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A Diophantine Equation is a polynomial equation with rational coefficients,  and we are typically interested in the set of its rational (or integral) solutions. A famous example is the Fermat equation 

       xⁿ + yⁿ = zⁿ.

In 1637 Pierre de Fermat claimed that this equation has no integral solution (x,y,z) with xyz≠0 if n>2. This claim,affectionately called Fermat's Last Theorem, has withstood all attempted proofs for more than 300 years, until Andrew Wiles settled the question in 1994.

Since lecture time is not sufficient to contain a proof of Wiles theorem, we will rather give a gentle introduction to the theory of Diophantine equations, assuming only a moderate amount of elementary number theory and algebra. We will start by studing a few simple classes of equations (Pell type equations, ternary quadratic, and a few cubic equations) using only elementary number theory.  We will then introduce more specialized techniques (p-adic numbers, diophantine approximation) and apply them to concrete problems. At the end we will have a close look at elliptic curves and prove, among other things, Mordell's Theorem.

Requirements

Elementary number theory and algebra, corresponding to the course Elemente der Algebra.

For the administration of the exercise points (50% will be required for the admission to the exam) you register on the  Moodle page for the lecture.

• Date: March 6, 2018 from 9 a.m. to 1 p.m.
• Room: E12, Helmholtzstraße 18

• Silverman, J. und Tate, J.: Rational Points on Elliptic Curves, Springer
• Ireland and Rosen: A classical Introduction to Modern Number Theory
• S. Müller-Stach, J. Piontkowski: Elementare und Algebraische Zahlentheorie
• S. Wewers: Algebraic number theory, lecture notes WS 2013/14 (only chapter 1)