Oeljeklaus–Toma manifolds and arithmetic invariants
- 115 Downloads
We consider Oeljeklaus–Toma manifolds coming from number fields with precisely one complex place. Our general theme is to relate the geometry to the arithmetic. We show that just knowing the fundamental group allows us to recover the number field. We also show that this fails if there are more complex places. The first homology turns out to be related to an interesting ideal. We compute the volume in terms of the discriminant and regulator of the number field. Is there a conceptual reason for this? We explore this and see what happens if we (entirely experimentally) regard them as “baby siblings” of hyperbolic manifolds coming from number fields.
Mathematics Subject Classification53C55
I would like to express my sincere gratitude to Victor Vuletescu for teaching me a lot of things, not all of them of mathematical nature. This note is a direct result of his inspiring ideas about the interplay of geometric and arithmetic conditions in Oeljeklaus–Toma manifolds. I also thank Chris Wuthrich for introducing me to SAGE. I would also like to thank the anonymous referee for a number of helpful remarks, which have led to a much clearer presentation.
- 2.Chinburg, T., Friedman, E., Jones, K.N., Reid, A.W.: The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Super. Pisa Cl. Sci. (4) 30(1), 1–40 (2001)Google Scholar
- 8.Gromov, M.: Hyperbolic manifolds (according to Thurston and Jørgensen). In: Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, , pp. 40–53. Springer, Berlin (1981)Google Scholar
- 17.Stein, W., et al.: Sage Mathematics Software (Version 4.7.2), The Sage Development Team. http://www.sagemath.org
- 19.The PARI Group, Bordeaux, PARI/GP version 2.7.0,2014. http://pari.math.u-bordeaux.fr/
- 23.Wall, C.T.C.: Geometries and geometric structures in real dimension \(4\) and complex dimension \(2\). Geometry and topology (College Park, Md., 1983/84), Lecture Notes in Math., vol. 1167, pp. 268–292. Springer, Berlin, (1985)Google Scholar