Supersingular curves of genus two and class numbers

Tomoyoshi Ibukiyama; Toshiyuki Katsura; Frans Oort

Compositio Mathematica (1986)

  • Volume: 57, Issue: 2, page 127-152
  • ISSN: 0010-437X

How to cite


Ibukiyama, Tomoyoshi, Katsura, Toshiyuki, and Oort, Frans. "Supersingular curves of genus two and class numbers." Compositio Mathematica 57.2 (1986): 127-152. <>.

author = {Ibukiyama, Tomoyoshi, Katsura, Toshiyuki, Oort, Frans},
journal = {Compositio Mathematica},
keywords = {polarization; characteristic p; Jacobian variety; supersingular elliptic curves; prescribed automorphism group; curves of genus two; Hasse-Witt- matrix; class numbers of quaternionic algebras},
language = {eng},
number = {2},
pages = {127-152},
publisher = {Martinus Nijhoff Publishers},
title = {Supersingular curves of genus two and class numbers},
url = {},
volume = {57},
year = {1986},

AU - Ibukiyama, Tomoyoshi
AU - Katsura, Toshiyuki
AU - Oort, Frans
TI - Supersingular curves of genus two and class numbers
JO - Compositio Mathematica
PY - 1986
PB - Martinus Nijhoff Publishers
VL - 57
IS - 2
SP - 127
EP - 152
LA - eng
KW - polarization; characteristic p; Jacobian variety; supersingular elliptic curves; prescribed automorphism group; curves of genus two; Hasse-Witt- matrix; class numbers of quaternionic algebras
UR -
ER -


  1. [1] M. Deuring: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Univ. Hamburg14 (1941) 197-272. Zbl0025.02003MR5125JFM67.0107.01
  2. [2] M. Eichler: Über die Idealklassenzahl total definiter Quaternionenalgebren. Math. Z.43 (1938) 102-109. Zbl0017.15003MR1545717JFM63.0093.02
  3. [3] M. Eichler: Über die Idealklassenzahl hyperkomplexer Systeme. Math. Z.43 (1938) 481-494. Zbl0018.20201MR1545733JFM64.0085.01
  4. [4] M. Eichler: Zur Zahlentheorie der Quaternionen-Algebren. J. Reine Angew. Math.195 (1955) 127-151. Zbl0068.03303MR80767
  5. [5] K. Hashimoto: Class numbers of positive definite ternary quaternion hermitian forms. Proceed. Japan Acad.59 Ser. A (1983) 490-493. Zbl0539.10020MR732595
  6. [6] K. Hashimoto and T. Ibukiyama: On class numbers of positive definite binary quaternion hermitian forms (I). J. Fac. Sci. Univ. Tokyo Sect IA, 27 (1980) 549-601, (II) ibid. 28 (1981) 695-699,(III) ibid. to appear. Zbl0493.10030MR603952
  7. [7] T. Hayashida: A class number associated with the product of an elliptic curve with itself. J. Math. Soc. Japan20 (1968) 26-43. Zbl0186.26501MR233804
  8. [8] J. Igusa: Class number of a definite quaternion with prime discriminant. Proc. Nat. Acad. Sci. U.S.A.44 (1958) 312-314. Zbl0081.03601MR98728
  9. [9] J. Igusa: Arithmetic variety of moduli for genus two. Ann. of Math.72 (1960) 612-649. Zbl0122.39002MR114819
  10. [10] T. Katsura and F. Oort: Families of supersingular abelian surfaces. In preparation. Zbl0636.14017
  11. [11] N. Koblitz: P-adic variation of the zeta-function over families defined over finite fields. Compositio Math.31 (1975) 119-218. Zbl0332.14008MR414557
  12. [12] S. Lang: Abelian Varieties. New York: Interscience-Wiley (1959). Zbl0098.13201MR106225
  13. [13] Yu. I. Manin: The theory of commutative formal groups over fields of finite characteristic. Russian Math. Surveys18 (1963) 1-83. Zbl0128.15603MR157972
  14. [14] D. Mumford: Abelian Varieties. Oxford University Press (1970). Zbl0223.14022MR282985
  15. [15] P. Norman and F. Oort: Moduli of abelian varieties. Ann. of Math.112 (1980) 413-439. Zbl0483.14010MR595202
  16. [16] N.O. Nygaard: Slopes of powers of frobenius on crystalline cohomology. Ann. Sci. Ecole Norm. Sup.14 (1981) 369-401. Zbl0519.14012MR654203
  17. [17] F. Oort: Subvarieties of moduli spaces. Invent. Math.24 (1974) 95-119. Zbl0259.14011MR424813
  18. [18] F. Oort: Which abelian surfaces are products of elliptic curves? Math. Ann.214 (1975) 35-47. Zbl0283.14007MR364264
  19. [19] G. Shimura: Arithmetic of alternating forms and quaternion hermitian forms. J. Math. Soc. Japan15 (1963) 33-65. Zbl0121.28102MR146172
  20. [20] T. Shioda: Some results on unirationality of algebraic surfaces. Math. Ann.230 (1977) 153-168. Zbl0343.14021MR572983
  21. [21] T. Shioda: Supersingular K3 surfaces. Lecture Notes in Math.732. Berlin- Heidelberg-New York: Springer (1979) 564-591. Zbl0414.14019MR555718
  22. [22] W.C. Waterhouse: Abelian varieties over finite fields. Ann. Sci. Ecole Norm. Sup.2 (1962) 521-560. Zbl0188.53001MR265369
  23. [23] N. Yui: On the Jacobian varieties of hyperelliptic curves over fields of characteristic p &gt; 2. Journ. of Alg.52 (1978) 378-410. Zbl0404.14008MR491717

Citations in EuDML Documents

  1. Tomoyoshi Ibukiyama, Toshiyuki Katsura, On the field of definition of superspecial polarized abelian varieties and type numbers
  2. Yuri G. Zarhin, Non-supersingular hyperelliptic jacobians
  3. Toshiyuki Katsura, Frans Oort, Families of supersingular abelian surfaces
  4. Eyal Z. Goren, Kristin E. Lauter, Class Invariants for Quartic CM Fields
  5. Stephen S. Kudla, Michael Rapoport, Cycles on Siegel threefolds and derivatives of Eisenstein series
  6. Everett W. Howe, Enric Nart, Christophe Ritzenthaler, Jacobians in isogeny classes of abelian surfaces over finite fields

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