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January  2014, 8(1): 25-59. doi: 10.3934/jmd.2014.8.25

Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows

Alexander Gorodnik 1, and  Frédéric Paulin 2,

1. 

School of Mathematics, University of Bristol, Bristol BS8 1TW
2. 

Département de mathématique, UMR 8628 CNRS, Bât. 425, Université Paris-Sud, 91405 ORSAY Cedex, France

Received  June 2013 Published  July 2014

In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral points on sections of quadrics. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.
Keywords: homogeneous variety, counting, norm form, exponential decay of correlation., Siegel weight, diagonalizable flow, Integral point, mixing, decomposable form.
Mathematics Subject Classification: Primary: 37A17, 37A45; Secondary: 14M17, 20G20, 14G05, 11E2.
Citation: Alexander Gorodnik, Frédéric Paulin. Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows. Journal of Modern Dynamics, 2014, 8 (1) : 25-59. doi: 10.3934/jmd.2014.8.25
References:
[1]

T. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer Verlag, New York-Heidelberg, 1976.  Google Scholar

[2]

M. Babillot, Points entiers et groupes discrets: De l'analyse aux systèmes dynamiques, in Rigidité, Groupe Fondamental et Dynamique, Panor. Synthèses, 13, Soc. Math. France, Paris, 2002, 1-119.  Google Scholar

[3]

B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), New Math. Mono., 11, Cambridge Univ. Press, 2008. doi: 10.1017/CBO9780511542749.  Google Scholar

[4]

Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble), 62 (2012), 1889-1942. doi: 10.5802/aif.2738.  Google Scholar

[5]

A. Borel, Ensembles fundamentaux pour les groupes arithmétiques, in Colloque sur la Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier-Villars, Paris, 1962, 23-40.  Google Scholar

[6]

A. Borel, Introduction aux Groupes Arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.  Google Scholar

[7]

A. Borel, Linear Algebraic Groups, 2nd edition, Grad. Texts Math., 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.  Google Scholar

[8]

A. Borel, Reduction theory for arithmetic groups, in Algebraic Groups and Discontinuous Subgroups (eds. A. Borel and G. D. Mostow) (Proc. Sympos. Pure Math. Boulder, Colo., 1965), Amer. Math. Soc., 1966, 20-25.  Google Scholar

[9]

A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2), 75 (1962), 485-535. doi: 10.2307/1970210.  Google Scholar

[10]

A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[11]

M. Borovoi and Z. Rudnick, Hardy-Littlewood varieties and semisimple groups, Invent. Math., 119 (1995), 37-66. doi: 10.1007/BF01245174.  Google Scholar

[12]

L. Clozel, Démonstration de la conjecture $\tau$, Invent. Math., 151 (2003), 297-328. doi: 10.1007/s00222-002-0253-8.  Google Scholar

[13]

H. Cohn, A Second Course in Number Theory, Wiley, 1962; reprinted as Advanced Number Theory, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1980.  Google Scholar

[14]

J.-L. Colliot-Thélène and F. Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compositio Math., 145 (2009), 309-363. doi: 10.1112/S0010437X0800376X.  Google Scholar

[15]

M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples, in Analyse Harmonique sur les Groupes de Lie (Sém. Nancy-Strasbourg 1976-1978), II, Lect. Notes Math., 739, Springer, Berlin, 1979, 132-178.  Google Scholar

[16]

W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179. doi: 10.1215/S0012-7094-93-07107-4.  Google Scholar

[17]

A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209. doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar

[18]

A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2), 143 (1996), 253-299. doi: 10.2307/2118644.  Google Scholar

[19]

A. Eskin and H. Oh, Representations of integers by an invariant polynomial and unipotent flows, Duke Math. J., 135 (2006), 481-506. doi: 10.1215/S0012-7094-06-13533-0.  Google Scholar

[20]

A. Eskin, Z. Rudnick and P. Sarnak, A proof of Siegel's weight formula, Internat. Math. Res. Notices, 5 (1991), 65-69. doi: 10.1155/S1073792891000090.  Google Scholar

[21]

W. T. Gan and H. Oh, Equidistribution of integer points on a family of homogeneous varieties: A problem of Linnik, Compositio Math., 136 (2003), 323-352. doi: 10.1023/A:1023256605535.  Google Scholar

[22]

A. Gorodnik and H. Oh, Rational points on homogeneous varieties and equidistribution of adelic periods, Geom. Funct. Anal., 21 (2011), 319-392. doi: 10.1007/s00039-011-0113-z.  Google Scholar

[23]

K. Györy, On the distribution of solutions of decomposable form equations, in Number Theory in Progress, Vol. 1 (Zakopane-Kościelisko, 1997) (eds. K. Györy, H. Iwaniec and J. Urbanowicz), de Gruyter, Berlin, 1999, 237-265.  Google Scholar

[24]

M. Hirsch, Differential Topology, Grad. Texts Math., 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[25]

D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $ PSL(2,\RR)$, J. Euro. Math. Soc., 11 (2009), 283-313. doi: 10.4171/JEMS/151.  Google Scholar

[26]

T. Kimura, Introduction to Prehomogeneous Vector Spaces, Transl. Math. Mono., 215, Amer. Math. Soc., Providence, RI, 2003.  Google Scholar

[27]

D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.  Google Scholar

[28]

D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.  Google Scholar

[29]

H. Koch, Number Theory: Algebraic Numbers and Functions, Grad. Stud. Math., 24, Amer. Math. Soc., Providence, RI, 2000.  Google Scholar

[30]

S. Lang, Algebraic Number Theory, Second edition, Grad. Texts Math., 110, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0853-2.  Google Scholar

[31]

D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math., 22 (1900), 293-335. doi: 10.2307/2369728.  Google Scholar

[32]

A. Nevo, Exponential volume growth, maximal functions on symmetric spaces, and ergodic theorems for semi-simple Lie groups, Erg. Theo. Dyn. Syst., 25 (2005), 1257-1294. doi: 10.1017/S0143385704000951.  Google Scholar

[33]

H. Oh, Hardy-Littlewood system and representations of integers by an invariant polynomial, Geom. Funct. Anal., 14 (2004), 791-809. doi: 10.1007/s00039-004-0475-6.  Google Scholar

[34]

H. Oh, Orbital counting via mixing and unipotent flows, in Homogeneous Flows, Moduli Spaces and Arithmetic (eds. M. Einsiedler, et al.), Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 339-375.  Google Scholar

[35]

E. Peyre, Obstructions au principe de Hasse et à l'approximation faible, Séminaire Bourbaki, Vol. 2003/2004, Astérisque, 299 (2005), 165-193.  Google Scholar

[36]

J. Parkkonen and F. Paulin, Équidistribution, comptage et approximation par irrationnels quadratiques, J. Mod. Dyn., 6 (2012), 1-40. doi: 10.3934/jmd.2012.6.1.  Google Scholar

[37]

J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature,, preprint, ().   Google Scholar

[38]

J. Parkkonen and F. Paulin, On the arithmetic of crossratios and generalised Mertens' formulas, to appear in Ann. Fac. Scien. Toulouse, arXiv:1308.5500, 2013. Google Scholar

[39]

V. Platonov and A. Rapinchuck, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.  Google Scholar

[40]

M. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[41]

I. Reiner, Maximal Orders, Academic Press, 1975.  Google Scholar

[42]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602.  Google Scholar

[43]

M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2), 100 (1974), 131-170. doi: 10.2307/1970844.  Google Scholar

[44]

W. M. Schmidt, Norm form equation, Ann. of Math. (2), 96 (1972), 526-551. doi: 10.2307/1970824.  Google Scholar

[45]

J.-P. Serre, Cours d'arithmetique, Collection SUP: "Le Mathématicien,'' 2 Presses Universitaires de France, Paris, 1970.  Google Scholar

[46]

C. L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2), 45 (1944), 577-622. doi: 10.2307/1969191.  Google Scholar

[47]

C. L. Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math. (2), 45 (1944), 667-685. doi: 10.2307/1969296.  Google Scholar

[48]

T. A. Springer, Linear algebraic groups, in Algebraic Geometry IV (eds. A. Parshin and I. Shavarevich), Encyc. Math. Scien., 55, Springer-Verlag, 1994, 1-121. doi: 10.1007/978-3-662-03073-8.  Google Scholar

[49]

J. L. Thunder, Decomposable form inequalities, Ann. of Math. (2), 153 (2001), 767-804. doi: 10.2307/2661368.  Google Scholar

[50]

V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants, Transl. Math. Mono., 179, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[51]

A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, 1965. Google Scholar

show all references

References:
[1]

T. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer Verlag, New York-Heidelberg, 1976.  Google Scholar

[2]

M. Babillot, Points entiers et groupes discrets: De l'analyse aux systèmes dynamiques, in Rigidité, Groupe Fondamental et Dynamique, Panor. Synthèses, 13, Soc. Math. France, Paris, 2002, 1-119.  Google Scholar

[3]

B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), New Math. Mono., 11, Cambridge Univ. Press, 2008. doi: 10.1017/CBO9780511542749.  Google Scholar

[4]

Y. Benoist and H. Oh, Effective equidistribution of $S$-integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble), 62 (2012), 1889-1942. doi: 10.5802/aif.2738.  Google Scholar

[5]

A. Borel, Ensembles fundamentaux pour les groupes arithmétiques, in Colloque sur la Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain; Gauthier-Villars, Paris, 1962, 23-40.  Google Scholar

[6]

A. Borel, Introduction aux Groupes Arithmétiques, Publications de l'Institut de Mathématique de l'Université de Strasbourg, XV, Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969.  Google Scholar

[7]

A. Borel, Linear Algebraic Groups, 2nd edition, Grad. Texts Math., 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.  Google Scholar

[8]

A. Borel, Reduction theory for arithmetic groups, in Algebraic Groups and Discontinuous Subgroups (eds. A. Borel and G. D. Mostow) (Proc. Sympos. Pure Math. Boulder, Colo., 1965), Amer. Math. Soc., 1966, 20-25.  Google Scholar

[9]

A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2), 75 (1962), 485-535. doi: 10.2307/1970210.  Google Scholar

[10]

A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.  Google Scholar

[11]

M. Borovoi and Z. Rudnick, Hardy-Littlewood varieties and semisimple groups, Invent. Math., 119 (1995), 37-66. doi: 10.1007/BF01245174.  Google Scholar

[12]

L. Clozel, Démonstration de la conjecture $\tau$, Invent. Math., 151 (2003), 297-328. doi: 10.1007/s00222-002-0253-8.  Google Scholar

[13]

H. Cohn, A Second Course in Number Theory, Wiley, 1962; reprinted as Advanced Number Theory, Dover Books on Advanced Mathematics, Dover Publications, Inc., New York, 1980.  Google Scholar

[14]

J.-L. Colliot-Thélène and F. Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compositio Math., 145 (2009), 309-363. doi: 10.1112/S0010437X0800376X.  Google Scholar

[15]

M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples, in Analyse Harmonique sur les Groupes de Lie (Sém. Nancy-Strasbourg 1976-1978), II, Lect. Notes Math., 739, Springer, Berlin, 1979, 132-178.  Google Scholar

[16]

W. Duke, Z. Rudnick and P. Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179. doi: 10.1215/S0012-7094-93-07107-4.  Google Scholar

[17]

A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209. doi: 10.1215/S0012-7094-93-07108-6.  Google Scholar

[18]

A. Eskin, S. Mozes and N. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2), 143 (1996), 253-299. doi: 10.2307/2118644.  Google Scholar

[19]

A. Eskin and H. Oh, Representations of integers by an invariant polynomial and unipotent flows, Duke Math. J., 135 (2006), 481-506. doi: 10.1215/S0012-7094-06-13533-0.  Google Scholar

[20]

A. Eskin, Z. Rudnick and P. Sarnak, A proof of Siegel's weight formula, Internat. Math. Res. Notices, 5 (1991), 65-69. doi: 10.1155/S1073792891000090.  Google Scholar

[21]

W. T. Gan and H. Oh, Equidistribution of integer points on a family of homogeneous varieties: A problem of Linnik, Compositio Math., 136 (2003), 323-352. doi: 10.1023/A:1023256605535.  Google Scholar

[22]

A. Gorodnik and H. Oh, Rational points on homogeneous varieties and equidistribution of adelic periods, Geom. Funct. Anal., 21 (2011), 319-392. doi: 10.1007/s00039-011-0113-z.  Google Scholar

[23]

K. Györy, On the distribution of solutions of decomposable form equations, in Number Theory in Progress, Vol. 1 (Zakopane-Kościelisko, 1997) (eds. K. Györy, H. Iwaniec and J. Urbanowicz), de Gruyter, Berlin, 1999, 237-265.  Google Scholar

[24]

M. Hirsch, Differential Topology, Grad. Texts Math., 33, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[25]

D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $ PSL(2,\RR)$, J. Euro. Math. Soc., 11 (2009), 283-313. doi: 10.4171/JEMS/151.  Google Scholar

[26]

T. Kimura, Introduction to Prehomogeneous Vector Spaces, Transl. Math. Mono., 215, Amer. Math. Soc., Providence, RI, 2003.  Google Scholar

[27]

D. Kleinbock and G. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinaĭ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, Amer. Math. Soc., Providence, RI, 1996, 141-172.  Google Scholar

[28]

D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.  Google Scholar

[29]

H. Koch, Number Theory: Algebraic Numbers and Functions, Grad. Stud. Math., 24, Amer. Math. Soc., Providence, RI, 2000.  Google Scholar

[30]

S. Lang, Algebraic Number Theory, Second edition, Grad. Texts Math., 110, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0853-2.  Google Scholar

[31]

D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math., 22 (1900), 293-335. doi: 10.2307/2369728.  Google Scholar

[32]

A. Nevo, Exponential volume growth, maximal functions on symmetric spaces, and ergodic theorems for semi-simple Lie groups, Erg. Theo. Dyn. Syst., 25 (2005), 1257-1294. doi: 10.1017/S0143385704000951.  Google Scholar

[33]

H. Oh, Hardy-Littlewood system and representations of integers by an invariant polynomial, Geom. Funct. Anal., 14 (2004), 791-809. doi: 10.1007/s00039-004-0475-6.  Google Scholar

[34]

H. Oh, Orbital counting via mixing and unipotent flows, in Homogeneous Flows, Moduli Spaces and Arithmetic (eds. M. Einsiedler, et al.), Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 339-375.  Google Scholar

[35]

E. Peyre, Obstructions au principe de Hasse et à l'approximation faible, Séminaire Bourbaki, Vol. 2003/2004, Astérisque, 299 (2005), 165-193.  Google Scholar

[36]

J. Parkkonen and F. Paulin, Équidistribution, comptage et approximation par irrationnels quadratiques, J. Mod. Dyn., 6 (2012), 1-40. doi: 10.3934/jmd.2012.6.1.  Google Scholar

[37]

J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature,, preprint, ().   Google Scholar

[38]

J. Parkkonen and F. Paulin, On the arithmetic of crossratios and generalised Mertens' formulas, to appear in Ann. Fac. Scien. Toulouse, arXiv:1308.5500, 2013. Google Scholar

[39]

V. Platonov and A. Rapinchuck, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.  Google Scholar

[40]

M. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[41]

I. Reiner, Maximal Orders, Academic Press, 1975.  Google Scholar

[42]

P. Sarnak, Asymptotic behavior of periodic orbits of the horocycle flow and Eisenstein series, Comm. Pure Appl. Math., 34 (1981), 719-739. doi: 10.1002/cpa.3160340602.  Google Scholar

[43]

M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2), 100 (1974), 131-170. doi: 10.2307/1970844.  Google Scholar

[44]

W. M. Schmidt, Norm form equation, Ann. of Math. (2), 96 (1972), 526-551. doi: 10.2307/1970824.  Google Scholar

[45]

J.-P. Serre, Cours d'arithmetique, Collection SUP: "Le Mathématicien,'' 2 Presses Universitaires de France, Paris, 1970.  Google Scholar

[46]

C. L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2), 45 (1944), 577-622. doi: 10.2307/1969191.  Google Scholar

[47]

C. L. Siegel, The average measure of quadratic forms with given determinant and signature, Ann. of Math. (2), 45 (1944), 667-685. doi: 10.2307/1969296.  Google Scholar

[48]

T. A. Springer, Linear algebraic groups, in Algebraic Geometry IV (eds. A. Parshin and I. Shavarevich), Encyc. Math. Scien., 55, Springer-Verlag, 1994, 1-121. doi: 10.1007/978-3-662-03073-8.  Google Scholar

[49]

J. L. Thunder, Decomposable form inequalities, Ann. of Math. (2), 153 (2001), 767-804. doi: 10.2307/2661368.  Google Scholar

[50]

V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants, Transl. Math. Mono., 179, Amer. Math. Soc., Providence, RI, 1998.  Google Scholar

[51]

A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, 1965. Google Scholar

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