×

On the arithmetic of cross-ratios and generalised Mertens’ formulas. (English. French summary) Zbl 1358.11112

Summary: We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension \(\leq 5\). We prove generalisations of Mertens’ formula for quadratic imaginary number fields and definite quaternion algebras over \(\mathbb Q\), counting results of quadratic irrationals with respect to two different natural complexities, and counting results of representations of (algebraic) integers by binary quadratic, Hermitian and Hamiltonian forms with error bounds. For each such statement, we prove an equidistribution result of the corresponding arithmetically defined points. Furthermore, we study the asymptotic properties of crossratios of such points, and expand M. Pollicott’s recent results on the Schottky-Klein prime functions [“The Schottky-Klein prime function and counting functions for Fenchel double crosses”, Preprint].

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
11E39 Bilinear and Hermitian forms
11R52 Quaternion and other division algebras: arithmetic, zeta functions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Babillot (M.).— On the mixing property for hyperbolic systems. Israel J. Math. 129 p. 61-76 (2002). · Zbl 1053.37001
[2] Beardon (A. F.).— The geometry of discrete groups. Grad. Texts Math. 91, Springer-Verlag (1983). · Zbl 0528.30001
[3] Benoist (Y.) and Quint (J.-F.).— Random walks on finite volume homogeneous spaces. Invent. Math. 187 p. 37-59 (2012). · Zbl 1244.60009
[4] Cohn (H.).— A classical invitation to algebraic numbers and class fields. Springer Verlag (1978). · Zbl 0395.12001
[5] Cohn (H.).— A second course in number theory. Wiley, 1962, reprinted as Advanced number theory, Dover (1980). · Zbl 0208.31501
[6] Cosentino (S.).— Equidistribution of parabolic fixed points in the limit set of Kleinian groups. Erg. Theo. Dyn. Syst. 19 p. 1437-1484 (1999). · Zbl 0947.30033
[7] Dal’Bo (F.), Otal (J.-P.), and Peigné (M.).— Séries de Poincaré des groupes géométriquement finis. Israel J. Math. 118 p. 109-124 (2000). · Zbl 0968.53023
[8] Delgove (F.) and Retailleau (N.).— Sur la classification des hexagones hyperboliques à angles droits en dimension 5. This volume. · Zbl 1331.51018
[9] Deuring (M.).— Algebren. 2nd. ed., Erg. Math. Grenz. 41, Springer Verlag (1968). · Zbl 0159.04201
[10] Elstrodt (J.), Grunewald (F.), and Mennicke (J.).— Groups acting on hyperbolic space: Harmonic analysis and number theory. Springer Mono. Math., Springer Verlag (1998). · Zbl 0888.11001
[11] Fenchel (W.).— Elementary geometry in hyperbolic space. de Gruyter Stud. Math. 11, de Gruyter (1989). · Zbl 0674.51001
[12] Gorodnik (A.) and Paulin (F.).— Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. J. Modern Dynamics 8 p. 25-59 (2014). · Zbl 1351.37012
[13] Gorodnik (A.) and Paulin (F.).— In preparation.
[14] Grotz (W.).— Mittelwert der Eulerschen \(\varphi \)-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern. Monatsh. Math. 88 p. 219-228 (1979). · Zbl 0413.12012
[15] Harcos (G.).— Equidistribution on the modular surface and L-functions. In “Homogeneous flows, moduli spaces and arithmetic”, M. Einsiedler et al eds., Clay Math. Proc. 10, Amer. Math. Soc., p. 377-387 (2010). · Zbl 1267.11057
[16] Hardy (G. H.) and Wright (E. M.).— An introduction to the theory of numbers. Oxford Univ. Press, sixth ed. (2008). · Zbl 1159.11001
[17] Huxley (M. N.).— Exponential sums and lattice points III. Proc. London Math. Soc. 87 p. 591-609 (2003). · Zbl 1065.11079
[18] Kellerhals (R.).— Quaternions and some global properties of hyperbolic 5-manifolds. Canad. J. Math. 55 p. 1080-1099 (2003). · Zbl 1054.57019
[19] Kim (I.).— Counting, mixing and equidistribution of horospheres in geometrically finite rank one locally symmetric manifolds. Preprint [arXiv:1103.5003], to appear in J. reine angew. Mathematik. · Zbl 1319.53051
[20] Krafft (V.) and Osenberg (D.).— Eisensteinreihen für einige arithmetisch definierte Untergruppen von \(\operatorname{SL}_2({\mathbb{H}})\). Math. Z. 204 p. 425-449 (1990). · Zbl 0725.11024
[21] Lang (S.).— Algebraic number theory. Grad Texts Math. 2nd ed., Springer Verlag, 1994. · Zbl 0811.11001
[22] Maclachlan (C.) and Reid (A.).— Parametrizing Fuchsian subgroups of the Bianchi groups. Canad. J. Math. 43 p. 158-181 (1991). · Zbl 0739.20020
[23] Mohammadi (A.) and Oh (H.).— Matrix coefficients, counting and primes for orbits of geometrically finite groups. Preprint [arXiv:1208.139], to appear in J. European Math. Soc.
[24] Nagell (T.).— On the number of representations of an A-number in an algebraic field. Ark. Mat. 4 p. 467-478 (1962). · Zbl 0107.04001
[25] Odoni (R. W. K.).— Representations of algebraic integers by binary quadratic forms and norm forms from full modules of extension fields. J. Numb. Theo. 10 p. 324-333 (1978). · Zbl 0411.12010
[26] Oh (H.).— Orbital counting via mixing and unipotent flows. In “Homogeneous flows, moduli spaces and arithmetic”, M. Einsiedler et al eds., Clay Math. Proc. 10, Amer. Math. Soc., p. 339-375 (2010). · Zbl 1343.11062
[27] Oh (H.) and Shah (N.).— Equidistribution and counting for orbits of geometrically finite hyperbolic groups. J. Amer. Math. Soc. 26 p. 511-562 (2013). · Zbl 1334.22011
[28] Oh (H.) and Shah (N.).— Counting visible circles on the sphere and Kleinian groups. Preprint [arXiv:1004.2129], to appear in “Geometry, Topology and Dynamics in Negative Curvature” (ICM 2010 satellite conference, Bangalore), C. S. Aravinda, T. Farrell, J.-F. Lafont eds, London Math. Soc. Lect. Notes. · Zbl 1368.52013
[29] Otal (J.-P.) and Peigné (M.).— Principe variationnel et groupes kleiniens. Duke Math. J. 125 p. 15-44 (2004). · Zbl 1112.37019
[30] Parker (J.) and Short (I.).— Conjugacy classification of quaternionic Möbius transformations. Comput. Meth. Funct. Theo. 9 p. 13-25 (2009). · Zbl 1158.15014
[31] Parkkonen (J.) and Paulin (F.).— Prescribing the behaviour of geodesics in negative curvature. Geom. & Topo. 14 p. 277-392 (2010). · Zbl 1191.53026
[32] Parkkonen (J.) and Paulin (F.).— Spiraling spectra of geodesic lines in negatively curved manifolds. Math. Z. 268 (2011) 101-142, Erratum: Math. Z. 276 p. 1215-1216 (2014). · Zbl 1228.53055
[33] Parkkonen (J.) and Paulin (F.).— On the representations of integers by indefinite binary Hermitian forms. Bull. London Math. Soc. 43 p. 1048-1058 (2011). · Zbl 1255.11022
[34] Parkkonen (J.) and Paulin (F.).— Équidistribution, comptage et approximation par irrationnels quadratiques. J. Mod. Dyn. 6 p. 1-40 (2012). · Zbl 1371.37009
[35] Parkkonen (J.) and Paulin (F.).— On the arithmetic and geometry of binary Hamiltonian forms. Appendix by Vincent Emery. Algebra & Number Theory 7 p. 75-115 (2013). · Zbl 1273.11065
[36] Parkkonen (J.) and Paulin (F.).— Skinning measure in negative curvature and equidistribution of equidistant submanifolds. Erg. Theo. Dyn. Sys. 34 p. 1310-1342 (2014). · Zbl 1321.37029
[37] Parkkonen (J.) and Paulin (F.).— Counting arcs in negative curvature. Preprint [arXiv:1203. 0175], to appear in “Geometry, Topology and Dynamics in Negative Curvature” (ICM 2010 satellite conference, Bangalore), C. S. Aravinda, T. Farrell, J.-F. Lafont eds, London Math. Soc. Lect. Notes.
[38] Parkkonen (J.) and Paulin (F.).— Counting common perpendicular arcs in negative curvature. Preprint [arXiv:1305.1332]. · Zbl 1375.37104
[39] Parkkonen (J.) and Paulin (F.).— Counting and equidistribution in Heisenberg groups. Preprint [arXiv:1402.7225]. · Zbl 1360.11067
[40] Pollicott (M.).— The Schottky-Klein prime function and counting functions for Fenchel double crosses. Preprint (2011).
[41] Roblin (T.).— Ergodicité et équidistribution en courbure négative. Mémoire Soc. Math. France, 95 (2003). · Zbl 1056.37034
[42] Sarnak (P.).— Reciprocal geodesics. Clay Math. Proc. 7 p. 217-237 (2007). · Zbl 1198.11039
[43] Serre (J.-P.).— Lectures on \(N_X(p)\). Chapman & Hall/CRC Research Notes Math. 11, CRC Press (2012). · Zbl 1238.11001
[44] Shimura (G.).— Introduction to the arithmetic theory of automorphic functions. Princeton Univ. Press (1971). · Zbl 0872.11023
[45] Vignéras (M. F.).— Arithmétique des algèbres de quaternions. Lect. Notes in Math. 800, Springer Verlag (1980). · Zbl 0422.12008
[46] Walfisz (A.).— Weylsche Exponentialsummen in der neueren Zahlentheorie. VEB Deutscher Verlag der Wissenschaften (1963). · Zbl 0146.06003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.