DescriptionLet K be a field, which for simplicity we take of characteristic different to 2 and 3. Then an elliptic curve over K is a projective curve defined by an equation of the form ZY2 = X3 + aZ2X + bZ3, where a,b in K are such that 4a3 + 27b2 ≠ 0.A distinguished feature of elliptic curves is that their points have the structure of an abelian group. This fact makes the theory of elliptic curves extremely interesting and rich, with links to various important notions of mathematics such as modular forms, Galois representations, and sphere packing. Moreover, elliptic curves have important applications, for example in cryptography. The main goal of this project is to learn the basics of the theory of elliptic curves, as for example at the level of [1], and then specialize in one of the following directions (the list is by all means not complete).
Directions:The group of rational (resp. integral) points of an elliptic curve: The main aim of this direction is to investigate various results concerning the group of rational (resp. integral) points of an elliptic curve. A good starting point for this direction is Chapter VIII (resp. Chapter IX) of [1].Elliptic Curves with Complex Multiplication: An elliptic curve with "many" endomorphisms is said to have Complex Multiplication (CM). The theory of complex multiplication is very rich. The main aim here is to understand the so-called "Main Theorem of Complex Multiplication" and the construction of the Grossencharacters attached to an elliptic curve with CM. A good starting pointy for this is chapter 2 of [2]. Elliptic Curves and Sphere Packing: . The goal in this direction is to investigate the use of elliptic curves for obtaining dense sphere packings. The fundamental work here is of Elkies and Shioda. A good starting point is [5], Chapter IV and the references there. Applications of Elliptic Curves to Cryptography: There are interesting applications of elliptic curves to cryptography. For example one can use elliptic curves in order to factorize integers (an algorithm due to Lenstra) or for primality proving (an algorithm due to Goldwasher and Kilian). Moreover one can implement cryptographic protocols using elliptic curves (elliptic curve cryptography). As a starting point one can take the book of Koblitz [4]. Elliptic Curves and Galois Representations: An important property of ellipitic curves is that they give rise to representations of Galois groups. The aim of this direction is to understand various issues of this connection such as the notion of bad reduction, the Neron-Ogg-Shafarevich Criterion, the Hasse-Weil L-function. A good starting point is Chapter VII of [1]. Elliptic Curves and Modular Forms: Elliptic Curves are closely related to the theory of modular forms, through the famous Shimura-Taniyama-Weil conjecture. A possible aim of this direction is to learn some basics of the theory of modular forms, understand the conjecture and report on what is known. A good starting point is Chapter V of [5]. The Congruent Number Problem (hard!) This is a very old, easily stated, but still unsolved problem which goes back to ancient Greeks. The question is to determine which integer is the area of a right triangle with sides of rational length. The main goal here is to understand the work of Tunnell, which relates the problem to the Birch and Swinnerton-Dyer Conjecture, and report on recent progress. It requires the theory of modular forms (of half integral weight!). A good reference is the book of Koblitz [3].
ResourcesThere are many good books on elliptic curves. Here are some to start with:
[2] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, 1986 Springer. [3] N. Koblitz, Introduction to Elliptic Curves and Modular Forms , Springer, 1984. [4] N. Koblitz, A Course in Number Theory and Cryptography , Springer, 2nd edn, 1987. [5] J. Milne, Elliptic Curves, available online from http://www.jmilne.org/math/Books/ectext5.pdf. Prerequisites
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