Quotient Rings

I hope I’ll have time to post more regular mathematics again soon.

As part of the BMC 2011 there will also be a small workshop on “Number Theory and Algebraic Geometry” organised by Minhyong Kim (UCL) and Frank Neumann (Leicester).The invited speakers for this workshop are:

Herbert Gangl (University of Durham)
Behrang Noohi (Kings College)
Lorenzo Ramero (University of Lille 1)
Andrei Yafaev (UCL)

Further information can be found here
.
If you plan to attend the BMC or the workshop, please register for the BMC through the general webpage.

The goal of the first part of the lectures is to define and get used to the definition of -categories proposed by Baez and Dolan in [1], and later modified by others. In the first lecture we mainly concentrated on defining opetopes, which will be the cell shapes used for the definition of an -category. Already at this point we did not (directly) take the approach of [1], which is to define the slicing operation on operads, and then apply this successively to the initial operad . However we might look at this approach later on. Instead we looked at the definition given in [2], which defines opetopes using complexes of combinatorial objects (similar to “meta-trees” in [1]). In the next lecture we plan to finish some of the discussion on opetopes, and to define (weak) -categories. To help understand the definition we will go through some examples and some low values of , for example working out how this gives a definition of a set for , of a (classical) category for . If there is time we might start looking at the result of Cheng [3], that opetopic 2-categories aren’t that different to bicategories.

[1] John C. Baez and James Dolan, Higher-Dimensional Algebra III: -Categories and the Algebra of Opetopes,
[2] Joachim Kock, Andr\’e Joyal, Michael Batanin, and Jean-Francois Mascari, Polynomial functors and opetopes.
[3] Eugenia Cheng, Opetopic bicategories: comparison with the classical theory.

James

James Haydon (UCL)

N-categories according to Baez and Dolan

on Thursday, 27, January, 2011.

All are welcome to attend.

The only difficulty is finding the seminar room: You must

Enter the Kathleen Lonsdale Building through the walkway from the north cloisters of the UCL main building (map). Go through the two sets of double doors and down the long corridor, then up to the second floor using the staircase or the lift. Dial my office number EXT. 31333 from the phone by the lift or the number of the seminar room M204, EXT. 33384 for access.

Contact: Qing Liu (Qing.Liu@math.u-bordeaux1.fr).
Topics: arithmetic geometry, algebraic geometry.
Salary: 2400 euros/month “brut”, 1958 euros/month “net”.
No teaching.

The former deadline to send the application file was the end of April but it is now postponed to mid-June.

The application file consists of:

- CV including a list of publications;
- electronic versions of the publications;
- research project;
- recommendation letters (to be sent directly by email).

and has to be sent to Qing Liu:

Qing.Liu@math.u-bordeaux1.fr

Since D_cris is not defined for Z_\ell-adic representations, Bloch-Kato finiteness condition cannot be defined at places over \ell via D_cris (or any analogous way) in general. It can be defined when the representation is Barsotti-Tate (with no assumption on ramification) and Fontaine-Laffaille case, and probably possible for crystalline representations at an unramified prime over \ell. Furthermore the definition of the local H^1_f(T) for a Galois stable lattice T in V is the preimage of H^1_f(V) in H^1(T). In particular, all torsion part of H^1(T) is in H^1_f(T). Thus, if “\ell is not p” and V is ramified, there could be some torsion class in H^1_f(T) which is in fact ramified. This is detected by the local Tamagawa number (which is why I didn’t discuss.)

And I do not know if T is Barsotti-Tate, the two possible definitions of H^1_f(T) should coincide… When T=Z_p(1) or T_p(A) for some abelian variety, they do coincide. In any case, what’s used in Fontain-Perrin-Riou is the one that contains all the torsion part of H^1(T).

At the end of the study group talk yesterday, Kevin Buzzard asked if I could see that for any non-zero element in the fundamental line \Delta_f (a 1-dimensional E-vector space) the canonical Euler-Poincare norm should be 1 at almost all places. Of course, the construction of the Euler-Poincare norm needs some serious conjectures, and we will assume them. By generalizing the example of abelian varieties too far, I guess I overlooked many (possible subtle) points. I don’t know which of them cause genuine difficulty and which of them are entirely due to my ignorance.

For the case when M=h^1(A)(1) where A is an abelian variety over a number field F, by choosing  Z-bases for the relevant Q-vector spaces (namely, Mordell-Weil groups of A and A^\vee, Lie algebra of the Neron model of A^/vee, homology lattice of the part fixed by “complex conjugation”), one obtain a non-zero element (call it \delta_0) of \Delta_f(M). One can compute te canonical norm of \delta_0 each rational odd(!) prime \ell, and the answer is expressed in terms of the sizes of the \ell-power torsion parts of A(F) and A^\vee, and (the \ell-part of) the Tamagawa number Tam^0(T_\ell), depending on the choice of the Lie algebra basis, where T_\ell is the Z_\ell-adic Tate module of A^\vee. (I’m following the notation of Fontaine-Perrin-Riou in II.4.5.) All of these numbers are well-known and are 1 for almost all \ell. (And with some more work, we can express the rational part of the L-value in terms of these.)

Now, take M=h^i(X)(r) for some smooth projective variety X over F, where my ignorance begins to reveal. My first question is whether one has a good replacement of Mordell-Weil groups; i.e., whether it is possible to “make a good sense of” motivic cohomology with integer coefficients with Bloch-Kato type finiteness condition and whether it’s finitely generated over the integer ring (thus, the torsion part is finite). I also need that the \ell-primary parts of integral motivic cohomology should map isomorphically onto the torsion part of H^i_f(T_\ell), where T_\ell is now the Z_\ell-adic realization (using Z_\ell-adic etale cohomology), or at least some product-formula-like cancellation of the ratios. My second question is whether the \ell-parts of the global Tamagawa numbers for T_\ell be 1 for almost all \ell. It seems plausible that something could be done on this, but I cannot see how.

So I end up asking the same question that Kevin Buzzard asked: whether it is possible, assuming all the conjecture needed to construct Euler-Poincare norms, to see that non-zero elements \Delta_f(M) should have norm 1 for almost all primes without assuming Bloch-Kato conjecture. Now, I won’t be surprised if the affirmative answer takes a serious work (or the answer is not affirmative?), though it is also quite possible that I’m just not seeing the obvious answer.

20/1 Deligne conjectures -Kevin Buzzard

27/1 Beilinson conjectures 1 – Andreas Holmstrom

3/2 Cancelled

10/2 Beilinson conjectures – 2 -Andreas Holmstrom

17/2 Bloch Kato – 1 – Cecilia Busuioc

24/2 Bloch-Kato- 2 – Wansu Kim

3/3 Bloch-Kato – 3 – Mahesh Kakde

10/3 Main conjecture (a la Kato) – Minhyong Kim

17/3 Noncommutative main conjecture -Mahesh Kakde