Arithmetic fake projective spaces and arithmetic fake grassmannians

We show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C. Wall) as the complex projective space of dimension n-1, and show that if n=5, there are at least four such subgroups, which are in fact torsion-free. This, in particular, leads to examples of a fake projective space of dimension 4. Analogous results for arithmetic fake grassmannians Gr(m,n) with n>3 odd are also obtained.Comment: 20 pages, the exposition has been improve

A CM construction for curves of genus 2 with p-rank 1

We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field \F_{p^2} of $p^2$ elements. The corresponding curves can be constructed using explicit CM constructions. In one of our algorithms, the group of \F_{p^2}-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over \F_{p^2} out of necessity: we show that curves of $p$-rank 1 over \F_p for large $p$ cannot be efficiently constructed using explicit CM constructions.Comment: 19 page

On rings of integers generated by their units

We give an affirmative answer to the following question by Jarden and Narkiewicz: Is it true that every number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? As a part of the proof, we generalize a theorem by Hinz on power-free values of polynomials in number fields.Comment: 15 page

Ideal class groups of cyclotomic number fields II

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions