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

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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

Abstract
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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2), 75 (1962), 485-535. doi: 10.2307/1970210.
[10]	A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.
[11]	M. Borovoi and Z. Rudnick, Hardy-Littlewood varieties and semisimple groups, Invent. Math., 119 (1995), 37-66. doi: 10.1007/BF01245174.
[12]	L. Clozel, Démonstration de la conjecture $\tau$, Invent. Math., 151 (2003), 297-328. doi: 10.1007/s00222-002-0253-8.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	M. Hirsch, Differential Topology, Grad. Texts Math., 33, Springer-Verlag, New York-Heidelberg, 1976.
[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.
[26]	T. Kimura, Introduction to Prehomogeneous Vector Spaces, Transl. Math. Mono., 215, Amer. Math. Soc., Providence, RI, 2003.
[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.
[28]	D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.
[29]	H. Koch, Number Theory: Algebraic Numbers and Functions, Grad. Stud. Math., 24, Amer. Math. Soc., Providence, RI, 2000.
[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.
[31]	D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math., 22 (1900), 293-335. doi: 10.2307/2369728.
[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.
[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.
[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.
[35]	E. Peyre, Obstructions au principe de Hasse et à l'approximation faible, Séminaire Bourbaki, Vol. 2003/2004, Astérisque, 299 (2005), 165-193.
[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.
[37]	J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, preprint, arXiv:1305.1332.
[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.
[39]	V. Platonov and A. Rapinchuck, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.
[40]	M. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.
[41]	I. Reiner, Maximal Orders, Academic Press, 1975.
[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.
[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.
[44]	W. M. Schmidt, Norm form equation, Ann. of Math. (2), 96 (1972), 526-551. doi: 10.2307/1970824.
[45]	J.-P. Serre, Cours d'arithmetique, Collection SUP: "Le Mathématicien,'' 2 Presses Universitaires de France, Paris, 1970.
[46]	C. L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2), 45 (1944), 577-622. doi: 10.2307/1969191.
[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.
[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.
[49]	J. L. Thunder, Decomposable form inequalities, Ann. of Math. (2), 153 (2001), 767-804. doi: 10.2307/2661368.
[50]	V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants, Transl. Math. Mono., 179, Amer. Math. Soc., Providence, RI, 1998.
[51]	A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, 1965.

show all references

References:

[1]	T. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer Verlag, New York-Heidelberg, 1976.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2), 75 (1962), 485-535. doi: 10.2307/1970210.
[10]	A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006.
[11]	M. Borovoi and Z. Rudnick, Hardy-Littlewood varieties and semisimple groups, Invent. Math., 119 (1995), 37-66. doi: 10.1007/BF01245174.
[12]	L. Clozel, Démonstration de la conjecture $\tau$, Invent. Math., 151 (2003), 297-328. doi: 10.1007/s00222-002-0253-8.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	M. Hirsch, Differential Topology, Grad. Texts Math., 33, Springer-Verlag, New York-Heidelberg, 1976.
[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.
[26]	T. Kimura, Introduction to Prehomogeneous Vector Spaces, Transl. Math. Mono., 215, Amer. Math. Soc., Providence, RI, 2003.
[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.
[28]	D. Kleinbock and G. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math., 138 (1999), 451-494. doi: 10.1007/s002220050350.
[29]	H. Koch, Number Theory: Algebraic Numbers and Functions, Grad. Stud. Math., 24, Amer. Math. Soc., Providence, RI, 2000.
[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.
[31]	D. N. Lehmer, Asymptotic evaluation of certain totient sums, Amer. J. Math., 22 (1900), 293-335. doi: 10.2307/2369728.
[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.
[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.
[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.
[35]	E. Peyre, Obstructions au principe de Hasse et à l'approximation faible, Séminaire Bourbaki, Vol. 2003/2004, Astérisque, 299 (2005), 165-193.
[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.
[37]	J. Parkkonen and F. Paulin, Counting common perpendicular arcs in negative curvature, preprint, arXiv:1305.1332.
[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.
[39]	V. Platonov and A. Rapinchuck, Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.
[40]	M. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972.
[41]	I. Reiner, Maximal Orders, Academic Press, 1975.
[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.
[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.
[44]	W. M. Schmidt, Norm form equation, Ann. of Math. (2), 96 (1972), 526-551. doi: 10.2307/1970824.
[45]	J.-P. Serre, Cours d'arithmetique, Collection SUP: "Le Mathématicien,'' 2 Presses Universitaires de France, Paris, 1970.
[46]	C. L. Siegel, On the theory of indefinite quadratic forms, Ann. of Math. (2), 45 (1944), 577-622. doi: 10.2307/1969191.
[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.
[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.
[49]	J. L. Thunder, Decomposable form inequalities, Ann. of Math. (2), 153 (2001), 767-804. doi: 10.2307/2661368.
[50]	V. E. Voskresenskiĭ, Algebraic Groups and their Birational Invariants, Transl. Math. Mono., 179, Amer. Math. Soc., Providence, RI, 1998.
[51]	A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, 1965.

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