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Rigidity of p ‐adic cohomology classes of congruence subgroups of GL( n , ℤ)
Author(s) -
Ash Avner,
Pollack David,
Stevens Glenn
Publication year - 2008
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/plms/pdm048
Subject(s) - mathematics , rigidity (electromagnetism) , cohomology , modulo , congruence (geometry) , pure mathematics , congruence relation , conjecture , congruence subgroup , deformation theory , algebra over a field , arithmetic , discrete mathematics , geometry , structural engineering , engineering
This paper provides foundations for studying p ‐adic deformations of arithmetic eigenpackets, that is, of systems of Hecke eigenvalues occurring in the cohomology of arithmetic groups with coefficients in finite‐dimensional rational representations. The concept of ‘arithmetic rigidity’ of an arithmetic eigenpacket is introduced and investigated. An arithmetic eigenpacket is said to be ‘arithmetically rigid’ if (modulo twisting) it does not admit a p ‐adic deformation containing a Zariski dense set of arithmetic specializations. The case of GL( n ) and ordinary eigenpackets is worked out, leading to the construction of a ‘universal p ‐ordinary arithmetic eigenpacket’. Tools for explicit investigation into the structure of the associated eigenvarieties for GL( n ) are developed. Of note is the purely algebraic Theorem 5.1, which keeps track of the specializations of the universal eigenpacket. We use these tools to prove that known examples of non‐selfdual cohomological cuspforms for GL(3) are arithmetically rigid. Moreover, we conjecture that, in general, arithmetic rigidity for GL(3) is equivalent to non‐selfduality.