AUTOMORPHISMS OF X(11) OVER CHARACTERISTIC 3, AND THE MATHIEU GROUP M 11 C. S. RAJAN Abstract. We show that the automorphism group of the curve X(11) is the Mathieu group M 11 , over a field of characteristic 3, containing a 11 th root of unity. As a consequence, we also exhibit a Galois covering of the affine line over characteristic 3, with Galois group the Mathieu group M 11 . 1. The automorphism group of the modular curve X(p), for a prime p ≥ 7, over a field of characteristic zero, is known to be the group PSL(2, F p ). However, over a field k of characteristic 3, containing a 11 th root of unity, it was shown by Adler in ([A]), that a larger group acts as automorphisms on the curve X(11). Adler exhibited an action of the Mathieu group M 11 , the smallest amongst the sporadic simple groups, on the curve X(11), over k. Our aim is to show that over such fields, Aut(X(11)) is precisely the group M 11 . Apart from using some properties of M 11 , and the M 11 action on X(11), our main tool is the Riemann-Hurwitz formula, together with a result of Henn ([H]), classifying curves with large automorphism groups. The main step is to show that the automorphism group leaves invariant the singular fibres of the map from X(11) to the quotient of X(11) by the M 11 action. 2. We recall some of the facts we need from ramification theory ([S]). Let k be an algebraically closed field of characteristic p. Let X, Y be smooth projective curves over k, respectively of genus g(X),g(Y ). Suppose f : X → Y is a branched Galois cover of degree n, with Galois group G. Let Q 1 , ··· ,Q l be the branch points of f in Y . For i =1, ··· ,l, let r i denote the number of closed points in the fiber f −1 (Q i ), and e i denote the exponent of ramification of any closed point P i lying above Q i . We have, (1) |G| = n = r i e i , i =1, ··· , l. For a ramified point P on X, let G(P ) be the isotropy subgroup of G, fixing P . We have the sequence of higher ramification groups of P , G(P )= G 0 (P ) ⊃ G 1 (P ) ⊃···⊃ G m (P ) = (1), where G 0 (P ) is the inertia group at P . G 1 (P ) is the wild part of the inertia group at P and is a p-Sylow subgroup of G 0 (P ). G 0 (P ) is the semi-direct product of a cyclic subgroup H and G 1 (P ), and H maps isomorphically onto the tame ramification group G 0 (P )/G 1 (P ). Further, for a point P i lying above Q i , the ramification index e i of P i is |G 0 (P )|. We have the Hurwitz formula, (2) Hurwitz formula : 2g(Y ) − 2 = (2g(X) − 2)|G| + l  i=1 r i e ′ i , 1991 Mathematics Subject Classification. Primary 14H45; Secondary 14E09, 20D08. 1