1. CMB Online first
||Homology of the Fermat tower and universal measures for Jacobi sums|
We give a precise description of the homology group of the Fermat
curve as a cyclic module over a group ring.
As an application, we prove the freeness of the profinite homology
of the Fermat tower.
This allows us to define measures, an equivalent of Anderson's
adelic beta functions,
in a similar manner to Ihara's definition of $\ell$-adic universal
power series for Jacobi sums.
We give a simple proof of the interpolation property using a
motivic decomposition of the Fermat curve.
Keywords:Fermat curves, Ihara-Anderson theory, Jacobi sums
Categories:11S80, 11G15, 11R18
2. CMB 2006 (vol 49 pp. 21)
||Evaluation of the Dedekind Eta Function |
We extend the methods of Van der Poorten and Chapman
explicitly evaluating the Dede\-kind eta function at quadratic
irrationalities. Via evaluation of Hecke
$L$-series we obtain new evaluations at points in
imaginary quadratic number fields with
class numbers 3 and 4. Further, we overcome the limitations
of the earlier methods and via modular equations provide
explicit evaluations where the class number is 5 or 7.