On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces

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• 作者：Mei-Chu Chang ; Igor E. Shparlinski
• 关键词：Integer points on hypersurfaces ; Multiplicative character sums ; Congruences
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：282
• 期：3-4
• 页码：935-954
• 全文大小：511 KB
• 参考文献：1.Baragar, A.: Asymptotic growth of Markoff–Hurwitz numbers. Compos. Math. 94, 1–18 (1994)MathSciNet MATH
  2.Baragar, A.: The exponent for the Markoff–Hurwitz equations. Pac. J. Math. 182, 1–21 (1998)MathSciNet CrossRef MATH
  3.Baragar, A.: The Markoff–Hurwitz equations over number fields. Rocky Mt. J. Math. 35, 695–712 (2005)MathSciNet CrossRef MATH
  4.Birch, B.J.: Forms in many variables. Proc. R. Soc. Ser. A 265, 245–263 (1961)MathSciNet CrossRef MATH
  5.Browning, T.D.: Quantitative Arithmetic of Projective Varieties, Progress in Mathemetics, vol. 277. Birkhäuser Verlag, Basel (2009)CrossRef
  6.Browning, T.D., Heath-Brown, R., Salberger, P.: Counting rational points on algebraic varieties. Duke Math. J. 132, 545–578 (2006)MathSciNet CrossRef MATH
  7.Cao, W.: On generalized Markoff–Hurwitz-type equations over finite fields. Acta Appl. Math. 112, 275–281 (2010)MathSciNet CrossRef MATH
  8.Chang, M.-C.: Factorization in generalized arithmetic progressions and applications to the Erdős–Szemerédi sum–product problems. Geom. Funct. Anal. 13, 720–736 (2003)MathSciNet CrossRef MATH
  9.Chang, M.-C.: An estimate of incomplete mixed character sums. In: Bárány, I., Solymosi, J. (eds.) An Irregular Mind, Bolyai Society Math. Studies, vol. 21, pp. 243–250. Springer, Berlin (2010)CrossRef
  10.Chang, M.-C.: Short character sums for composite moduli. J. d’Analyse Math. 123, 1–33 (2014)MathSciNet CrossRef MATH
  11.Cochrane, T., Shi, S.: The congruence \(x_1x_2 \equiv x_3x_4 ~(\text{ mod } m)\) and mean values of character sums. J. Number Theory 130, 767–785 (2010)MathSciNet CrossRef
  12.Goutet, P.: An explicit factorisation of the zeta functions of Dwork hypersurfaces. Acta Arith. 144, 241–261 (2010)MathSciNet CrossRef MATH
  13.Harris, M., Shepherd-Barron, N., Taylor, R.: A family of Calabi–Yau varieties and potential automorphy. Ann. Math. 171, 779–813 (2010)MathSciNet CrossRef MATH
  14.Heath-Brown, D.R.: The density of rational points on nonsingular hypersurfaces. Proc. Indian Acad. Sci. Math. Sci. 104, 13–29 (1994)MathSciNet CrossRef MATH
  15.Heath-Brown, D.R., Pierce, L.: Counting rational points on smooth cyclic covers. J. Number Theory 132, 1741–1757 (2012)MathSciNet CrossRef MATH
  16.Heath-Brown, D.R., Pierce, L.: Burgess bounds for short mixed character sums. J. Lond. Math. Soc. 91, 693–708 (2015)MathSciNet CrossRef MATH
  17.Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc, Providence, RI (2004)MATH
  18.Katz, N.M.: Another look at the Dwork family. In: Tschinkel, Y., Zarhin, Y. (eds.) Algebra, Arithmetic, and Geometry. Honor of Yu. I. Manin, Progress in Mathematics, vol. II, 270, pp. 89–126. Birkhäuser Boston Inc, Boston (2009)
  19.Kloosterman, R.: The zeta function of monomial deformations of Fermat hypersurfaces. Algebra Number Theory 1, 421–450 (2007)MathSciNet CrossRef MATH
  20.Li, W.-C.W.: Number Theory with Applications. World Scientific, Singapore (1996)CrossRef
  21.Marmon, O.: The density of integral points on complete intersections. Q. J. Math. 59, 29–53 (2008)MathSciNet CrossRef MATH
  22.Marmon, O.: The density of integral points on hypersurfaces of degree at least four. Acta Arith. 141, 211–240 (2010)MathSciNet CrossRef MATH
  23.Postnikov, A.G.: On the sum of characters with respect to a modulus equal to a power of a prime number. Izv. Akad. Nauk SSSR. Ser. Mat. 19, 11–16 (1955). (in Russian)MathSciNet
  24.Postnikov, A.G.: On Dirichlet \(L\) -series with the character modulus equal to the power of a prime number. J. Indian Math. Soc. 20, 217–226 (1956)MathSciNet MATH
  25.Salberger, P.: On the density of rational and integral points on algebraic varieties. J. Reine Angew. Math. 606, 123–147 (2007)MathSciNet MATH
  26.Salberger, P.: Counting rational points on projective varieties. Preprint (2013)
  27.Shparlinski, I.E.: On the distribution of points on the generalised Markoff–Hurwitz and Dwork hypersurfaces. Int. J. Number Theory 10, 151–160 (2014)MathSciNet CrossRef MATH
  28.Tschinkel, Y.: Algebraic varieties with many rational points. In: Darmon, H., Ellwood, D.A., Hassett B., Tschinkel, Y. (eds.) Arithmetic Geometry. Clay Math. Proc., vol. 8, pp. 243–334. American Mathematical Society, Providence, RI (2009)
  29.Wooley, T.D.: Translation invariance, exponential sums, and Warings problem. In: Proceedings of International Congress of Mathematicians, Seoul, pp. 505–529 (2014)
  30.Yu, Y.-D.: Variation of the unit root along the Dwork family of Calabi–Yau varieties. Math. Ann. 343, 53–78 (2009)MathSciNet CrossRef MATH
  
• 作者单位：Mei-Chu Chang (1)
  Igor E. Shparlinski (2)
  
  1. Department of Mathematics, University of California, Riverside, CA, 92521, USA
  2. Department of Pure Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

We use bounds of mixed character sums modulo a square-free integer q of a special structure to estimate the density of integer points on the hypersurface $$\begin{aligned} f_1(x_1) + \cdots + f_n(x_n) =a x_1^{k_1} \ldots x_n^{k_n} \end{aligned}$$for some polynomials \(f_i \in {\mathbb {Z}}[X]\) and nonzero integers a and \(k_i\), \(i=1, \ldots , n\). In the case of $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^2\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$the above hypersurface is known as the Markoff–Hurwitz hypersurface, while for $$\begin{aligned} f_1(X) = \cdots = f_n(X) = X^n\quad \text{ and }\quad k_1 = \cdots = k_n =1 \end{aligned}$$it is known as the Dwork hypersurface. Our results are substantially stronger than those known for general hypersurfaces. Keywords Integer points on hypersurfaces Multiplicative character sums Congruences Mathematics Subject Classification 11D45 11D72 11L40 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (30) References1.Baragar, A.: Asymptotic growth of Markoff–Hurwitz numbers. Compos. Math. 94, 1–18 (1994)MathSciNetMATH2.Baragar, A.: The exponent for the Markoff–Hurwitz equations. Pac. J. Math. 182, 1–21 (1998)MathSciNetCrossRefMATH3.Baragar, A.: The Markoff–Hurwitz equations over number fields. Rocky Mt. J. Math. 35, 695–712 (2005)MathSciNetCrossRefMATH4.Birch, B.J.: Forms in many variables. Proc. R. Soc. Ser. A 265, 245–263 (1961)MathSciNetCrossRefMATH5.Browning, T.D.: Quantitative Arithmetic of Projective Varieties, Progress in Mathemetics, vol. 277. Birkhäuser Verlag, Basel (2009)CrossRef6.Browning, T.D., Heath-Brown, R., Salberger, P.: Counting rational points on algebraic varieties. Duke Math. J. 132, 545–578 (2006)MathSciNetCrossRefMATH7.Cao, W.: On generalized Markoff–Hurwitz-type equations over finite fields. Acta Appl. Math. 112, 275–281 (2010)MathSciNetCrossRefMATH8.Chang, M.-C.: Factorization in generalized arithmetic progressions and applications to the Erdős–Szemerédi sum–product problems. Geom. Funct. Anal. 13, 720–736 (2003)MathSciNetCrossRefMATH9.Chang, M.-C.: An estimate of incomplete mixed character sums. In: Bárány, I., Solymosi, J. (eds.) An Irregular Mind, Bolyai Society Math. Studies, vol. 21, pp. 243–250. Springer, Berlin (2010)CrossRef10.Chang, M.-C.: Short character sums for composite moduli. J. d’Analyse Math. 123, 1–33 (2014)MathSciNetCrossRefMATH11.Cochrane, T., Shi, S.: The congruence \(x_1x_2 \equiv x_3x_4 ~(\text{ mod } m)\) and mean values of character sums. J. Number Theory 130, 767–785 (2010)MathSciNetCrossRef12.Goutet, P.: An explicit factorisation of the zeta functions of Dwork hypersurfaces. Acta Arith. 144, 241–261 (2010)MathSciNetCrossRefMATH13.Harris, M., Shepherd-Barron, N., Taylor, R.: A family of Calabi–Yau varieties and potential automorphy. Ann. Math. 171, 779–813 (2010)MathSciNetCrossRefMATH14.Heath-Brown, D.R.: The density of rational points on nonsingular hypersurfaces. Proc. Indian Acad. Sci. Math. Sci. 104, 13–29 (1994)MathSciNetCrossRefMATH15.Heath-Brown, D.R., Pierce, L.: Counting rational points on smooth cyclic covers. J. Number Theory 132, 1741–1757 (2012)MathSciNetCrossRefMATH16.Heath-Brown, D.R., Pierce, L.: Burgess bounds for short mixed character sums. J. Lond. Math. Soc. 91, 693–708 (2015)MathSciNetCrossRefMATH17.Iwaniec, H., Kowalski, E.: Analytic Number Theory. Amer. Math. Soc, Providence, RI (2004)MATH18.Katz, N.M.: Another look at the Dwork family. In: Tschinkel, Y., Zarhin, Y. (eds.) Algebra, Arithmetic, and Geometry. Honor of Yu. I. Manin, Progress in Mathematics, vol. II, 270, pp. 89–126. Birkhäuser Boston Inc, Boston (2009)19.Kloosterman, R.: The zeta function of monomial deformations of Fermat hypersurfaces. Algebra Number Theory 1, 421–450 (2007)MathSciNetCrossRefMATH20.Li, W.-C.W.: Number Theory with Applications. World Scientific, Singapore (1996)CrossRef21.Marmon, O.: The density of integral points on complete intersections. Q. J. Math. 59, 29–53 (2008)MathSciNetCrossRefMATH22.Marmon, O.: The density of integral points on hypersurfaces of degree at least four. Acta Arith. 141, 211–240 (2010)MathSciNetCrossRefMATH23.Postnikov, A.G.: On the sum of characters with respect to a modulus equal to a power of a prime number. Izv. Akad. Nauk SSSR. Ser. Mat. 19, 11–16 (1955). (in Russian)MathSciNet24.Postnikov, A.G.: On Dirichlet \(L\)-series with the character modulus equal to the power of a prime number. J. Indian Math. Soc. 20, 217–226 (1956)MathSciNetMATH25.Salberger, P.: On the density of rational and integral points on algebraic varieties. J. Reine Angew. Math. 606, 123–147 (2007)MathSciNetMATH26.Salberger, P.: Counting rational points on projective varieties. Preprint (2013)27.Shparlinski, I.E.: On the distribution of points on the generalised Markoff–Hurwitz and Dwork hypersurfaces. Int. J. Number Theory 10, 151–160 (2014)MathSciNetCrossRefMATH28.Tschinkel, Y.: Algebraic varieties with many rational points. In: Darmon, H., Ellwood, D.A., Hassett B., Tschinkel, Y. (eds.) Arithmetic Geometry. Clay Math. Proc., vol. 8, pp. 243–334. American Mathematical Society, Providence, RI (2009)29.Wooley, T.D.: Translation invariance, exponential sums, and Warings problem. In: Proceedings of International Congress of Mathematicians, Seoul, pp. 505–529 (2014)30.Yu, Y.-D.: Variation of the unit root along the Dwork family of Calabi–Yau varieties. Math. Ann. 343, 53–78 (2009)MathSciNetCrossRefMATH About this Article Title On the density of integer points on generalised Markoff–Hurwitz and Dwork hypersurfaces Journal Mathematische Zeitschrift Volume 282, Issue 3-4 , pp 935-954 Cover Date2016-04 DOI 10.1007/s00209-015-1571-z Print ISSN 0025-5874 Online ISSN 1432-1823 Publisher Springer Berlin Heidelberg Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords Integer points on hypersurfaces Multiplicative character sums Congruences 11D45 11D72 11L40 Authors Mei-Chu Chang (1) Igor E. Shparlinski (2) Author Affiliations 1. Department of Mathematics, University of California, Riverside, CA, 92521, USA 2. 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