The first part of this course is an introduction to algebraic geometry, concentrating on algebraic curves. We will study affine and projective algebraic varieties, Hilbert’s Nullstellensatz, the local structure of algebraic curves, divisors, differentials and the Riemann–Roch theorem.
In the second part of the course we will study the zeta function of a curve over a finite field, proving rationality, the functional equation, and the Riemann hypothesis.
There will be some homework sets, which will count for 25% of the final grade. The final examination will count for 75% of the final grade.
Algebra 1, 2, 3.
The main reference for the first part of the course will be W. Fulton, Algebraic Curves, available freely online here.
Other books covering the material include: M. Reid, Undergraduate Algebraic Geometry, available here.
J. Silverman, The Arithmetic of Elliptic Curves (chapters 1 and 2 only)
For the second part of the course, lecture notes will be provided.