Computational Arithmetic - Geometry for Algebraic Curves

ABOUT THE COURSE: We want to count the number of roots of an algebraic system (over finite fields). This is a very difficult question in general(eg. #P-hard). However, there are fast algorithms known for special cases. In this course we will focus on the "2-variable" case, i.e. curves. This case already demands significant theory and has an amazing list of applications in computer science. We will cover some important aspects of this 20th century mathematics in a self-contained way, and see as many applications as time permits.INTENDED AUDIENCE: Computer Science & Engineering, Mathematics, Electronics, Physics, & similar disciplines.PREREQUISITES: A very good grasp in at least one of these: Algebra, Number theory, Geometry, Calculus, Analysis, or Topology,.INDUSTRY SUPPORT: Discrete Optimization, Cryptography/ Cyber Security, Coding theory, Computer Algebra, Symbolic Computing Software, Learning Software

Week 1:Algebraic-Geometry via Zariski Topology
Week 2:Variety—affine or projective. Rational Function and Germs.
Week 3:Morphism, Function and Birationality. Tangent Space and Smoothness.
Week 4:Resolving Singularity via Valuation. Discrete Valuation Ring (dvr).
Week 5:Abstract Curve and its Smooth Model.
Week 6:Approximate Valuation Theorem.
Week 7:Divisor Group. L-sheaf or Line Bundle.
Week 8:Principal Divisor and Degree. Class Group and Genus.
Week 9:Adele, Differential and the Riemann-Roch Theorem.
Week 10:Hasse-Weil Local Zeta Function. Functional Equation.
Week 11:Galois Cover and proving Weil’s Riemann Hypothesis.
Week 12:Character Sum. Torsion Point. Cohomology in the zeta function.