Given an indefinite binary quaternionic Hermitian form with coefficients in a maximal order of a definite quaternion algebra over , 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 by , as tends to . We compute the volumes of hyperbolic -manifolds 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 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.
Jouni Parkkonen. Frédéric Paulin. "On the arithmetic and geometry of binary Hamiltonian forms." Algebra Number Theory 7 (1) 75 - 115, 2013. https://doi.org/10.2140/ant.2013.7.75