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arXiv · 1105.2290

On the arithmetic and geometry of binary Hamiltonian forms

Jouni Parkkonen·Frédéric Paulin
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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 5-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 5 to describe the reduction theory of both definite and indefinite binary quaternionic Hermitian forms.

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Jouni Parkkonen, Frédéric Paulin. 2013-02-21. On the arithmetic and geometry of binary Hamiltonian forms. https://doi.org/10.1112/blms%2Fbdr041

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