Download Filip Najman: Arithmetic geometry (60 HOURS) Arithmetic

Filip Najman: Arithmetic geometry (60 HOURS)
Arithmetic geomoetry is a branch of mathematics which uses the methods of algebraic geometry and applies it to problems coming from number
theory. In arithmetic geometry, we study the properties of the set of solutions of a polynomial equation or a set of polynomial equations, but over
”arithmetically interesting” fields, which are far from being algebraically
closed, such as the rational numbers or over finite fields.
In this course we will first introduce the necessary concepts from number
theory and commutative algebra: finite fields, p-adic fields, p-adic integers,
quadratic forms.
We will show the connection between curves and their function fields,
introduce divisors and prove the Rimeann-Roch theorem. We will use the
methods that we developed to prove results about elliptic curves, curves of
genus > 2 and Abelian varieties and show how these methods can be used
to solve Diophantine equations.
1. W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry,
W. A. Benjamin 1969.
2. J-P. Serre, A Course in Arithmetic. Springer-Verlag, 1996.
3. I. R. Shafarevich, Basic Algebraic Geometry I, Springer-Verlag, 2013.
4. H. Stichtenoth, Algebraic Function Fields and Codes, Springer, 2008.
5. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag,
2009.
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