## MATH-GA 2150.001 Advanced Topics in Algebra :Introduction to Algebraic Geometry and Elliptic Curves

Fall 2015, 5:10pm-7pm, WWH 517

Office hours : Tuesdays, 5:10pm-7pm or by appointment

Prerequisites: Elements of linear algebra and the theory of rings and fields.

The objects of study in algebraic geometry are systems defined by polynomial equations. Here are some examples:
1. x^2+y^2+z^2-w^2=0;
2. y^2=x^3-2x
3. x^2+y^2=u^2, x^2+z^2=v^2, y^2+z^2=w^2, x^2+y^2+z^2=t^2.

The first example gives a projective quadric; the second one defines an elliptic curve. Rational solutions of the third system of equations provide a rectangular box such that the lengths of the edges, face diagonals, and long diagonals are rational numbers; the existence of of a nontrivial solution to these equations is still unknown.

We will start this introductory course with some topics from the commutative algebra, such as ideals in polynomial rings and the famous Nullstellensatz theorem. We will also discuss some projective geometry in dimension two. A large part of the course will be devoted to the study of elliptic curves over various fields: finite fields, fields of rational or complex numbers. For elliptic curves defined over finite fields we will also discuss applications to cryptography.

### Homework

• Homework after the lecture on September, 9, 2015, answers
• Homework after the lecture on September, 16, 2015.
• Homework after the lecture on September, 30, 2015.
• Homework after the lecture on October, 7, 2015.
• Homework after the lecture on October, 14, 2015.
• Homework after the lecture on November, 4, 2015.
• Homework after the lecture on November, 11, 2015.
• Homework after the lecture on November, 18, 2015.

### References :

• D. Perrin, Algebraic geometry: an introduction, Springer-Verlag London, 2008
• M. Hindry, Arithmetics, Springer, London, 2011
• K. Ireland, M. Rosen, A classical introduction to modern number theory, Springer-Verlag, New York, 1990
• L. Washington, Elliptic curves: Number theory and cryptography, Chapman and Hall, 2008
• J. Silverman, The arithmetic of elliptic curves, Springer, Dordrecht, 2009.