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8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES

  • Jung, Hwan-Yup (Department of Mathematics Education Chungbuk National University) ;
  • Yue, Qin (Department of Mathematics Nanjing University of Aeronautics and Astronautics)
  • Received : 2010.07.13
  • Published : 2011.11.01

Abstract

For imaginary quadratic number fields F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$, where ${\varepsilon}{\in}${-1,-2} and distinct primes $p_i{\equiv}1$ mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})4$ and binary quadratic forms such as $p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$, where $h+(2p_1)$ is the narrow class number of $\mathbb{Q}(\sqrt{2p_1})$. The results are also very useful for numerical computations.

Keywords

References

  1. P. Barrucand and H. Cohn, Note on primes of type $x^{2}+32y^{2}$, class number, and residuacity, J. Reine Angew. Math. 238 (1969), 67-70.
  2. P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, Would Sci., Singapore 1988.
  3. P. E. Conner and J. Hurrelbrink, On the 4-rank of the tame kernel $K_{2}$(O) in positive definite terms, J. Number Theory 88 (2001), no. 2, 263-282. https://doi.org/10.1006/jnth.2000.2626
  4. F. Gerth III, Counting certain number fields with prescibed l-class numbers, J. Reine Angew. Math. 337 (1982), 195-207.
  5. F. Gerth III, The 4-class ranks of quadratic fields, Invent. Math. 77 (1984), no. 3, 489-515. https://doi.org/10.1007/BF01388835
  6. F. Gerth III and S. W. Graham, Application of a character sum estimate to a 2-class number density, J. Number Theory 19 (1984), no. 2, 239-247. https://doi.org/10.1016/0022-314X(84)90108-2
  7. G. Hardy and E. Wright, An Introduction to the Theory of Numbers, Fifth edition, London, 1979.
  8. E. Hecke, Lecture on the Theory of Algebraic Numbers, GTM 77, Springer-Verlag, 1981.
  9. J. Hurrelbrink and Q. Yue, On ideal class groups and units in terms of the quadratic form $x^{2}+32y^{2}$, Chinese Ann. Math. Ser. B 26 (2005), no. 2, 239-252. https://doi.org/10.1142/S0252959905000208
  10. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM 84, Springer-Verlag, 1972.
  11. J. Neukirch, Class Field Theory, Springer, Berlin, 1986.
  12. P. Stevenhagen, Divisibity by 2-powers of certain quadratic class numbers, J. Number Theory 43 (1993), no. 1, 1-19. https://doi.org/10.1006/jnth.1993.1001
  13. X. Wu and Q. Yue, 8-ranks of class groups of some imaginary quadratic number fields, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 2061-2068. https://doi.org/10.1007/s10114-007-0965-1
  14. Q. Yue, On tame kernel and class group in terms of quadratic forms, J. Number Theory 96 (2002), no. 2, 373-387. https://doi.org/10.1016/S0022-314X(02)92808-8
  15. Q. Yue, 8-ranks of class groups of quadratic number fields and their densities, Acta Matematica Sinica (Eng. Ser.), to apppear.
  16. Q. Yue and J. Yu, The densities of 4-ranks of tame kernels for quadratic fields, J. Reine Angew. Math. 567 (2004), 151-173.

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