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By P. F. Hsieh, A. W. J. Stoddart

ISBN-10: 3540053697

ISBN-13: 9783540053699

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More generally, note that the groups are interrelated - if then there is a natural surjection The kernel of this map can be studied. For instance, for our situation, with S = {q}, T = {p,q}, the kernel is the critical subgroup. References [1] L. Bartholdi, Endomorphic presentations of branch groups, J. Algebra 268 (2003), 419– 443. Cannon, Handbook of MAGMA Functions, Sydney: School of Mathematics and Statistics, University of Sydney (1993). Boston, Some Cases of the Fontaine-Mazur Conjecture II, J.

The first possibility implies that P centralizes a Sylow s-subgroup. Repeating this argument for each prime s dividing implies that P centralizes R, whence the normal closure of P does as well. Thus, G centralizes R, a contradiction to the minimality of R. So we may assume that R commutes with G and so by minimality has prime order r. We consider the possibilities for the inertia groups of G on X/R. These are precisely where I is an inertia group of RG on X. We identify GR/R with G. First consider the case where Then IR / R contains P, whence by the structure of inertia groups and the subgroup structure of G, IR / R = P.

Igusa, Arithmetic variety of moduli for genus two, Annals Math. 72 (1960), 612-649. [Mor] C. J. Moreno, Algebraic Curves over Finite Fields, Cambridge Tracts in Mathematics 97, CUP, 1991. Automorphisms of the modular curve [Raj] C. Rajan, Automorphisms of X(11) over characteristic 3 and the Mathieu group Ramanujan Math. Soc. 13 (1998), 63–72. 37 J. [Rit] C. Ritzenthaler, Automorphismes des courbes modulaires X(n) en caractéristique p, Manuscripta Math. 109 (2002), 49–62. [Ser] Jean-Pierre Serre, Local Fields, GTM 67, 1979.

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Analytic Theory of Differential Equations by P. F. Hsieh, A. W. J. Stoddart

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