Programs
- Fundamental Algebra and Analysis I
- Fundamental Algebra and Analysis II
- Geometry and Topology
- Abstract Algebra I
- Abstract Algebra II
- Ordinary Differential Equations
- Measure Theory and Real Analysis
- Probability Theory
- Functions of One Complex Variable
- Functional Analysis
- Numerical Analysis
- Category Theory
- Number Theory
- Algebraic Topology
- Representation Theory
- Differential Geometry and Differential Manifolds
- Partial Differential Equations
- Stochastic Processes
- Mathematical Modeling
- Numerical Algebra and Optimization
- Numerical Simulation
- Regression and Generalized Linear Models
- Scheme Theory
- Sheaf Theory and Cohomology
- Riemannian Geometry
- Elliptic Curves
- Topics in Several Complex Variables
- Geometric Measure Theory
- Harmonic Analysis
- Analytic Theory of Biological Dynamics
Elliptic Curves
Prerequisites:
Basic Algebraic Number Theory including p-adic numbers and algebraic number fields, Algebra including basic ommutative algebra, Galois theory and finite fields.
Course description:
Elliptic curves is a classical area of mathematics which has been further developed in the last 70 years. The theory combines geometric, analytic, arithmetic aspects and insights. Elliptic curves show up and are used in many areas of modern math research, in math physics, in coding and cryptography, in AI. Various famous diophantine problems can be interpreted and solved in terms of properties of elliptic curves over number fields.
The course presents main features of elliptic curves over complex numbers, including elliptic functions, elliptic curves over p-adic numbers, elliptic curves over algebraic number fields,
and related theories such as complex multiplication theory, elliptic surfaces, zeta- and L-functions of elliptic curves over number fields.
Learning objectives:
(a) to introduce the main strands of the theory of elliptic curves;, their objects, concepts, ideas;
(b) to teach various techniques and methods used in the study of elliptic curves and applications;
(c) to describe connections with areas of modern number theory.
By the end of the course, students should be able
(a) to have a good knowledge of the main constructions, results and topics about elliptic curves;
(b) to understand a wide range of concepts and methods used in the study of elliptic curves and their applications;
(c) to develop skills and techniques used in the study of elliptic curves and their applications.
Detailed topics covered:
Algebraic Curves. Riemann-Roch theorem.
Curves of genus 0. Curves of genus 1.
Weierstrass equation of elliptic curve.
Discriminant and j-invariant.
Legendre equation of elliptic curve.
Mordell-Weil theorem.
Complex multiplication theory.
Elliptic surfaces.
The Neron model of elliptic curve.
Zeta- and L-function of elliptic curve. Main problems and conjectures about elliptic curves.
References:
Joseph H. Silvemann, The arithmetic of elliptic curves, 2nd ed., Springer, 2016, ISBN 978-0-387-09493-9 e-ISBN 978-0-387-09494-6
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curvers, Springer 1994, ISBN 978-0-387-94328-2 ISBN 978-1-4612-0851-8 (eBook)