Elliptic curves and number theory
This course provides a comprehensive introduction to the arithmetic theory of elliptic curves and their deep connections with number theory. Topics include the basic theory of elliptic curves over various fields, the group law, torsion points, isogenies, and reduction modulo primes. Special emphasis is placed on elliptic curves over number fields, and on their applications in Diophantine equations and modern cryptography.
Key highlights include the Mordell-Weil theorem, the structure of the rational points, complex multiplication, the theory of heights, and the role of elliptic curves in proving Fermat’s Last Theorem via modularity. Students will also explore computational aspects, such as point counting over finite fields and algorithms used in elliptic curve cryptography. Throughout the course, classical results and modern perspectives will be blended to offer a unified view of the subject.
This course is ideal for students with a background in algebra, algebraic number theory, and a basic understanding of algebraic geometry. It prepares students for research in number theory, arithmetic geometry, and cryptography.
Key highlights include the Mordell-Weil theorem, the structure of the rational points, complex multiplication, the theory of heights, and the role of elliptic curves in proving Fermat’s Last Theorem via modularity. Students will also explore computational aspects, such as point counting over finite fields and algorithms used in elliptic curve cryptography. Throughout the course, classical results and modern perspectives will be blended to offer a unified view of the subject.
This course is ideal for students with a background in algebra, algebraic number theory, and a basic understanding of algebraic geometry. It prepares students for research in number theory, arithmetic geometry, and cryptography.
Lecturer
Date
16th September ~ 11th December, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday,Tuesday,Thursday | 09:50 - 11:25 | A3-3-201 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Basic algebraic number theory, linear algebra, basis algebraic geometry
Reference
1. J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer, 2nd Edition, 2009.
2. J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer, 1994.
3. L. Washington, Elliptic Curves: Number Theory and Cryptography, Chapman & Hall/CRC, 2nd Edition, 2008.
4. N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics 97, Springer, 1993.
5. J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.
6. J. Neukirch, Algebraic Number theory.
7. R. Hartshorne, Algebraic Geometry. GTM52
2. J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer, 1994.
3. L. Washington, Elliptic Curves: Number Theory and Cryptography, Chapman & Hall/CRC, 2nd Edition, 2008.
4. N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Graduate Texts in Mathematics 97, Springer, 1993.
5. J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer, 1992.
6. J. Neukirch, Algebraic Number theory.
7. R. Hartshorne, Algebraic Geometry. GTM52
Audience
Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
Chinese