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The Mathieu Group M 11 and the Modular Curve X (11)
Author(s) -
Adler A.
Publication year - 1997
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611597000014
Subject(s) - mathematics , elliptic curve , modular form , inflection point , invariant (physics) , modular group , group (periodic table) , automorphism group , modular elliptic curve , modular curve , automorphism , pure mathematics , combinatorics , geometry , quarter period , mathematical physics , chemistry , organic chemistry
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 We also comment on the relation of our work to some unpublished work of Bott and Tate. 1991 Mathematics Subject Classification : 11F32, 11G20, 14G10, 14H10, 14N10, 20B25, 20C34.