Potential automorphy and change of weight

Abstract

We prove an automorphy lifting theorem for $l$-adic representations where we impose a new condition at $l$, which we call “potentential diagonalizability.” This result allows for “change of weight” and seems to be substantially more flexible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of $l$-adic representations of the absolute Galois group of a totally real field is potentially automorphic and hence is pure and its $L$-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

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    @book {MR1007299, MRKEY = {1007299},
      AUTHOR = {Arthur, James and Clozel, Laurent},
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      TITLE = {\'{E}léments de géométrie algébrique. {IV}. \'{E}tude locale des schémas et des morphismes de schémas. {III}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      VOLUME = {28},
      YEAR = {1966},
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      }
  • [harris:manin] Go to document M. Harris, "Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications," in Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin. Vol. II, Boston, MA: Birkhäuser, 2009, vol. 270, pp. 1-21.
    @incollection {harris:manin, MRKEY = {2641185},
      AUTHOR = {Harris, Michael},
      TITLE = {Potential automorphy of odd-dimensional symmetric powers of elliptic curves and applications},
      BOOKTITLE = {Algebra, Arithmetic, and Geometry: in Honor of {Y}u. {I}. {M}anin. {V}ol. {II}},
      SERIES = {Progr. Math.},
      VOLUME = {270},
      PAGES = {1--21},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2009},
      MRCLASS = {11F80 (11F70 11G05 22E55)},
      MRNUMBER = {2641185},
      MRREVIEWER = {Neil P. Dummigan},
      DOI = {10.1007/978-0-8176-4747-6_1},
      ZBLNUMBER = {1234.11068},
     }
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      AUTHOR = {Harris, Michael and Shepherd-Barron, Nick and Taylor, Richard},
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      JOURNAL = {Ann. of Math.},
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      MRCLASS = {11F80 (11G18 11G35 11R34 14J32)},
      MRNUMBER = {2630056},
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     }
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      AUTHOR = {Harris, Michael and Taylor, Richard},
      TITLE = {The Geometry and Cohomology of Some Simple {S}himura Varieties},
      SERIES = {Ann. of Math. Stud.},
      NUMBER = {151},
      TITLENOTE = {with an appendix by Vladimir G. Berkovich},
      PUBLISHER = {Princeton Univ. Press},
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      PAGES = {viii+276},
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      MRCLASS = {11G18 (11F70 11S37 14G35 22E45)},
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      JOURNAL = {Amer. J. Math.},
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      AUTHOR = {Langlands, R. P.},
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      AUTHOR = {Mazur, B.},
      TITLE = {Deforming {G}alois representations},
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      AUTHOR = {Nekov{á}{\v{r}},
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      AUTHOR = {Serre, {\relax J-P}},
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      PAGES = {xvi+177 pp. (not consecutively paged)},
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      AUTHOR = {Tate, J.},
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      FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
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Authors

Thomas Barnet-Lamb

Brandeis University, Waltham, MA

Toby Gee

Imperial College London, London, UK

David Geraghty

Boston College, Chestnut Hill, MA

Richard Taylor

Institute for Advanced Study, Princeton, NJ