Is there something like a Lefschetz hyperplane formula for Arakelov motivic cohomology? More systematically, for what cohomologies and over which base schemes do we have Lefschetz???
In Soulé’s conjecture, there is a problem of choosing the measure giving the volume of the R/Z summands. Can we relate this measure to Deligne’s period conjecture, where the periods are supposed to be rather explicit things?
Does Mihailescu imply Vandiver? If so we know things about K-theory of rings of integers that wasn’t known before.
Conjectural ranks of motivic cohomology groups can be found in at least two ways. One is via Soule’s ICM 1983 conjecture on vanishing orders of Hasse-Weil zeta in terms of algebraic K-groups (compute vanishing orders numerically, assume that very very small numbers are equal to zero). The other is via the computation of Deligne cohomology.
We have a reasonable candidate for the abelian category of 1-motives, I think??? Can we use this or maybe something else such based on the Cisinski-Deglise paper, maybe by gluing arguments for non-fields, to make non-conjectural the framework of Scholbach in this case? Can we for example define and study the intermediate extension functor??? Note that most standard conjectures, maybe all, are known for curves.
Check proof of etale vanishing. Mimick?
One could hope for computations using descent. Not obvious to what extent descent is useful for computations. One situation where one could hope for usefulness is the situation of a regular model with some singular fibers, then one could blow up the fibers maybe and get smooth things, relating the two via some descent for blowups/normalizations. This could reduce some problems to the smooth case, maybe.
Another idea: Noetherian induction? See for example Quillen 4.1 for A1-invariance of K’-groups
Idea: Use Deligne 1-motives and the Orgogozo embedding to address at least elliptic curves. Or, try to work with a (triangulated or abelian) category of arithmetic 1-motives, which would contain the motive of all arithmetic surfaces, and maybe also all motives of products of curves/arithmetic surfaces. Can we get such a cat by gluing together cats of 1-motives over fields? Are cats of 1-motives known to have motivic t-structures, and can we then use this to get t-structure on arithmetic 1-motives? If so, can we use such a t-structure to prove some finiteness statements, or some algebraic cycle statements, or improve on the results in Jakob’s thesis, for example find an unconditional intermediate extension functor and interpret zeta functions and L-functions of arithmetic surfaces?
http://mathoverflow.net/questions/79740/cohomology-of-complete-intersections
Notes from Ramakrishnan survey: About the Beilinson conj: For varieties X over Q of positive dimension, every positive result (outside the critical values) have exhibited only a suitable lattice contained in the image of r, having the expected relation to the associated L-value, but nothing at all is known about the full image OR about the kernel. (Any result about the kernel would be very interesting.) In principle, maybe the K-group is too big for the conjecture? Or is rational equivalence too fine (in codim > 1)?? For homologically trivial cycles, the height pairing give rise to another equivalance, but is it strictly coarser?? These are very hard questions.
The arithmetic surface computation with real coeffs boils down do questions about K1. Ramakrishnan’s Hilbert-Blumenthal paper is found in CMS conf proc no 7, ISBN 0-8218-6012-7, check the library.
Use the formal properties of H-hat to compute things. One might hope for implications for properties of the regulator. Example: The regulator being and iso for all cellular varieties, when it should be so.
Note that my thesis gluing chapter should give a localization theorem for H-hat as well, reducing a computation over a number ring it to H-hat of varieties over finite fields and over number fields.
One could also try curves. See computation showing that only three groups should be nonvanishing in this case.
Another example: Could we give a new proof of Borel’s rank theorem using only the Spec Z case and Galois descent for Arakelov motivic cohom? For this, look maybe at qfh-descent instead? Or some related topology.
Note: TCMM Lemma 3.3.28, p 79 gives a case where the coohmology of the base injects into the cohomology of the top. This might indicate that descent in general allows to conclude things about the target scheme from knowing stuff on the top, so maybe descent is useless for going the other way around?? But maybe we can cover the top by nice things, and use descent.
Beilinson-Soule devissage argument by Kahn in K-th handbook, page 365.
For computations in K-theory, check Harder’s article mentioned by Jakob.
Jakob: The number ring example is Example 4.7. Using the localization sequence, I think one could push things a little bit further: given a regular proper model E / Z, almost all special fibers E_p / F_p are smooth. There are some results of Harder (Die Kohomologie S-arithmetischer Gruppen …) on K-theory of smooth projective curves over F_p, I think, this should help. Secondly, a few fibers are non-smooth, but the general theory of elliptic curves tells (I believe?) what types of singularities can occur. The normalization of the bad fibers are again smooth over F_p, so there will probably just individual additional summands \oplus Z in the K-theory groups, where the number of copies of the Z’s should depend on how many points get collapsed in the normalization map \widetilde{E_p} \r E_p of the bad fibers. I bet this idea is well-known to experts, but it could be instructive to spell out the localization sequence and see what remains.
Alg K-theory and CFT for arithmetic surfaces, Bloch, Annals. He should prove that CH^2(X) is finite, and so zero when tensored with the reals.
Jakob about elliptic curves: Use localization for K’-theory and the paper of Harder, maybe.
Notes, 25 May 2011. Kahn pp 364 seems to say that for B-S vanishing, the case of general regular schemes of finite type over Z can be reduced to the case of regular projective schemes which are generically smooth over a suitable ring of integers, and with strict semistable reduction outside the smooth locus. This uses the Gysin sequence, one step being that the conjecture is true for X iff it is true for some open U in X. It also uses the following: By de Jong, any regular scheme contains an open U such that there is a finite etale map V –> U such that V is contained in a scheme of the type described above (gen smooth, elsewhere semistable). Quote: “A transfer argument finishes the proof”.
References given by Kahn for a more detailed version of this devissage argument: 1. B. Kahn Some finiteness results in étale cohomology, J. Number Theory 99 (2003), 57–73. 2. T. Geisser Applications of de Jong’s theorem on alterations, in Resolution of Singularities In tribute to Oscar Zariski, Progr. in Math. 181, Birkhäuser, 2000, 299-314.
nLab page on Compute Arakelov motivic cohomology