# Graduate Number Theory Seminar:Kolyvagin's Application of Euler Systems to Elliptic Curves

## Practical Information

Organizers: Kiran Kedlaya (kedlaya@math.mit.edu) and Brian Osserman (osserman@math.mit.edu)
When: Spring 2000, 3 hours/week, TuTh 1-2:30
Where: MIT, Rm 2-102

## Seminar Description

Format: a semester-long seminar studying Kolyvagin's application of Euler systems to elliptic curves (i.e., to the conjecture of Birch and Swinnerton-Dyer). The seminar will be focused around the expository paper Kolyvagin's Work on Modular Elliptic Curves by Professor Gross. All participants will be expected to give lectures, and to prepare TeX lecture handouts. In spirit it will be a continuation of Professor Mazur's Math 254 at Harvard, but in practice people who did not take this should have no trouble participating.

Topics: Kolyvagin's actual proof, and background material in class field theory and the theories of complex multiplication, modular curves, and modular forms as appropriate. If time permits, we may look at aspects of Kolyvagin's work not covered in Gross' paper.

Prerequisites: A solid level of comfort with basic algebraic number theory and elliptic curve theory. Some familiarity with Galois cohomology and the main results of class field theory will probably be necessary, but a high degree of comfort will not be assumed.

## Lecture Schedule

Introduction - 1 lecture (2/3)
Overview of the main result, the argument for it, and the background material that goes into it.

Complex Multiplication - 2 lectures (2/8, 2/10)
Background: Introduction to the theory of lattices with complex multiplication. Definition of ring class groups and ring class fields, the j function, and a proof that it generates the ring class field.

Basics of modular curves and Heegner points - 2 lectures (2/15, 2/17)
Background: Introduction to modular curves as quotients of upper half plane, the j and j_N functions generate their field of functions, the modular equation, modular curves as moduli spaces, Weil parametrizations of elliptic curves, and the definition of Heegner points.
Paper: The construction of y_n (p. 238, from beginning of section 3)

Characteristic polynomials and restrictions on l - 1 lecture (2/24)
Background: Weil pairing to compute characteristic polynomials of Frob(l), complex conjugation on torsion points.
Paper: Basic pre-Euler system properties of y_n (pp. 239-240, through the reduction of prop 3.6 to prop 3.7)

Hecke correspondence, Eichler-Shimura congruence - 2 lectures (2/29, 3/2)
Background: An introduction to the basics of modular curve and modular form theory, including modular forms parametrizing elliptic curves, the Hecke correspondence and the Eichler-Shimura congruence.
Paper: The Euler system axioms (p. 240, prop 3.7)

Construction of cohomology classes - 1 lecture (3/7)
Paper:The construction of the various cohomology classes associated to the y_n. Triviality conditions for them (pp. 241-242, all of section 4).

Complex conjugation's action on the cohomology classes - 2 lectures (3/9, 3/14)
Background: L functions of elliptic curves, their functional equations, and the sign of said functional equations. The Fricke involution, and its action on Heegner points.
Paper: Analysis of the action of complex conjugation on the cohomology classes (pp. 243-244, all of section 5)

Local triviality of d(n) - 2 lectures (3/16, 4/4)
Background: Neron models, modular curves over finite fields
Paper: Proof that the d(n) are locally trivial at all places not dividing n, conditions for local triviality at places dividing n (pp. 245-246, all of section 6)

Derivation of local pairing - 2 lectures (4/6, 4/11)
Paper: Applications of Tate local duality and other results from Galois cohomology to construct a local pairing <,> between E(K_lambda)/pE(K_lambda) and H^1(K_lambda, E)[p] (pp. 247-248, all of section 7)

Sum of local invariants is 0, application - 1 lecture (4/13)
Background: A review of the results from global class field theory that the sum of local invariants is 0.
Paper: The proof that a cohomology class trivial at all but one place forces the Selmer group to be locally trivial at that place (pp. 248-249, all of secion 8)

Concrete Selmer group work - 1 lecture (4/20)
Paper: Construction of the pairing [,], relating it to both local and global vanishing of the Selmer group (pp. 250-252, all of section 9)

Finishing the theorem - 1 lecture (4/27)
Paper: Using Chebotarev's density theorem to find l satisfying all desired conditions, giving local vanishing of the Selmer group, which is then related to global bounds via the pairing [,] (pp. 252-254, all of section 10)

## References

Here are the relevant references for background material for the course (in order of when material will come up in the seminar):

• Basic elliptic curve theory: Joseph Silverman, Arithmetic of Elliptic Curves
• Complex multiplication theory, and the modular equation: David Cox, Primes of the Form x^2+ny^2
• Complex multiplication theory: Joseph Silverman, Advanced Topics in the Arithmetic of Elliptic Curves
• Modular curves, basics through Hecke correspondence and Eichler-Shimura congruence, and Galois cohomology: Cornell, Silverman, and Stevens, Modular Forms and Fermat's Last Theorem (essays by David Rohrlich and Joseph Silverman)
• Hecke correspondence and Eichler-Shimura congruence: Anthony Knapp, Elliptic Curves
• Class field theory: Cassels and Frohlich, Algebraic Number Theory
• Advanced results from Galois cohomology: J. S. Milne, Arithmetic Duality Theorems
Other available material includes:

Kiran's senior thesis, a full development of complex multiplication theory, available from his math page.

J. S. Milne has course notes on nearly all of the relevant topics, available online from his math page.

Also, for those interested in further reading on Euler systems, there is Karl Rubin's forthcoming book, already available online from his web site.

There is also a Galois cohomology cheat sheet (PS) or DVI.

If you have any other suggestions for references, please email me.