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Math Department

Darrin Doud

282 TMCB
Department of Mathematics
Brigham Young University
Provo, UT 84602
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Galois representations with conjectural connections to arithmetic cohomology

with Avner Ash and David Pollack
Duke Mathematical Journal 112(2002), 521-579.

Abstract In this paper we extend a conjecture of Ash and Sinnott relating niveau one Galois representations to the mod p cohomology of congruence subgroups of SL(n,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 which 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.

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Cited by

  • Avner Ash, Smith theory and Hecke operators, J. Algebra, 259 (2003), 43--58.
  • Darrin Doud,Three-dimensional Galois representations with conjectural connections to arithmetic cohomology. Number theory for the millennium, I (Urbana, IL, 2000), 365--375, A K Peters, Natick, MA, 2002.
  • Hyunsuk Moon, The non-existence of certain mod p Galois representations, Bull. Korean Math. Soc., 40 (2003), 537--544.
  • Luis Dieulefait and Nuria Vila, On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335--361.
  • David P. Roberts, Frobenius Classes in Alternating Groups, Rocky Mountain J. of Math., 34 (2004) 1483--1496.
  • Avner Ash, David Pollack and Dayna Soares, SL3(F2)-extensions of Q and arithmetic cohomology modulo 2, Experimental Mathematics, 13 (2004), 298--307.
  • Darrin Doud, Wildly Ramified Galois Representations and a generalization of a conjecture of Serre, Experimental Mathematics, 14 (2005), 119-127.
  • Avner Ash, David Pollack, and Warren Sinnott, A6-extensions of Q and the mod p cohomology of GL3(Z). J. Number Theory, 115 (2005), 176--196.
  • Florian Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Harvard University Ph.D. Thesis, 2006.
  • Darrin Doud and Michael W. Moore, Even icosahedral Galois representations of prime conductor. J. Number Theory, 118 (2006), 62--70.
  • Yuichiro Taguchi, On the finiteness of various Galois representations. Primes and knots, 249--261, Contemp. Math., 416, Amer. Math. Soc., Providence, RI, 2006.
  • Darrin Doud, Supersingular Galois representations and a generalization of a conjecture of Serre, Experimental Mathematics, 16 (2007), 119--128.
  • Hyunsuk Moon, On four-dimensional mod 2 Galois representations and a conjecture of Ash et al, Bull. Korean Math. Soc., 44(2007), 173--176.
  • Fred Diamond, A correspondence between representations of local Galois groups and Lie-type groups, L-functions and Galois representations, London Math. Soc. Lecture Note Ser. (3), 320, Cambridge Univ. Press, Cambridge, 2007, 187--206.
  • Hyunsuk Moon and Yuichiro Taguchi, On the finiteness and non-existence of certain mod 2 Galois representations of quadratic fields, Diophantine analysis and related fields--DARF 2007/2008, AIP Conf. Proc, 976 (2008) 169--175.
  • Hyunsuk Moon and Yuichiro Taguchi, On the finiteness and non-existence of certain mod 2 Galois representations of quadratic fields, Kyungpook Math. J., 48 (2008), 323-330.
  • Avner Ash, David Pollack, Glenn Stevens, Rigidity of p-adic cohomology classes of congruence subgroups of GL(n,Z), Proc. Lond. Math. Soc., (3), 96 (2008), 367--388.
  • Darrin Doud, Distinguishing contragredient Galois representations in characteristic two, Rocky Mountain J. Math., 38 (2008), 835--848.
  • Avner Ash and David Pollack, Everywhere unramified automorphic cohomology for SL3(Z), Int. J. Number Theory, 4 (2008), 663--675.
  • Michael M. Schein, Weights in Serre's conjecture for GLn via the Bernstein-Gelfand-Gelfand complex, J. Number Theory, 128 (2008), 2808--2822.
  • Michael M. Schein, Weights in generalizations of Serre's conjecture and the mod p local Langlands correspondence, in ``Symmetries in Algebra and Number Theory, I. Kersten and R. Meyer, eds., p. 115-138, Georg-August Universitat, Gottingen, 2008.
  • Meghan DeWitt and Darrin Doud, Finding Galois representations corresponding to certain Hecke eigenclasses, Int. J. Number Theory , 5 (2009), 1--11.
  • David Pollack and Robert Pollack, A construction of rigid analytic cohomology classes for congruence subgroups of SL3(Z), Canad. J. Math., 61 (2009), 674--690.
  • Florian Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Duke Math. J., 149 (2009), 37--116.
  • Rebecca Torrey, On Serre's Conjecture Over Imaginary Quadratic Fields, Ph.D. Thesis, King's College, London, University of London, 2009.
  • Darrin Doud and Russell Ricks, LLL reduction and a conjecture of Gunnells. Proc. Amer. Math. Soc. 138 (2010), 409--415.
  • Kevin Buzzard, Fred Diamond, and Frazier Jarvis, On Serre's conjecture for mod l Galois representations over totally real fields, Duke Math. J., 155 (2010), 105--161.
  • Avner Ash, Paul Gunnells, Mark McConnell, Torsion in the cohomology of congruence sugroups of SL(4,Z) and Galois representations, Journal of Albebra 325 (2011), 404-415.
  • Toby Gee, Automorphic lifts of prescribed types, Mathematische Annalen, 350 (2011), 107-144.
  • Florian Herzig and J. Tilouine, Conjecture de type de Serre et formes compagnons pour GSp4, preprint, arxiv.org, 2008.
  • Toby Gee and David Geraghty, Companion forms for unitary and symplectic groups, preprint, Arxiv.org, 2009.
  • Nicolas Bergeron and Akshay Venkatesh, The Asymptotic Growth of Torsion Homology for arithmetic groups, preprint, 2010.
  • Avner Ash, Direct sums of mod p characters of GQ and the homology of GLn(Z), preprint, 2010.
  • Simon Marshall and Werner Muller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, preprint, 2011.
  • Matthew Emerton, Toby Gee, and Florian Herzig, Weight cycling and Serre-type conjectures for unitary groups, arXiv preprint, 2011.
  • Rebecca Torrey, On Serre's conjecture over imaginary quadratic fields, arXiv preprint, 2011.
  • Adam Mohamed, Weight reduction for cohomological mod p modular forms over imaginary quadratic fields, ArXiv prepring, 2011.

Maintained by Darrin Doud.

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