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Brigham Young University
Math Department

Darrin Doud

322 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.
  • Andrzej Dąbrowski, Hipoteza Serre'a o modularności i nowe dowody Wieldiego Twierdzenia Fermata, Wiad. Mat. 45 (2009), 3-24.
  • 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.
  • Yuichiro Taguchi, Introduction to Serre's modularity conjecture on Galois Representations (in Japanese), RIMS Kokyuoku Bessatsu, B19 (2010), 7-22.
  • Avner Ash, Paul Gunnells, Mark McConnell, Torsion in the cohomology of congruence sugroups of SL(4,Z) and Galois representations, Journal of Algebra 325 (2011), 404-415.
  • Toby Gee, Automorphic lifts of prescribed types, Mathematische Annalen, 350 (2011), 107-144.
  • Adam Mohamed, Some explicit aspects of modular forms over imaginary quadratic fields, Ph.D. Thesis, Universitat Duisburg-Essen, Essen, 2011.
  • Toby Gee and David Geraghty, Companion forms for unitary and symplectic groups, Duke Math. Journal, 161 (2012), 247-303.
  • Władysław Narkievicz, Rational Number Theory in the 20th century. From PNT to FLT., Springer Monographs in Mathematics, Springer, London, 2012.
  • Rebecca Torrey, On Serre's conjecture over imaginary quadratic fields, J. Number Theory 132 (2012), 637-656.
  • Becky E. Hall, Computing homology using generalized Gröbner bases, J. Symbolic Computation 54 (2013), 59-71.
  • Nicolas Bergeron and Akshay Venkatesh, The Asymptotic Growth of Torsion Homology for arithmetic groups, J. Inst. Math. Jussieu 12 (2013), 391-447.
  • Florian Herzig and J. Tilouine, Conjecture de type de Serre et formes compagnons pour GSp4, J. Reine Angew. Math 676 (2013), 1-32.
  • Simon Marshall and Werner Muller, On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds, Duke Math J. 162 (2013), 863-888.
  • Avner Ash, Direct sums of mod p characters of GQ and the homology of GLn(Z), Communications in Algebra 41 (2013, 1751-1775.
  • Matthew Emerton, Toby Gee, and Florian Herzig, Weight cycling and Serre-type conjectures for unitary groups, Duke Math. J. 162 (2013), 1649-1722.
  • Avner Ash and Darrin Doud, Reducible Galois representations and the homology of GL(3,Z), Int. Math. Res. Not. 2014 (2014), 1379-1408.
  • Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Local-global compatibility for l=p, II, Annales scientifiques de l'ENS 47 (2014), 165-179.
  • Avner Ash, Paul Gunnells, and Mark McConnell, Mod 2 homology for GL(4) and Galois representations, J. Number Theory 146 (2015), 4-22.
  • Avner Ash and Darrin Doud, Highly reducible Galois representations attached to the homology of GL(n,Z), Proc. Am. Math. Soc. 143 (2015), 3801-3813.
  • Peter Scholze, On torsion in the cohomology of locally symmetric varieties, Annals of Mathematics 182 (2015), 945-1066.
  • Jared Weinstein, Reciprocity Laws and Galois Representations: Recent Breakthroughs, Bulletin of the AMS 53 (2016), 1--39.
  • Avner Ash and Darrin Doud, Relaxation of strict parity for reducible Galois representations attached to the homology of GL(3,Z), Int. J. Number Theorem 12 (2016), 361-381.
  • James Newton and Jack A. Thorne, Torsion Galois representations over CM fields and Hecke algebras in the derived category, Forum Math. Sigma 4 (2016), 88 pp.
  • Avner Ash and Darrin Doud, Galois representations attached to tensor products of arithmetic cohomology, J. Algebra 465 (2016), 81-99.
  • Adam Mohamed, Weight reduction for cohomological mod p modular forms over imaginary quadratic fields, ArXiv preprint, 2011.
  • Stefano Morra and Chol Park, Serre weights for three-dimensional ordinary Galois representations, preprint, 2015
  • Toby Gee, Florian Herzig, and David Savitt, General Serre Weight Conjectures, ArXiv preprint, 2015.

Maintained by Darrin Doud.

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