2017, 11: 501-550. doi: 10.3934/jmd.2017020

Asymptotic distribution of values of isotropic here quadratic forms at S-integral points

1. 

Department of Mathematical Sciences, Seoul National University, Kwanak-ro 1, Kwanak-gu, Seoul 08826, Republic of Korea

2. 

Department of Mathematics, Jacobs University, 28759 Bremen, Germany

Received  June 09, 2016 Revised  August 02, 2017 Published  November 2017

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. Suppose we are given a finite set of places
$S$
over
${\mathbb{Q}}$
containing the Archimedean place and excluding the prime
$2$
, an irrational isotropic form
${\mathbf q}$
of rank
$n\geq 4$
on
${\mathbb{Q}}_S$
, a product of
$p$
-adic intervals
$\mathsf{I}_p$
, and a product
$\Omega$
of star-shaped sets. We show that unless
$n=4$
and
${\mathbf q}$
is split in at least one place, the number of
$S$
-integral vectors
$\mathbf v \in {\mathsf{T}} \Omega$
satisfying simultaneously
${\mathbf q}(\mathbf v) \in I_p$
for
$p \in S$
is asymptotically given by
$\begin{split}\lambda({\mathbf q}, \Omega) |\,\mathsf{I}\,| \cdot \| {\mathsf{T}} \|^{n-2}\end{split}$
as
${\mathsf{T}}$
goes to infinity, where
$|\,\mathsf{I}\,|$
is the product of Haar measures of the
$p$
-adic intervals
$I_p$
. The proof uses dynamics of unipotent flows on
$S$
-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an
$S$
-arithmetic variant of the
$ \alpha$
-function introduced in [10], and an
$S$
-arithemtic version of a theorem of Dani-Margulis [7].
Citation: Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai. Asymptotic distribution of values of isotropic here quadratic forms at S-integral points. Journal of Modern Dynamics, 2017, 11: 501-550. doi: 10.3934/jmd.2017020
References:
[1]

E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, No. 3, Interscience, New York, (1957). doi: 10.1002/9781118164518.

[2]

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

[3]

A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math., 83 (1992), 347-372.

[4]

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

[5]

R. Cheung, Integrability Estimates on the Space of $S$-Arithmetic Lattices with Applications to Quadratic Forms, Ph.D. Thesis, Yale University, 2016.

[6]

H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, second edition, with a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, edited and prepared for publication by T. D. Browning, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511542893.

[7]

S. Dani and G. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., 16, Part 1, Amer. Math. Soc., Providence, RI, (1993), 91-137.

[8]

H. Davenport and D. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. (3), 9 (1959), 544-555. doi: 10.1112/plms/s3-9.4.544.

[9]

J. Dieudonn, Sur les functions continues $p$-adiques, Bull. Sci. Math., (2), 68 (1994), 79-95.

[10]

A. EskinG. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984.

[11]

A. EskinG. Margulis and S. Mozes, Quadratic forms of signature $(2,2)$ and eignevalue spacings on rectangular $2$-tori, Ann. of Math. (2), 161 (2005), 679-725. doi: 10.4007/annals.2005.161.679.

[12]

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York, 1959.

[13]

G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math., 46 (1915), 263-283.

[14]

M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, in Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, 275–290.

[15]

D. Kleinbock and G. Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519-581. doi: 10.4171/CMH/102.

[16]

E. Landau, Neue Untersuchungen über die Pfeiffer'sche Methode zur Abschätzung von Gitterpunktanzahlen, in Sitzungsber. D. Math-naturw. Classe der Kaiserl. Akad. d. Wissenschaften, 2, Abteilung, Wien, (1915), 469-505.

[17]

G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes, (French summary), C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249-253.

[18]

G. A. Margulis and G. Tomanov, Invariant measures of unipotent groups over local fields on homogeneous spaces, Inv. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565.

[19]

A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727.

[20]

V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, (1994).

[21]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382. doi: 10.1215/S0012-7094-95-07710-2.

[22]

D. Ridout, Indefinite quadratic forms, Mathematika, 5 (1958), 122-124. doi: 10.1112/S0025579300001443.

[23]

W. Schmidt, Approximation to algebraic numbers, Enseignement Math. (2), 17 (1971), 187-253.

[24]

J.-P. Serre, A Course in Arithmetic, translated from the French, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, (1973).

[25]

J.-P. Serre, Lie Algebras and Lie Groups, Lectures Notes in Mathematics, Vol. 1500, Springer-Verlag Berlin Heidelberg New York, (1965).

[26]

J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Inst. Hautes. Études Sci. Publ. Math., 54 (1981), 323-401.

[27]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, (1997), Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, (2000), 265-297.

show all references

References:
[1]

E. Artin, Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, No. 3, Interscience, New York, (1957). doi: 10.1002/9781118164518.

[2]

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

[3]

A. Borel and G. Prasad, Values of isotropic quadratic forms at $S$-integral points, Compositio Math., 83 (1992), 347-372.

[4]

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

[5]

R. Cheung, Integrability Estimates on the Space of $S$-Arithmetic Lattices with Applications to Quadratic Forms, Ph.D. Thesis, Yale University, 2016.

[6]

H. Davenport, Analytic Methods for Diophantine Equations and Diophantine Inequalities, second edition, with a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman, edited and prepared for publication by T. D. Browning, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. doi: 10.1017/CBO9780511542893.

[7]

S. Dani and G. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, in I. M. Gel'fand Seminar, Adv. Soviet Math., Part 1, Amer. Math. Soc., 16, Part 1, Amer. Math. Soc., Providence, RI, (1993), 91-137.

[8]

H. Davenport and D. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. (3), 9 (1959), 544-555. doi: 10.1112/plms/s3-9.4.544.

[9]

J. Dieudonn, Sur les functions continues $p$-adiques, Bull. Sci. Math., (2), 68 (1994), 79-95.

[10]

A. EskinG. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2), 147 (1998), 93-141. doi: 10.2307/120984.

[11]

A. EskinG. Margulis and S. Mozes, Quadratic forms of signature $(2,2)$ and eignevalue spacings on rectangular $2$-tori, Ann. of Math. (2), 161 (2005), 679-725. doi: 10.4007/annals.2005.161.679.

[12]

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York, 1959.

[13]

G. H. Hardy, On the expression of a number as the sum of two squares, Quart. J. Math., 46 (1915), 263-283.

[14]

M. N. Huxley, Integer points, exponential sums and the Riemann zeta function, in Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, 275–290.

[15]

D. Kleinbock and G. Tomanov, Flows on $S$-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82 (2007), 519-581. doi: 10.4171/CMH/102.

[16]

E. Landau, Neue Untersuchungen über die Pfeiffer'sche Methode zur Abschätzung von Gitterpunktanzahlen, in Sitzungsber. D. Math-naturw. Classe der Kaiserl. Akad. d. Wissenschaften, 2, Abteilung, Wien, (1915), 469-505.

[17]

G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes, (French summary), C. R. Acad. Sci. Paris. Sér. I Math., 304 (1987), 249-253.

[18]

G. A. Margulis and G. Tomanov, Invariant measures of unipotent groups over local fields on homogeneous spaces, Inv. Math., 116 (1994), 347-392. doi: 10.1007/BF01231565.

[19]

A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U.S.A., 15 (1929), 724-727.

[20]

V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, (1994).

[21]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382. doi: 10.1215/S0012-7094-95-07710-2.

[22]

D. Ridout, Indefinite quadratic forms, Mathematika, 5 (1958), 122-124. doi: 10.1112/S0025579300001443.

[23]

W. Schmidt, Approximation to algebraic numbers, Enseignement Math. (2), 17 (1971), 187-253.

[24]

J.-P. Serre, A Course in Arithmetic, translated from the French, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, (1973).

[25]

J.-P. Serre, Lie Algebras and Lie Groups, Lectures Notes in Mathematics, Vol. 1500, Springer-Verlag Berlin Heidelberg New York, (1965).

[26]

J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Inst. Hautes. Études Sci. Publ. Math., 54 (1981), 323-401.

[27]

G. Tomanov, Orbits on homogeneous spaces of arithmetic origin and approximations, in Analysis on Homogeneous Spaces and Representation Theory of Lie Groups, Okayama-Kyoto, (1997), Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, (2000), 265-297.

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