On the arithmetic and geometry of binary Hamiltonian forms

@article{Parkkonen2011OnTA,
  title={On the arithmetic and geometry of binary Hamiltonian forms},
  author={Jouni Parkkonen and Fr'ed'eric Paulin},
  journal={Algebra \& Number Theory},
  year={2011},
  volume={7},
  pages={75-115}
}
Given an indefinite binary quaternionic Hermitian form f with coefficients in a maximal order of a definite quaternion algebra over 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 +∞. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using… 
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