Holmstrom Numerical investigation of zeta values

Check the formulations of Soule vs Flach-Morin. Flach said that the measure on Deligne cohom should come from integral de Rham cohomology.

Note that Tim Dokchitser mentioned some new stuff maybe in Magma, which should be useful for playing with zeta value computations.

It should be worthwhile to test Soule’s conjecture numerically - is it possible??? Check what would be involved and discuss at Sardinia maybe.

For which varieties can we compute Deligne cohomology easily? This should be a lot of varieties. Would be cool if we could include some positive-dimensional non-cellular examples. See Scholbach thesis, Lemma 2.6.2 (p 31) for short exact sequences with Betti and de Rham, in the smooth and proper case.

Concrete calculations relating to Haar measure???

For the idea of volumes, see also intro in Gillet, in Current trends volume (ed Ribet). It seems like Pic-hat can be described in terms of adeles in a way which makes one believe that there is a common root of Soule’s idea and the idea of Tamagawa numbers. See p 215.

Srinivas suggested after Lichtenbaums talk in Oberwolfach that one should be able to do some calculations. Here are some suggestions. “I don’t expect higher arithmetic Chow gps to be zero for these examples, or for toric varieties, but probably there are torsion phenomena.

• Families of rational surfaces or del Pezzo surfaces over Z. Take H1 of a model of such a variety.
• Cubic surface in P3 defined over Z, maybe more difficult
• First thing to try: Rank 2 projective module over ROI in a number field (this always has some nice structure, I think free thing plus something generated by something in the class gp), take projective bundle of this. Could also blowup something containing the closed fiber. Fibrewise, these (maybe all examples here) give Tate motives, but globally there is some glueing.
• K3 surfaces may be very hard
• Modular elliptic curve - could use knowledge of L-fn of modular forms/curves. Maybe for modular elliptic curves get things which could be birational to products of things.
• Maybe the Fermat quartic

From the Freiburg research proposal: Special values of zeta functions and L-functions should be related to stable homotopy groups of spheres (Salch). He has suggested in talks that in some sense chromatic calculations of height n should correspond to special values of L-functions attached to certain Galois representations of dimension n, the case n = 1 corresponding to the fact that Bernoulli numbers show up both in the image of the J-homomorphism and in the special values of the Riemann zeta function. The latter fact can also be “explained” by the fact that the K-theory spectrum of F1 should be the sphere spectrum, and that the Riemann zeta values are related to K-theory of the integers. A six functors formalism for SH, involving Spec F1 and Spec Z, should make it possible to make all this precise, and also study the case n=2, which would involve 2-dimensional Galois representations and elliptic cohomology theories.

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