We prove an operator-theoretic reconstruction of the Beilinson-Bloch regulator for compact Riemann surfaces, using loop operators and the Connes-Karoubi character for Fredholm modules. The proof includes a new computation of the Connes-Karoubi character for Steinberg symbols of the circle, which relies on the Helton-Howe determinant theory, but not on the Carey-Pincus theory of joint torsion.
A key computation in our work is the evaluation of the Connes--Karoubi character @math on Steinberg symbols @math . This is known to equal the joint torsion of Toeplitz operators [Prop. 1] Carey-Pincus:1999 , thanks to a combination of recent results of Kaad @cite_8 and Migler @cite_26 that enable the joint torsion, in this case, to be matched with the Connes--Karoubi character. (For details, see the remark at the end of .) However, since our work does not concern joint torsion, we choose to execute a direct, and considerably shorter, computation of @math , which employs the determinant theory of Helton and Howe @cite_4 , but is otherwise fairly elementary. The validity of the reconstruction formula for the full @math -group is, to the best of our knowledge, a new result.
@cite_26 @cite_4 @cite_8
1687736749 2075162834
A determinant in algebraic @math -theory is associated to any two almost commuting Fredholm operators. On the other hand, one can calculate a homologically defined invariant known as joint torsion. We answer in the affirmative a conjecture of Richard Carey and Joel Pincus, namely that these two invariants agree. In particular, this implies that joint torsion is norm continuous, depends only on the images of the operators modulo trace class, and satisfies the expected Steinberg relations. Moreover, we show that the determinant invariant of two commuting operators can be computed simply as a determinant on a finite dimensional vector space. In the context of 2-summable Fredholm modules, we prove that the Connes-Karoubi multiplicative character coincides with Brown's determinant invariant on algebraic K -theory. We study representation zeta functions of finitely generated, torsion-free nilpotent groups which are groups of rational points of unipotent group schemes over rings of integers of number fields. Using the Kirillov orbit method and @math -adic integration, we prove rationality and functional equations for almost all local factors of the Euler products of these zeta functions. We further give explicit formulae, in terms of Dedekind zeta functions, for the zeta functions of class- @math -nilpotent groups obtained from three infinite families of group schemes, generalizing the integral Heisenberg group. As an immediate corollary, we obtain precise asymptotics for the representation growth of these groups, and key analytic properties of their zeta functions, such as meromorphic continuation. We express the local factors of these zeta functions in terms of generating functions for finite Weyl groups of type @math . This allows us to establish a formula for the joint distribution of three functions, or statistics'', on such Weyl groups. Finally, we compare our explicit formulae to @math -adic integrals associated to relative invariants of three infinite families of prehomogeneous vector spaces.