Kolyvagin's Euler system and Ciperiani-Wiles

UCLA NTG Participating Seminar:

Kolyvagin's Euler system for elliptic curves and its application in recent work of Ciperiani-Wiles

Fall 2007
Mondays, 3-4:15PM, MS 6627

In a recent preprint [CW], Ciperiani and Wiles begin a program whose aim it is to prove that any genus one curve C defined over Q has a solvable point. An important role in this is the investigation of the Tate-Shafarevich group Sha(E, Q) where E is the elliptic curve (over Q) that is the Jacobian of C.

The main aim of the seminar will be to study Kolyvagin's Heegner point Euler system and learn about Kolyvagin's proof, cf. [Kol1], [Kol2], of the finiteness of Sha(E, Q) provided that the so-called basic Heegner has infinite order, and its application to BSD. We will follow some notes by T. Weston [Wes] which in turn are based on the expository article [Gro] by B. Gross. The idea is to present the proof of Kolyvagin's result in as much detail a possible, also recalling some background on Galois cohomology, elliptic curves with complex multiplication and modular curves. The final portion of the seminar aims at covering some parts of the preprint [CW] of Ciperiani and Wiles.

The organization of the seminar was greatly facilitated by the notes of Weston [Wes1], by the notes from a seminar organized by Kedlaya and Osserman [Sem] and by Gross' expository article [Gro].

For a full description of the seminar, see the seminar flyer.

Gebhard Böckle's farewell lunch pictures

Practical Information

Organizers: Chandrashekhar Khare and Gebhard Böckle
Location: MS 6627
Time: Mondays, 3PM - 4:15PM

Lecture Schedule

Week
Date

Speaker & Description

1
October 1 Gebhard Böckle, Introduction and statement of BSD

Formulation of the conjecture of Birch and Swinnerton-Dyer, including a recall of the necessary invariants of an elliptic curve needed to state it. Statement of the main results of Kolyvagin. Deduction of the consequences for BSD modulo some auxiliary results due to Gross-Zagier, Waldschmidt, etc. (which we will not cover in the seminar). Maybe a very rough sketch of Kolyvagin's argument.

If time permits, I shall try to give some glimpses on the work of Çiperiani and Wiles.

2
October 8 Patrick Allen, Local Galois cohomology

This is [Wes], § 2. We will present the results in the same three steps Weston does it. That way, we see very clearly what's important. The aim of the talk will be to explain, motivate, prove, etc. some of these results.
Details Slides Audio

3
October 15 Tommaso Centeleghe, Global Galois cohomology and interpretations of Selmer groups

This talk has two goals. One is to introduce the notion of global Selmer groups, and the other is to interpret them in our situation as groups describing certain Galois extensions.
Details Slides Audio

4
October 22 Jared Weinstein, Bounding Selmer groups by local conditions

After the concrete interpretations above, we now want to bound the size of the Selmer groups S^a by restricting the Frobenius elements at suitable places. Most of the talk should be dedicated to the proofs of [Wes], 3.4 and 3.5, perhaps after briefly recalling the notation.
Details Slides Audio

5
October 29 Jared Weinstein, Reduction of the main result to the construction of certain cohomology classes

We will present the remaining parts of [Wes], § 4 (recalling various things from previous talks). The upshot is this: if, by some magic (= Kolyvagin), we can produce suitable cohomology classes, then the proof of Theorem 4.1 (our main goal) will be completed. The proof is thus 'reduced' to proving Props. 4.8 and 4.12.
Slides Audio

6
November 5 Craig Citro, Heegner points on modular curves

Heegner points will be introduced and their role in the explicit construction of points on elliptic curves over imaginary qudratic fields will be discussed. The present talk should lay foundations and cover [Wes], § 5, up to and including Proposition 5.1.
Details Slides Audio

7
November 14 Craig Citro, Heegner points on elliptic curves and Kolyvagin's Euler system

Note unusual date (due to Veteran's Day)!

The first aim of this talk to to complete the list of properties of the points y_n, namely [Wes], Prop. 5.2, and thereby explain the remainder of [Wes], § 5. The larger part of the talk should introduce Kolyvagin's Euler system for elliptic curves (formed by the classes c(n)) and prove its basic properties.
Details Slides Audio

8
November 19 Miljan Brakocevic, Unramifiedness of Kolyvagin's classes at bad primes of E

This talk should give the proof of Lemma 6.6 in the case where v divides the level. This is well-documented in [And], but might require further literature on Néron models (see also [Si2], § IV and references therein) and X₀(N) over Spec (Z). The proof in [Gro], Prop. 6.2 is quite short.
Details Slides Audio

9
November 26 Davide Reduzzi, Ramification of Kolyvagin's classes

Now comes a crucial part: Analyze the ramification of the classes c(n) at places dividing n. This is [Wes], § 6.3, except for Lemma 6.6. The main result is Lemma 6.8. One can also consult [Gro], Prop 6.2, and [And], Prop 2.2(2).
Slides Audio

10
December 3 Patrick Allen, The completion of Kolyvagin's proof and examples

Finish! It's now easy to prove the missing propositions and thus complete the results. The reference is [Wes], § 6.4. This is perhaps also a good point to review the entire proof! It would be very nice to give some details on the example in [Wes], § 7. Or to dig up some further examples, e.g. [Wat].
Slides Audio

11
December 10 Gebhard Böckle, The results of Ciperiani-Wiles I

Here's what I'll talk about.
Slides Audio

11
December 10 Gebhard Böckle, The results of Ciperiani-Wiles II

Here's what I'll talk about.
Slides Audio

Pictures of talks presented using William Stein's gallery sotware.

References

Some of the references are available here.

[And] F. Andreatta, A criterion for local triviality, in [SEM], under "Néron models and local triviality of classes."
[Cla] P. Clark, Lecture notes on Eichler-Shimura, in [SEM], under "Hecke correspondence, Eichler-Shimura correspondence"
[CW] M. Çiperiani, A. Wiles, Solvable points on genus one curves
[Cox] D. Cox, Primes of the form x² + ny²
[Dar] H. Darmon, Rational points on modular elliptic curves
[DI] F. Diamond, J. Im, Modular forms and modular curves, from Seminar on Fermat's Last Theorem, CMS Conference Proceedings vol. 17.
[Edi] B. Edixhoven, On the Manin constants of modular elliptic curves, from Arithmetic algebraic geometry (van der Geer, Oort, Steenbrink eds.), Birhaüser, 1991.
[Gro] B. Gross, Kolyvagin's work on modular elliptic curves, in L-functions and arithmetic (Coates, Taylor eds), CUP, 1991.
[Ke1] K. Kedlaya, Complex multiplication and explicit class field theory, available here.
[Ke2] K. Kedlaya, Complex multiplication lectures, in [SEM]
[Ko1] V. Kolyvagin, Finiteness of E(Q) and Sha(E, Q) for a subclass of Weil curves, (transl.) Math. USSR-Izv. 32 (1989), no. 3, 523--541.
[Ko2] V. Kolyvagin, The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves, (transl.) Math. USSR-Izv. 33 (1989), no. 3, 473--499.
[Kna] A. Knapp, Elliptic curves.
[NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields.
[Neu] J. Neukirch, Algebraic number theory.
[Pap] M. Papikian, On Tate local duality, in [SEM], under "Constructing the local pairing."
[SEM] MIT Seminar on Kolyvagin's application of Euler systems to elliptic curves.
[Si1] J. Silverman, The arithmetic of elliptic curves.
[Si2] J. Silverman, Advanced topics in the arithmetic of elliptic curves.
[Wat] M. Watkins, Some remarks on Heegner point comptutations.
[Wes1] T. Weston, The Euler system of Heegner points.
[Wes2] T. Weston, The modular curves X₀(N) and X₁(N).

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