A family of Calabi-Yau varieties and potential automorphy

Abstract

We prove potential modularity theorems for $l$-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral $j$-invariant defined over a totally real field.

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      MRCLASS = {11F80 (11G40)},
      MRNUMBER = {2004c:11082},
      MRREVIEWER = {Ravi K. Ramakrishna},
      DOI = {10.1017/S1474748002000038},
      ZBLNUMBER = {1047.11051},
      }
  • [mero] R. Taylor, "On the meromorphic continuation of degree two $L$-functions," Doc. Math., iss. Extra Vol., pp. 729-779, 2006.
    @article {mero, MRKEY = {2290604},
      AUTHOR = {Taylor, Richard},
      TITLE = {On the meromorphic continuation of degree two {$L$}-functions},
      JOURNAL = {Doc. Math.},
      FJOURNAL = {Documenta Mathematica},
      YEAR = {2006},
      NUMBER = {Extra Vol.},
      PAGES = {729--779},
      ISSN = {1431-0635},
      MRCLASS = {11R39 (11F80 11G40)},
      MRNUMBER = {2008c:11154},
      MRREVIEWER = {M. Ram Murty},
      ZBLNUMBER = {1138.11051},
      }
  • [tay] R. Taylor, "Automorphy of some $l$-adic lifts of automorphic mod $l$ representations II," Publ. Math. I.H.E.S., vol. 108, pp. 183-239, 2008.
    @article{tay,
      author={Taylor, Richard},
      TITLE={Automorphy of some $l$-adic lifts of automorphic mod $l$ representations II},
      JOURNAL={Publ. Math. {\rm I.H.E.S.}},
      VOLUME={108},
      YEAR={2008},
      PAGES={183--239},
      ZBLNUMBER={1169.11021},
      }
  • [tw] Go to document R. Taylor and A. Wiles, "Ring-theoretic properties of certain Hecke algebras," Ann. of Math., vol. 141, iss. 3, pp. 553-572, 1995.
    @article {tw, MRKEY = {1333036},
      AUTHOR = {Taylor, Richard and Wiles, Andrew},
      TITLE = {Ring-theoretic properties of certain {H}ecke algebras},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {141},
      YEAR = {1995},
      NUMBER = {3},
      PAGES = {553--572},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G18 (11D41 11F11 11R34 13C40)},
      MRNUMBER = {96d:11072},
      MRREVIEWER = {Karl Rubin},
      DOI = {10.2307/2118560},
      ZBLNUMBER = {0823.11030},
      }
  • [ty] Go to document R. Taylor and T. Yoshida, "Compatibility of local and global Langlands correspondences," J. Amer. Math. Soc., vol. 20, iss. 2, pp. 467-493, 2007.
    @article {ty, MRKEY = {2276777},
      AUTHOR = {Taylor, Richard and Yoshida, Teruyoshi},
      TITLE = {Compatibility of local and global {L}anglands correspondences},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {20},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {467--493},
      ISSN = {0894-0347},
      MRCLASS = {11R39 (11F70 11F80 14G35 22E55)},
      MRNUMBER = {2007k:11193},
      MRREVIEWER = {Alexey A. Panchishkin},
      DOI = {10.1090/S0894-0347-06-00542-X},
      ZBLNUMBER = {05120650},
      }
  • [wiles] Go to document A. Wiles, "Modular elliptic curves and Fermat’s last theorem," Ann. of Math., vol. 141, iss. 3, pp. 443-551, 1995.
    @article {wiles, MRKEY = {1333035},
      AUTHOR = {Wiles, Andrew},
      TITLE = {Modular elliptic curves and {F}ermat's last theorem},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {141},
      YEAR = {1995},
      NUMBER = {3},
      PAGES = {443--551},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11G05 (11D41 11F11 11F80 11G18)},
      MRNUMBER = {96d:11071},
      MRREVIEWER = {Karl Rubin},
      DOI = {10.2307/2118559},
      ZBLNUMBER = {0823.11029},
      }

Authors

Michael Harris

Centre de Mathématiques de Jussieu
Université Paris 7 Denis Diderot
Case Postale 7012
2, place Jussieu
F-75251 Paris Cedex 05
France

Nick Shepherd-Barron

DPMMS
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
Cambridge CB3 0WB
England

Richard Taylor

Department of Mathematics
One Oxford Street
Harvard University
Cambridge, MA 02138
United States