Kolyvagin's Application of Euler Systems to Elliptic Curves

When: Spring 2000, 3 hours/week, TuTh 1-2:30

Where: MIT, Rm 2-102

**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.

- Introductory lecture (PS) or DVI
- Complex Multipication lectures (PS) or DVI
- Intro to Modular Curves and Heegner Points (PS) or DVI
- Characteristic Polynomials of Galois Actions on Torsion Points (PS) or DVI
- Hecke Correspondence, Eichler-Shimura Congruence (PS) or DVI
- Construction of Cohomology Classes (PS) or DVI
- Neron Models and Local Triviality of Classes (PS) or DVI
- Constructing the Local Pairing (PS) or DVI
- Applying the Local Pairing to Selmer Groups (PS) or DVI
- Concrete Selmer Group Manipulations (PS) or DVI

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)

- 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*

Course notes from Mazur's seminar, providing a good background, on Tom Weston's page.

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.