×

Local densities and explicit bounds for representability by a quadratic form. (English) Zbl 1090.11023

A classical theorem of V. A. Tartakovski (W. Tartakowsky) [Bull. Acad. Sci. Leningr. (7) (Izv. Akad. Nauk SSSR) 2, 111–122, 165–196 (1929; JFM 56.0882.04)] states that a positive definite integral quadratic form \(f\) in \(n\geq 5\) variables represents all sufficiently large integers that are represented by the genus of \(f\). This result extends to forms in 4 variables when the integers considered are restricted to those divisible by bounded powers of the primes at which \(f\) is anisotropic. Additional restrictions are needed in the ternary case, where there can exist infinite families of integers that are represented by a genus but not by all spinor genera within the genus. In this case, the results of W. Duke and R. Schulze-Pillot [Invent. Math. 99, No. 1, 49–57 (1990; Zbl 0692.10020)] show, in particular, that all sufficiently large integers primitively represented by the spinor genus of \(f\) are represented by \(f\) itself.
In the present paper, the author gives explicit lower bounds for an integer \(m\) (suitably restricted for the cases \(n=3,4\)) which ensure that \(m\) is represented by \(f\), provided that it is represented by the genus of \(f\). The main emphasis is on the cases \(n=3\) and \(n=4\) since reasonable bounds in these cases were not previously known (for \(n\geq5\) such bounds can be found in papers of G. L. Watson [Philos. Trans. R. Soc. Lond., Ser. A 253, 227–254 (1960; Zbl 0102.28102)] and J. S. Hsia and M. I. Icaza [Acta Arith. 89, 235–253 (1999; Zbl 0936.11021)]; previous results for \(n=4\) were obtained by R. Schulze-Pillot [Arch. Math. 77, 129–137 (2001; Zbl 0995.11030)]). The bounds obtained here also help to describe the general representation behavior when \(n=3\). In that case, an effective version of the result of Duke and Schulze-Pillot within any fixed square class is given, including an asymptotic formula for the number of representations by \(f\) of the integers within the square class. The author notes that the results of this paper can be extended to arbitrary totally real number fields, and all the local analysis in the paper is carried out in that generality.
The strength of the results obtained here is demonstrated by resolving the well known and long-standing conjecture that the only positive integers that fail to be represented by the form \(x^2+3y^2+5z^2+7w^2\) are 2 and 22.

MSC:

11E12 Quadratic forms over global rings and fields
11D09 Quadratic and bilinear Diophantine equations
11E25 Sums of squares and representations by other particular quadratic forms
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11Y50 Computer solution of Diophantine equations

Software:

QFLib
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. N. Andrianov and V. G. Zhuravlëv, Modular Forms and Hecke Operators , Transl. Math. Monogr. 145 , Amer. Math. Soc., Providence, 1995.
[2] A. O. L. Atkin and J. Lehner, Hecke operators on \(\Gamma_0(m)\) , Math. Ann. 185 (1970), 134–160. · Zbl 0177.34901
[3] F. Van der Blij, On the theory of quadratic forms , Ann. of Math. (2) 50 (1949), 875–883. JSTOR: · Zbl 0034.31103
[4] J. W. S. Cassels, Rational Quadratic Forms , London Math. Soc. Monogr. 13 , Academic Press, London, 1978. · Zbl 0395.10029
[5] B. A. Cipra, On the Niwa-Shintani theta-kernel lifting of modular forms , Nagoya Math. J. 91 (1983), 49–117. · Zbl 0523.10014
[6] J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic \(L\)-functions , Ann. of Math. (2) 151 (2000), 1175–1216. JSTOR: · Zbl 0973.11056
[7] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups , 3rd ed., Grundlehren Math. Wiss. 290 , Springer, New York, 1999. · Zbl 0915.52003
[8] P. Deligne, La conjecture de Weil, I , Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307. · Zbl 0287.14001
[9] W. Duke and R. Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids , Invent. Math. 99 (1990), 49–57. · Zbl 0692.10020
[10] J. Hanke, QFLib package for Pari/GP, available from http://www.math.duke.edu/\(\tilde\;\)jonhanke
[11] ——–, On a local-global principle for integral quadratic forms , preprint, 2003, http://www.math.duke.edu/\(\tilde\;\)jonhanke J. S. Hsia, Representations by spinor genera , Pacific J. Math 63 (1976), 147–152.
[12] J. S. Hsia, and M. I. Icaza, Effective version of Tartakowski’s theorem , Acta Arith. 89 (1999), 235–253. · Zbl 0936.11021
[13] J. Igusa, Forms of Higher Degree , Tata Inst. Fund. Res. Lect. Math. 59 , Tata Inst. Fund. Res., Mumbai, 1978. · Zbl 0417.10015
[14] Y. Kitaoka, Arithmetic of Quadratic Forms , Cambridge Tracts in Math. 106 , Cambridge Univ. Press, Cambridge, 1993. · Zbl 0785.11021
[15] M. Kneser, Darstellungsmasse indefiniter quadratischer Formen , Math Z. 77 (1961), 188–194. · Zbl 0100.03601
[16] ——–, Quadratische Formen , newly revised and edited with R. Scharlau, Springer, Berlin, 2002.
[17] S. S. Kudla and S. Rallis, On the Weil-Siegel formula , J. Reine Angew. 387 (1988), 1–68. · Zbl 0644.10021
[18] –. –. –. –., On the Weil-Siegel formula, II: The isotropic convergent case , J. Reine Angew. 391 (1988), 65–84. · Zbl 0644.10022
[19] J. Milnor and D. Husemoller, Symmetric Bilinear Forms , Ergeb. Math. Grenzgeb. (2) 73 , Springer, New York, 1973. · Zbl 0292.10016
[20] O. T. O’Meara, Introduction to Quadratic Forms , reprint of 1973 ed., Classics Math., Springer, Berlin, 2000.
[21] K. Ono and K. Soundararajan, Ramanujan’s ternary quadratic form , Invent. Math. 130 (1997), 415–454. · Zbl 0930.11022
[22] S. Rallis, \(L\)-functions and the oscillator representation , Lecture Notes in Math. 1254 , Springer, Berlin, 1987. · Zbl 0605.10016
[23] R. Schulze-Pillot, Darstellung durch Spinorgeschlechter ternärer quadratischer Formen , J. Number Theory 12 (1980), 529–540. · Zbl 0443.10017
[24] –. –. –. –., Thetareihen positiv definiter quadratischer Formen , Invent. Math. 75 (1984), 283–299. · Zbl 0533.10021
[25] –. –. –. –., On explicit versions of Tartakovski’s theorem , Arch. Math. 77 (2001), 129–137. · Zbl 0995.11030
[26] –. –. –. –., Exceptional integers for genera of integral ternary positive definite quadratic forms , Duke Math. J. 102 (2000), 351–357. · Zbl 0948.11019
[27] C. L. Siegel, Über die analytische Theorie der quadratischen Formen , Ann. of Math. (2) 36 (1935), 527–606. JSTOR: · Zbl 0012.19703
[28] G. Shimura, On modular forms of half integral weight , Ann. of Math. (2) 97 (1973), 440–481. JSTOR: · Zbl 0266.10022
[29] G. L. Watson, Quadratic Diophantine equations , Philos. Trans. Roy. Soc. London Ser. A 253 (1960/1961), 227–254. JSTOR: · Zbl 0102.28102
[30] –. –. –. –., Transformations of a quadratic form which do not increase the class-number , Proc. London Math. Soc. (3) 12 (1962), 577–587. · Zbl 0107.26901
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.