Meeting with Mazur

William A. Stein

29 December 2000

Abstract

In this note, I summarizes the content of a meeting between Mazur and I this afternoon.

We discussed three key topics.

1. My definition of visibility of Mordell-Weil groups.

2. The Beilinson-Mazur notion of motivic nature of Galois cohomology of finite modules.

3. The paper [Merel-Stein] The field generated by the points of small prime order on an elliptic curve

I’ll discuss the key points of our conversation on each of these topics in the following three sections.

1 Visibility of Mordell-Weil

After explaining my definition and mentioning some questions, we discussed ”level raising nonvanishing”. This lead to a discussion about modular symbols mod-p. We thought about modular symbols as functions P^{1}(Q) → R, where R is some additive group. This is the cohomological point of view.

Then we talked about deformation theory for modular symbols. Given a modular symbol modulo p, who can one classify all lifts to a modular symbol modular p^{2} of some level? This lead to some questions. What is the subspace of Map(P^{1}(Q) → Z/pZ) generated by all modular symbols? Is it dense in the uncountable space of all such maps? Here the topology is such that points are close if all components of large height of their difference are 0. Is the space of modular symbols a vector space (of course?). Given a function on a million points, does it extend to a modular symbol?

These questions suggest a possible approach to level-raising nonvanishing. Try to extend a piece of the symbol mod p^{2} in such a way that the the special value is nonzero but the extended symbol is congruent mod p to the original symbol. I have no idea if this could ever work, but at least it is an idea!

2 Beilinson-Mazur cohomology

Next we talked about Beilinson-Mazur. First, some background. Start with all great recent K-theory results of people like Suslin and Vojovasky. Let’s summarize. What are they all telling us? One gets a surprising amount of H^{!}(K, −−) = galois cohomology that is accessible via K-theory.

Let’s compare galois cohomology and K theory, as the Milnor conjecture does. On the one hand, galois cohomology is in some sense shallow. It depends only on the Galois group and is somewhat elementary and topological. In contrast, K-theory is much deeper and more algebraic in flavor, behaving well with respect to base change. Kato, in constructing his Euler system, goes from K-theory to Galois cohomology.

Suslin and Vojovsky connect Étale cohomology with torsion to K-theory with torsion.

Beilinson: Maybe this whole issue has to do with ”finite motives”. For a long time Beilinson has been trying to find a definition of finite motives. We have usual motives, which are defined over Q and are constructed from algebraic varieties. But how do you define something like μ_{p} in a motive context? Here is Beilinson’s latest idea. Take an isogeny – the isogeny is part of the structure of the finite motive.

μ_{p} → E → F

Is motivic cohomology enough to give Galois cohomology? Follow chance observation that there’s a lot of visible stuff out there.

[[Somewhere in my notes I have a definition of the motivic cohomology of μ_{p} presented as above... You use points over fields L for which μ_{p}(L) = 1. ]] Suggestion: Let’s try to determine if motivic cohomology of μ_{p} (as presented) equals Galois cohomology over other fields.

1. Finite fields – everything is trivial

2. Local fields – surely it’s true, but actually do it! instead of squinting at it from a distance.

We’ll conclude that there is no local obstruction; thus if it is false that Galois and motivic are equal, then it’s false for deep reasons.

Let’s take as a first example μ_{3}. We probably can’t computationally prove that all of the cohomology is motivic, but we can at least try to catch one random element. Let’s take our random element to by 2 ∈ Q^{∗}/(Q^{∗})^{3}.

WARNING: I see no application. We are realizing the philosophy that ”Motivic elements RULE.” How can this be if there isn’t a motivic way to think about all of Galois cohomology?

3 Merel-Stein

Mazur had a number of comments regarding the paper by Merel and me. Here they are:

1. The title of section 1 could be much better. It’s a waste as it is, because it could be the first sentence. Inform!

2. Which Hecke operators (refering to the second sentence).

3. In the statement of Proposition 1, we should either put C^{∗} or take the star off of Z/pZ.

4. Before stating Proposition 1, say: “We prove blah [statement of prop], and this conclusion is contrary to fact, as we will see in Proposition 2.”

5. Reword the sentence in the proof that begins “According to Corollary 3”

6. Make sure, when refering to Kato’s theorem, that the discussion in section 1.5 of Loic’s other paper properly refers to the correct section of Scholl’s paper.

7. Delete the Remark 1 after the proposition because it is so unclear, and I don’t even know what it means. There are many possible interpretations.

8. Rephrase Proposition 2 to say: “There exists a cyclic subgroup C such that (E, C) does not descend to Q(\sqrt{p}), in the sense that ————–. [[Mazur said something about “every cyclic subgroup descends”...]]

9. “We show first it is” should be “We show first that it is”.

10. The CM argument doesn’t make sense to either Mazur or me! What if you twist E? Then the field of definition of the torsion BLOWS UP! It can’t be contained in L_{p}.

11. Long ago Mazur wrote an email to Cassels (or SD?) and proved that if E has CM then #E(Q) divides 12, or something like that.

12. In proof of Prop 2, say that C′_{1} is defined over Q(\sqrt{p}).

13. The paragraph “Let C_{1} be a cyclic...” is badly written. Fix.

14. Put the argument for the non-CM case first!

15. In section 3, I haven’t defined S_{2}(Γ_{0}(p); Z) yet!

16. In “Verifying condition 2”, I should write Q[χ] instead of Q(χ) and define it before using it instead of defining it after using it!

17. Get rid of the comma in the definition of χ-twisted winding element.

18. In “Verifying condition 3”, change “except p = 389” to “except possibly p = 389”.