On the arithmetic and geometry of binary Hamiltonian forms
Abstract
Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.
 Publication:

arXiv eprints
 Pub Date:
 May 2011
 arXiv:
 arXiv:1105.2290
 Bibcode:
 2011arXiv1105.2290P
 Keywords:

 Mathematics  Number Theory;
 11E39;
 20G20;
 11R52;
 53A35;
 11N45;
 15A21;
 11F06;
 20H10
 EPrint:
 With an appendix by Vincent Emery. Revised version