In this paper, we introduce certain Krein-space operators induced by free product algebras induced by both primes and directed graphs. We study operator-theoretic properties of such operators by computing free-probabilistic data containing number-theoretic data.
We generalize Kronecker’s solution of Pell’s equation to CM fields whose Galois group over is an elementary abelian 2-group. This is an identity which relates CM values of a certain Hilbert modular function to products of logarithms of fundamental units. When is imaginary quadratic, these CM values are algebraic numbers related to elliptic units in the Hilbert class field of . Assuming Schanuel’s conjecture, we show that when has degree greater than 2 over these CM values are transcendental....
In this note we give a new proof of the theorem of Kronecker-Weber based on Kummer theory and Stickelberger’s theorem.
Using Hausmann and Vogel's homology sphere bundle interpretation of algebraic K-theory, we construct K-theory invariants by a theory of characteristic classes for flat bundles. It is shown that the Borel classes are detected this way, as well as the rational K-theory of integer group rings of finite groups.