Euler Systems — MA 841 Fall 2014 Professor Robert Pollack Office

Euler Systems — MA 841 Fall 2014

Professor Robert Pollack Oﬃce MCS 232 E-mail [email protected] Oﬃce hours M 3–4; W 11–12 Lecture MW 12–1:30; PSY B51

Course website http://math.bu.edu/people/rpollack/Teach/841fall2014.html

Course textbook: There is no textbook for this course. I will be drawing from a number of diﬀerent sources in preparing my lectures, but I will keep the course website up-to-date with the main references I am using for each lecture. Also, the end of this syllabus contains a list of many of the key references for the course.

Course Material: The Birch and Swinnerton-Dyer conjecture (one of the Clay Millenium Problems) relates the rank of an elliptic curve to the order of vanishing of its L-series at s = 1. The striking aspect of this conjecture is that the rank of an elliptic curve is an algebraic invariant defined by the minimal number of generators of the curve’s rational points. On the other hand, the L-series of an elliptic curve is an analytic invariant of the curve being a holomorphic function defined by an infinite product. And there lies the rub — to prove this conjecture one needs to find a way to pass information back and forth between the algebraic world and the analytic world. Although this conjecture remains open, it has been largely settled for curves of rank 0 and 1. The key tool used in proving this conjecture in these low rank cases is Euler systems – the primary object of study of this course. Euler systems sit somewhere in between the algebraic and analytic worlds. An element in an Euler system lives in Galois cohomology groups which makes these elements algebraic in nature. But from another viewpoint, these elements are analytic in nature as they “know” the special values of the L-series of the associated elliptic curve. It is precisely this dual algebraic/analytic role of Euler systems which makes them so useful in attacking questions like the Birch and Swinnerton-Dyer conjecture.

Course Format: In structuring the format of MA841, I am anticipating that the audience will be comprised of students with a wide range of backgrounds from first year students to students planning to graduate. For this reason, my general plan is to subdivide the course into three parts with the pace of the course increasing with each part. The first part will give a proof the “Weak Mordell-Weil Theorem”. The second part of the course will study the Euler system of Heegner points and sketch a proof of the implication “analytic rank 1 implies algebraic rank 1” (assuming the Gross-Zagier formula). The third part will study the more modern aspects of Euler systems. Possibilities include the relation of Euler systems to Kolyvagin systems, Kato’s Euler system, and the very new construction of Euler systems as in David Loeffler’s talk at Glennfest. In what follows is a sketch of the material in each part of the course. The length of each part of the course will depend greatly on the background of the students in the course. I would also like to emphasize that this outline is just a sketch. I will modify it (possibly significantly) depending on the audience of the class. PART ONE (“Weak Mordell-Weil”) (1) Overview of the basics of elliptic curves I will briefly discuss the basics of elliptic curves like the group law, the torsion sub- group, and the statement of the Mordell-Weil theorem. This course will not include a complete treatment of the basics of the arithmetic of elliptic curves in any sense — but I will try to outline the key points that will be needed for the course so that students unfamiliar with this material can take these as a blackbox to be opened and digested on their own time.

(2) Overview of Galois cohomology I will recall the construction of group (and Galois) cohomology via derived functors and also give explicit descriptions of the relevant H1 groups and explicit descriptions of all of the standard functorial maps (e.g. inﬂation, restriction and the connecting homomorphism). Again, this course will not be giving a complete treatment of the foundations of Galois cohomology, but will hopefully provide enough details and references for students to ﬁll in background on this material as needed during the course.

(3) Selmer groups and Tate-Shafarevich groups I will give a detailed description of Selmer and Tate-Shafarevich groups as in [5, Chapter X] and [1].

(4) Proof of the weak Mordell-Weil Theorem I will give a complete proof the weak Mordell-Weil theorem using Selmer groups and some basic facts from algebraic number theory.

PART TWO (“analytic rank 1 implies algebraic rank 1”) (1) Modular curves and modular parametrizations I will give an overview of the basics of modular curves as moduli spaces of elliptic curves over C with some level structure. We will also need some more delicate information about modular curves (e.g. that they are actually algebraic varieties over Q and have nice integral models), but much of these facts will be taken as black boxes for the course. We will also need to examine the modularity of elliptic curves; that is, each elliptic curve is a quotient of some modular curves.

(2) Heegner points I will review the theory of CM elliptic curves as in [6, Chapter 1] which give rise to special points on modular curves. The image of these special points under an elliptic curve’s modular parametrization gives rise to the Euler system of Heegner points. (3) Local and global duality I plan to cover the statements of Tate local duality (which is a generalization of local class ﬁeld theory) and Poitou-Tate global duality. This is a necessary ingredient to use Euler systems to bound the size of Selmer groups.

(4) Proof of “analytic rank 1 implies algebraic rank 1” I’ll give a sketch of this argument and give detailed proofs of the Euler system parts of the argument (as in [2]).

PART THREE (“Euler systems and beyond”) The details of the third part of this course will be hashed out later in the semester. Possibilities include studying material from [4] and [3]. Homework: Exercises will be assigned during the lecture as details to be ﬁlled in, challenge questions or explicit examples to be worked out. These exercises will be collected every two weeks. Grading: Each student will be assessed through their participation in the course and their completion of the homework sets.

References [1] R. Greenberg, Iwasawa theory for elliptic curves, in Arithmetic theory of elliptic curves (Cetraro, 1997), 51–144, Lecture Notes in Math., 1716, Springer, Berlin, 1999. [2] B. Gross, Kolyvagin’s work on modular elliptic curves. L-functions and arithmetic (Durham, 1989), 235–256, London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press, Cambridge, 1991. [3] B. Mazur, K. Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799. [4] K. Rubin, Euler systems and modular elliptic curves, in Galois representations in arithmetic algebraic geometry (Durham, 1996), 351–367, Cambridge Univ. Press, Cambridge, 1998. [5] J. Silverman, The arithmetic of elliptic curves. Second edition. Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009. [6] J. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994.