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

Published online by Cambridge University Press:  01 January 1997

A Adler
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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 We also comment on the relation of our work to some unpublished work of Bott and Tate.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

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