The Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 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 invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.