Algorithmische Zahlentheorie).
- Class: Computational Aspects of Algebraic and Analytic Number Theory
We begin with a brief introduction to algebraic number theory and study in particular the splitting of primes and ideals in number fields. A second focus is on computational aspects of zeta- and L-functions as they appear in algebraic number theory and the theory of modular forms; here we are concerned with special values, functional equations, the analytic class number formula (related to the ideal class group), and the localization of zeros (with respect to the two unsolved Millennium problems from number theory, namely the Riemann hypothesis and the conjecture of Birch & Swinnerton-Dyer). Excellent literature on these topics: "Algebraic Number Theory" by Fröhlich & Taylor, "Zetafunktionen und quadratische Körper" by Zagier, "Riemann's Zeta-Function" by Edwards.
- Seminar 3
Summer semester 2021
- Class: Local Methods and their Applications
The first part of the class will be an Introduction to p-adic numbers and local fields. The second half will present the following applications: Factoring polynomials over the rationals, computing subfields of numberfields and computing Galois groups of number fields. (Literature: Neukirch: Algebraic Number Theory, Cohen: A Course in Computational Algebraic Number Theory)
- Class: Local-Global Principles
The idea behind a local-global principle is to use local information for solving a global problem. We study its fruitful (and, often enough, also explicit) applications to quadratic forms (in form of the celebrated theorem of Hasse-Minkowski), the so-called circle method (developed by Hardy-Littlewood & Ramanujan), and elliptic curves (with respect to their arithmetic, the conjecture of Birch & Swinnerton-Dyer, and the congruent number problem). Excellent literature on these topics: "Introduction to Elliptic Curves and Modular Forms" by Koblitz, "The Hardy-Littlewood Method" by Vaughan, "Quadratische Formen" by Kneser.