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On Some l ‐Adic Representations of Gal( ( Q ¯ / Q ) Attached to Noncongruence Subgroups
Author(s) -
Scholl A. J.
Publication year - 2006
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s002460930601856x
Subject(s) - mathematics , cohomology , galois module , dimension (graph theory) , image (mathematics) , pure mathematics , element (criminal law) , group (periodic table) , galois group , reductive group , combinatorics , algebra over a field , group theory , chemistry , organic chemistry , artificial intelligence , computer science , political science , law
The l ‐adic parabolic cohomology groups attached to noncongruence subgroups of SL 2 (Z) are finite‐dimensional l ‐adic representations ofGal ( Q ¯ / K )for some number field K . We exhibit examples (with K = Q) for which the primitive parts give Galois representations whose images are open subgroups of the full group of symplectic similitudes (of arbitrary dimension). The determination of the image of the Galois group relies on Katz's classification theorem for semisimple subalgebras of sl n containing a principal nilpotent element, for which we give a short conceptual proof, suggested by I. Grojnowski. 2000 Mathematics Subject Classification 11F80 (primary), 11G18 (secondary).