Hostname: page-component-7bb8b95d7b-dvmhs Total loading time: 0 Render date: 2024-09-13T22:20:27.587Z Has data issue: false hasContentIssue false
  • English
  • Français

Modular correspondences on X(11)

Published online by Cambridge University Press:  20 January 2009

Allan Adler
Allan Adler
Affiliation:
36 Rolens Drive, Apt. C4Kingston, RI 02881USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we show how to give a geometric interpretation of the modular correspondence T3 on the modular curve X(11) of level 11 using projective geometry. We use Klein's theorem that X(11) is isomorphic to the nodal curve of the Hessian of the cubic threefold Λ defined by V2W + W2X + X2Y + Y2Z + Z2V = 0 in P4(C) and geometry which we learned from a paper of W. L. Edge. We show that the correspondence T3 is essentially the correspondence which associates to a point p of the curve X(11) the four points where the singular locus of the polar quadric of p with respect to Λ meets X(11). Our control of the geometry is good enough to enable us to compute the eigenvalues of T3 acting on the cohomology of X(11). This is the first example of an explicit geometric description of a modular correspondence without valence. The results of this article will be used in subsequent articles to associate two new abelian varieties to a cubic threefold, to desingularize the Hessian of a cubic threefold and to study self-conjugate polygons formed by the quadrisecants of the nodal curve of the Hessian.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 3 , October 1992 , pp. 427 - 435
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

REFERENCES

1.Adler, Allan, Characterization of modular correspondences by geometric properties, Pacific J. Math., to appear.Google Scholar
2.Adler, Allan, On V2 W + W2 X + X2 Y + Y2 Z + Z2 V = 0 and related topics, preprint, 1978.Google Scholar
3.Adler, Allan, On the automorphism group of a certain cubic threefold, Amer. J. Math. 100 (1979), 1275–1280.CrossRefGoogle Scholar
4.Adler, Allan, Invariants of PSL 2(F p) acting on C n for n = 2m ± 1, in preparation.Google Scholar
5.Adler, Allan and Ramanan, S., Moduli of Abelian Varieties, to appear.Google Scholar
6.Edge, W. L., Klein's encounter with the simple group of order 660, Proc. London Math. Soc. 24 (1972), 647–668.CrossRefGoogle Scholar
7.Hecke, Erich, Über ein Fundamentalproblem aus der Theorie der elliptischen Modulfunktionen, Math. Werke, 525–547.Google Scholar
8.Hurwitz, Adolf, Über einige besondere homogene lineare differentialgleichungen, Math. Werke Bd. I. 153–162.CrossRefGoogle Scholar
9.Klein, Felix, Über die Transformation elfter Ordnung der elliptischen Modulfunktionen, Ges. Math. Abh., III, 140–165.CrossRefGoogle Scholar
10.Klein, Felix, Über gewisse Teilwerte der θ-Funktion, Math. Annalen, 17 (1881) [Ges. Math. Abh., III, 186–197].Google Scholar
11.Klein, Felix and Fricke, Robert, Vorlesungen über die Theorie der elliptischen Modulfunktionen, Vol. II, Ch. V (Johnson Reprint, New York 1976).Google Scholar