MATH524 THEORY OF FUNCTION FIELDS
| Course Code: | 2360524 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. ÖMER KÜÇÜKSAKALLI |
| Offered Semester: | Fall Semesters. |
Course Objectives
The objective of this course is to introduce students to the theory of algebraic function fields, with a focus on the arithmetic and geometric properties of curves over finite fields. The course develops key tools such as valuations, divisors, and differentials, leading up to a detailed study of the Riemann-Roch Theorem and its applications. Students will explore rational, elliptic, and hyperelliptic function fields, and examine the interplay between algebraic and arithmetic properties via the congruence zeta function and L-functions, including their associated functional equations. The course aims to build a conceptual and computational foundation for further study in algebraic geometry, number theory, or arithmetic geometry. There are many analogies between number theory over algebraic number fields and over algebraic function fields. One of our goals is to elaborate on these connections. The content of the course is relatively simple and essential to modern number theory and algebraic geometry.
Course Content
Valuations. Divisors, repartitions, differentials. Riemann-Roch Theorem. Rational function fields, elliptic and hyperelliptic function fields. Congruence zeta function, the functional equation for the L-functions.
Course Learning Outcomes
By the end of this course, students will be able to:
- Define and compute valuations on function fields and describe their properties.
- Understand and compute differentials and analyze their relationship to divisors.
- Use the Riemann-Roch Theorem to compute the dimension of spaces of functions and differentials associated with a divisor.
- Interpret the genus of a function field and relate it to geometric and arithmetic properties.
- Analyze and describe rational, elliptic, and hyperelliptic function fields.
- Define the congruence zeta function of a curve over a finite field and compute it in specific examples.
- State and apply the functional equation satisfied by the zeta function or L-function of a function field.
- Understand the significance of L-functions in number-theoretic and geometric contexts.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution