SYLLABUS

Fundamentals.

- Algebraic curves & function fields
- The Riemann-Roch theorem
- Zeta function of curves over finite fields
- The relevant Riemann hypothesis and its proof

Applications.

- L-functions
- Exponential sums
- Counting points on curves (methods: torsion-based or p-adic)
- Integer factoring via curves
- Algebraic geometric codes
- Hyperelliptic curve cryptography
- Computing roots of unity in finite fields

Schedule.

[Apr-2023] [pdf] Exponential sums. Isogeny, torsion. Cohomological interpretation of the zeta function.

[Mar-Apr-2023] [pdf] Zeta function. Riemann Hypothesis proof.

[Feb-Mar-2023] [pdf] Divisors. Riemann-Roch Theorem.

[Feb-2023] [pdf] Algebraic aspects. Approximation Theorem.

[Jan-2023] [pdf] Introduction. Geometric aspects. Smooth iff DVR.