Rigidity of p-adic cohomology classes of congruence subgroups of GL(n, ℤ)

• Published in: Proceedings of the London Mathematical Society 2008-03, Vol.96 (2), p.367-388
• Main Authors: Ash, Avner, Pollack, David, Stevens, Glenn
• Format: Article
• Language: English
• ISSN: 0024-6115; 1460-244X
• DOI: 10.1112/plms/pdm048

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.