6533b873fe1ef96bd12d4da9
RESEARCH PRODUCT
On the arithmetic and geometry of binary Hamiltonian forms
Frédéric PaulinJouni Parkkonensubject
AMS : 11E39 20G20 11R52 53A35 11N45 15A21 11F06 20H10representation of integersHyperbolic geometry20H10Geometry15A2101 natural sciencesHyperbolic volume[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR]11E39 20G20 11R52 53A35 11N45 15A21 11F06 20H10symbols.namesake11E390103 physical sciencesEisenstein seriesCongruence (manifolds)group of automorphs0101 mathematics20G20Quaternion11R52[MATH.MATH-GR] Mathematics [math]/Group Theory [math.GR]Mathematicsreduction theoryDiscrete mathematicsAlgebra and Number TheoryQuaternion algebraMathematics - Number TheorySesquilinear formta111010102 general mathematicsHamilton-Bianchi groupHermitian matrix53A35[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]11F06[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]symbols010307 mathematical physicsMathematics::Differential Geometry[MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG]Hamilton–Bianchi group11N45binary Hamiltonian formhyperbolic volume[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]description
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2011-05-11 |