Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T14:36:32.046Z Has data issue: false hasContentIssue false
  • English
  • Français

The Mathieu group M11 and the modular curve X(11)

Published online by Cambridge University Press:  01 January 1997

A Adler
Get access

Abstract

In this paper, we prove that the modular curve $X(11)$ over a field of characteristic 3 admits the Mathieu group $M_{11}$ as an automorphism group. We also examine some aspects of the geometry of the curve $X(11)$ in characteristic 3. In particular, we show that every point of the curve is a point of inflection, the curve has 110 hyperflexes and there are no inflectional triangles and 11232 inflectional pentagons, of which 144 are self-conjugate. The hyperflexes correspond to the supersingular elliptic curves. We comment on the relationship of Ward's quadrilinear invariant for $M_{12}$ to our work and announce for the first time the equations for Klein's A-curve of level 11. We also comment on the relation of our work to some unpublished work of Bott and Tate.

1991 Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10, 14N10, 20B25, 20C34.

Keywords

$\ell$-adic cohomologyHecke correspondencefunny curveGalois grouphyperflexinflectional pentagoninflectional tangentinvariantJacobian varietyMathieu groupmodular curvemodular representationtwisted curvezeta function
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 74 , Issue 1 , January 1997 , pp. 1 - 28
Copyright
London Mathematical Society 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)