We will cover the topics necessary to start research in arithmetic geometry. This course will give an overview to the following topics: 1. Fppf descent theory. 2. Algebraic groups. 3. Étale cohomology and Hochschild-Serre spectral sequence. 4. Brauer groups and Brauer-Manin obstruction to the existence of rational points.

This course serves as an introduction to arithmetic geometry, which combines algebras, number theory and algebraic geometry. It is a fruitful, fancy and fascinating topic and draws a lots of interests in current research . We aim to equip students with basic tools for further research in this area.

Students are assumed to be familiar with basic language of algebraic geometry (including chapter II,III of Hartshorne's book)

B. Poonen, Rational points on varieties.

U. Görtz and T. Wedhorn, Algebraic Geometry I and II. S. Bosch, W. Lütkebohmert and M. Raynaud, Néron model. B. Conrad, https://math.stanford.edu/~conrad/papers/adelictop.pdf L. Fu, Etale cohomology theory. J.-L. Colliot-Thélène and A. Skorobogatov, The Brauer-Grothendieck groups P. Gille and T. Szamuely, Central simple algebras and Galois Cohomology