6533b835fe1ef96bd129feb3
RESEARCH PRODUCT
Counting and equidistribution in Heisenberg groups
Frédéric PaulinJouni Parkkonensubject
Mathematics - Differential GeometryPure mathematicsGeneral MathematicsHyperbolic geometryMathematics::Number Theory[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]11E39 11F06 11N45 20G20 53C17 53C22 53C55chainEquidistribution theorem01 natural sciencesHeisenberg groupequidistributioncommon perpendicularIntegerLight cone0103 physical sciencesHeisenberg groupcubic point0101 mathematicsCygan distanceMertens formulaComplex projective planeMathematicsDiscrete mathematicsAMS codes: 11E39 11F06 11N45 20G20 53C17 53C22 53C55Mathematics - Number TheorySesquilinear formHeisenberg groups010102 general mathematicsHermitian matrixcomplex hyperbolic geometry[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT]sub-Riemannian geometry[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]counting010307 mathematical physicsdescription
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2014-02-28 |