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Eisenstein Series of 3/2 Weight and Eligible Numbers of Positive Definite Ternary Forms

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Abstract

A general algorithm is given for the number of representations for a positive integer n by the genus of a positive definite ternary quadratic form with form ax2 + by2 + cz2. Using this algorithm, we study several nontrivial genera of positive ternary forms with small discriminants in the paper. As a conclusion we prove that f1 = x2 + y2 + 7z2 represents all eligible numbers congruent to 2 mod 3 except 14 * 72k which was conjectured by Kaplansky in [K]. Our method is to use Eisenstein series of weight 3/2.

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References

  1. J. A. Antoniadis, M. Bungert and G. Frey, Properties of twists of elliptic curves, J.reine angew.Math. 405 (1990), 1–28.

    MathSciNet  MATH  Google Scholar 

  2. V. A. Bykovskii, A trace formula for the scalar product of Hecke series and its application, J. of Mathematical Sciences, 89 (1998),No.1, 915–932.

    Article  MathSciNet  Google Scholar 

  3. J. E. Cremona, Algorithms for modular elliptic curves, Cambridge Univ. Press 1992

  4. W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), 73–90.

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Duke and R. Schulze-Pillot, Representations of integers by positive definite ternary quadratic forms and equidistribution of lattice points on ellipsoids. Invent. Math. 99 (1990), 49–57.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Hsia and M. Jochner, Almost strong approximations for definite quadratic spaces, Invent. Math. 129 (1997), 471–487.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), 385–401.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. Jones, The regularity of a genus of positive ternary quadratic forms, Trans. Amer. Math. Soc. 33 (1931), 111–124.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Jones, The arithmetic theory of quadratic forms, Mathematical Association of America, 1950.

  10. I. Kaplansky, The first nontrivial genus of positive definite ternary forms, Math. Comp. 64 (1995), 341–345.

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Ono and K. Soundararajan, Ramanujan’s ternary quadratic form, Invent. Math. (1997), 415-454.

  12. D. Y. Pei, Eisenstein series of weight 3/2: I, II, Trans. Amer. Math. Soc. 274 (1982), 573–606 and 283 (1984), 589-603.

    MathSciNet  MATH  Google Scholar 

  13. D.Y.Pei, G.Rosenberger and Xueli Wang, The eligible numbers of positive definite ternary forms, to appear in Math.Zeit.(2000).

  14. G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), 440–481.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Schulze-Pillot, Thetareihen positiver definiter quadratischer Formen, Invent. Math. 75 (1984), 283–299.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. epartment of Mathematics, angzhou Teacher’s College, Guangzhou, 510405, P. R. China

    Dingyi Pei

  2. Fachbereich Mathematik, Universität Dortmund, 44221, Dortmund, Germany

    Gerhard Rosenberger

  3. Department of Mathematics, Guangzhou Teacher’s College, Guangzhou, 510405, P. R. China

    Xueli Wang

Authors
  1. Dingyi Pei
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  2. Gerhard Rosenberger
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  3. Xueli Wang
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Correspondence to Dingyi Pei.

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Pei, D., Rosenberger, G. & Wang, X. Eisenstein Series of 3/2 Weight and Eligible Numbers of Positive Definite Ternary Forms. Results. Math. 39, 292–319 (2001). https://doi.org/10.1007/BF03322691

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  • DOI: https://doi.org/10.1007/BF03322691

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