Galois representations with conjectural connections to arithmetic cohomology

By A. Ash, D. Doud, D. Pollack. 2001

86 citations recorded by the source corpus.

In this paper we extend a conjecture of A. Ash and W. Sinnott relating niveau 1 Galois representations to the mod p cohomology of congruence subgroups ofSLn(Z) to include Galois representations of higher niveau. We then present computational evidence for our conjecture in the case n= 3 in the form of three-dimensional Galois representations which appear to correspond to cohomology eigenclasses as predicted by the conjecture. Our examples include Galois representations with nontrivial weight and level, as well as irreducible three-dimensional representations that are in no obvious way related to lower-dimensional representations. In addition, we prove that certain symmetric square representations are actually attached to cohomology eigenclasses predicted by the conjecture.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ac6f25b17de1223d3acd2b4a0c4ee6eda702650f

How researchers see this paper

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In Geometry and Topology

1. M. Emerton
2. T. Gee
3. David Savitt
4. F. Herzig
5. Akshay Venkatesh
6. Nicolas Bergeron
7. David Geraghty
8. C. Breuil

In Mathematical Physics

1. T. Gee
2. M. Emerton
3. David Savitt
4. F. Herzig
5. Akshay Venkatesh
6. Nicolas Bergeron
7. M. Kisin
8. C. Breuil

Citations in the corpus

• On torsion in the cohomology of locally symmetric varieties (2013, unknown): “[7], [8].”
• The asymptotic growth of torsion homology for arithmetic groups (2010, background): “The idea of such a conjecture is due to A. Ash and others and was formulated precisely for G = SLn in [2].”
• The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations (2008, methodology): “ we obtain for the conjecture is of two kinds. First, when n= 3, Doud and Pollack independently verified for us computationally (up to convincing bounds) for several explicit, tame ρ(taken mostly from [ADP02]) that W(ρ) contains those weights predicted by conjecture 1.1 but missing in the predictions”
• Weight cycling and Serre-type conjectures for unitary groups (2011, result): “ecisely those predicted by [Her09]. (The same set of weights is also predicted to be a subset of the modular weights by Doud’s modification [Dou07] of the conjecture of Ash, Doud, Pollack, and Sinnott [ADP02], [AS00].) We remark that other results have recently been obtained in [GG12] (in the ordinar”
• General Serre weight conjectures (2015, methodology): “met Haluk S¸engu¨n has independently reproduced the calculations of [Tor12] in unpublished work, and has found complete agreement, except that the two “missing” weights were also obtained.) The paper [ADP02] explicitly carries out calculations for the weight part of Serre’s conjecture (Conj. 2.1.5) ”