Topics:

In the first part we study integral and rational solutions to an exemplary cubic equation. An incredible idea for any algebraic equation is to take the number of solutions of such an equation modulo different primes, to combine these numbers to a complex power series and study the resulting analytic functions called Zeta-functions and L-function. Along this way our example touches some of the deepest problems in analytic number theory.

In the second part we study very symmetric functions on the complex plane called modular forms. These functions appear frequently in different areas of mathematics and also in quantum field theory, which gives us many nice examples to study. The final twist that solves our initial exemplary problem is that modular forms are related by an integral transformation to L-functions. As an outlook: Proving this Tanyama Shimura-Conjecture was the main structural result in Andrew Wiles' proof of Fermat last theorem.

Script:

The lecture will essentially cover the first two thirds of "Neal Koblitz: Introduction to Elliptic Curves and Modular Forms".

Prerequisites:

Helpful prerequisites are Elementary Number Theory and Complex Analysis. Both are not necessary, but if the student is willing to close respective gaps by themselves.

Time:

Lecture is Monday 10:15-11:45 in H3 and Wednsday 14:15-15:45 in H2.

Exercises are Monday 12:15-13:45 Uhr in 415. (on 16.10. and 23.10. there are no exercises because there are no exercise sheets at this point)

Exams are Wednesday 14.02.18, 12-14 in H2 and Wednesday 14.03.18, 10-12.

Requirements:

Requirement for the exam are the regular turn-in of exercise sheets and the regular attendence in the exercise class, including several presentations of solutions. exercise sheets can be turned in by groups of at most three students, and every student should be able to present every turned-in solution.

The exercise sheets are given online at this place each Monday and are to be turned in at the following Monday in the lecture.

Exercise Sheets:

• Sheet 1 (due Monday 23.10. in the lecture)
• Sheet 2 (due Monday 30.10. in the lecture)
• Sheet 3 (due Monday 6.11. in the lecture)
• Sheet 4 (due Monday 13.11. in the lecture)
• Sheet 5 (due Monday 20.11. in the lecture)
• Sheet 6 (due Monday 27.11. in the lecture)
• Sheet 7 (due Monday 4.12. in the lecture)
• Sheet 8 (due Monday 11.12. in the lecture)
• Sheet 9 (due Monday 18.12. in the lecture)
• Sheet 10 (due Monday 8.1. in the lecture)
• Sheet 11 (due Monday 15.1. in the lecture)
• Sheet 12 (due Monday 22.1. in the lecture)