The action of Aut(C) on symplectic K-theory of the integers

Soren Galatius, University of Copenhagen

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The symplectic group Sp2g(Z) is the automorphism group a free abelian group Z2g preserving a unimodular skew-symmetric form. These groups appear in topology in relation to mapping class groups of oriented 2-manifolds, and more generally from automorphisms of closed oriented manifolds of dimension 2 mod 4. They are perfect groups for g > 2, and the symplectic K-theory KSp*(Z) may be defined as homotopy groups of the Quillen plus construction of BSp2g(Z) in the direct limit where g goes to infinity. Work of Karoubi from the 1970s calculates these groups in terms of usual algebraic K-theory of Z, at least after completing at an odd prime p (much more recent work does this for p=2). For instance, KSp22(Z) completed at p=691 is Z/691 + Z691.
The groups Sp2g(Z) also arise in algebraic geometry, where they are related to uniformization of principally polarized abelian varieties. This relationship may be exploited to construct a model for p-completed KSp(Z) on which the group of field automorphisms of the complex numbers naturally acts. I will discuss recent work with T. Feng and A. Venkatesh (arXiv:2007.15078) in which we determine p-completed KSp*(Z) as a representation of Aut(C) for odd p. In the case of KSp22(Z) completed at p=691 for instance, we see that the action on Z/691 is trivial and the action on Z691 is cyclotomic, but that the representation is a non-split extension between the two.