On the arithmetic and geometry of binary Hamiltonian forms

TL;DR

This paper investigates the asymptotic behavior of representations of integers by binary quaternionic Hermitian forms, utilizing hyperbolic geometry and Eisenstein series to compute volumes of related hyperbolic 5-manifolds.

Contribution

It provides a precise asymptotic count of integer representations by quaternionic forms and computes hyperbolic 5-manifold volumes using Eisenstein series and reduction theory.

Findings

Asymptotic equivalent for the number of representations of integers by quaternionic forms

Explicit volume calculations of hyperbolic 5-manifolds using Eisenstein series

Application of hyperbolic geometry to reduction theory of quaternionic 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 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.