The existence of curves of genus two with elliptic differentials

Journal of Number Theory

Volumn 64, Issue 1, 1997, Pages 130-161

  Kani, Ernst ^a  

^a QUEEN S UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0031139072     PISSN: 0022314X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jnth.1997.2105     Document Type: Article

References (19)

1. 0(m), Math. Ann. 185 (1970), 134-160.
2. [De] M. Deuring, Die Anzahl der Typen von Maximalordnungen einer definiten Quaternionenalgebra mit primer Grundzahl, Jahresber. Deutsch. Math.-Verein 54 (1950), 24-41.
3. [Di] L. E. Dickson, "History of the Theory of Numbers," Vol. I, Carnegie Institute, Washington, 1919 (Chelsea, Reprint, 1971).
4. [Ei] M. Eichler, Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math. 5 (1954), 355-366.
5. [Fr] G. Frey, On elliptic curves with isomorphic torsion structures and corresponding curves of genus 2, in "Conference on Elliptic Curves and Modular Forms, Hong Kong, December 18-21, 1993," pp. 79-98, International Press, 1995.
6. [FK] G. Frey and E. Kani, Curves of genus 2 covering elliptic curves and an arithmetical application, in "Arithmetic Algebraic Geometry" (G. van der Geer, F. Oort, and J. Steenbrink, Eds.), pp. 153-176, Progress in Mathematics, Vol. 89, Birkhäuser, Boston, 1991.
7. [G1] J. W. L. Glaisher, On the square of the series in which the coefficients are the sums of the divisors of the exponents, Messenger of Math. 14 (1884/5), 156-163.
8. [He] E. Hecke, Analytische Arithmetik der positiven quadratischen Formen, Kgl. Danske Viden. Selskab. Math.-fys. Med. XIII 12 (1940); in "Mathematische Werke," pp. 789-918, Vandenhoeck & Ruprecht, Göttingen, 1983.
9. [IKO] T. Ibukiyama, T. Katsura, and F. Oort, Supersingular curves of genus two and class numbers, Comp. Math. 57 (1986), 127-152.
10. [Ka] E. Kani, The number of curves of genus two with elliptic differentials, J. Reine Angew. Math., to appear.
11. [Ki] I. Kiming, On certain problems in the analytical arithmetic of quadratic forms arising from the theory of curves of genus 2 with elliptic differentials, Manuscripta Math. 87 (1995), 101-129.
12. [La] S. Lang, "Introduction to Modular Forms," Springer-Verlag, New York, 1976.
13. [Mc] P. J. McCarthy, "Introduction to Arithmetical Functions," Universitext Springer-Verlag, New York, 1986.
14. [Mi] T. Miyake, "Modular Forms," Springer-Verlag, Berlin, 1986.
15. [Ra] S. Ramanujan, "Collected Papers," Cambridge Univ. Press, Cambridge, UK, 1927 (Chelsea Reprint, 1962).
16. [Sch] B. Schoeneberg, "Elliptic Modular Functions," Springer-Verlag, Berlin, 1974.
17. [Sh] G. Shimura, "Introduction to the Arithmetic Theory of Automorphic Functions," Iwanami Shoten and Princeton Univ. Press, Princeton, 1971.
18. [Si] J. Silverman, "The Arithmetic of Elliptic Curves," Springer-Verlag, New York, 1986.
19. [SB] H. P. F. Swinnerton-Dyer and B. J. Birch, Elliptic curves and modular functions, in "Modular Functions of One Variable IV" (B. J. Birch and W. Kuyk, Eds.), pp. 2-32, Springer Lecture Notes, Vol. 476, Springer-Verlag, Berlin/New York, 1975.