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On the change of root numbers under twisting and applications


Author: Ariel Pacetti
Journal: Proc. Amer. Math. Soc. 141 (2013), 2615-2628
MSC (2010): Primary 11F70
DOI: https://doi.org/10.1090/S0002-9939-2013-11532-7
Published electronically: April 24, 2013
MathSciNet review: 3056552
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Abstract: The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can, for each odd prime $ p$, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is either Steinberg, principal series or supercuspidal at $ p$ by analyzing the change of sign under a suitable twist. We also explain the case $ p=2$, where twisting, in general, is not enough.


References [Enhancements On Off] (What's this?)

  • [AL70] A. O. L. Atkin and J. Lehner.
    Hecke operators on $ \Gamma _{0}(m)$.
    Math. Ann., 185:134-160, 1970. MR 0268123 (42:3022)
  • [AL78] A. O. L. Atkin and Wen Ch'ing Winnie Li.
    Twists of newforms and pseudo-eigenvalues of $ W$-operators.
    Invent. Math., 48(3):221-243, 1978. MR 508986 (80a:10040)
  • [BR99] Pilar Bayer and Anna Rio.
    Dyadic exercises for octahedral extensions.
    J. Reine Angew. Math., 517:1-17, 1999. MR 1728550 (2001a:11191)
  • [Car86] Henri Carayol.
    Sur les représentations $ l$-adiques associées aux formes modulaires de Hilbert.
    Ann. Sci. École Norm. Sup. (4), 19(3):409-468, 1986. MR 870690 (89c:11083)
  • [Dav00] Harold Davenport.
    Multiplicative number theory, volume 74 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, third edition, 2000.
    Revised and with a preface by Hugh L. Montgomery. MR 1790423 (2001f:11001)
  • [Del73] P. Deligne.
    Les constantes des équations fonctionnelles des fonctions $ L$.
    In Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pages 501-597. Lecture Notes in Math., Vol. 349. Springer, Berlin, 1973. MR 0349635 (50:2128)
  • [Dem05] Lassina Dembélé.
    Explicit computations of Hilbert modular forms on $ {\mathbb{Q}}(\sqrt {5})$.
    Experiment. Math., 14(4):457-466, 2005. MR 2193808 (2006h:11050)
  • [Dem08] Lassina Dembélé.
    An algorithm for modular elliptic curves over real quadratic fields.
    Experiment. Math., 17(4):427-438, 2008. MR 2484426 (2010a:11119)
  • [GK80] Paul Gérardin and Philip Kutzko.
    Facteurs locaux pour $ {\rm GL}(2)$.
    Ann. Sci. École Norm. Sup. (4), 13(3):349-384, 1980. MR 597744 (82i:22020)
  • [Hen79] Guy Henniart.
    Représentations du groupe de Weil d'un corps local, volume 2 of Publications Mathématiques d'Orsay 79 [Mathematical Publications of Orsay 79].
    Université de Paris-Sud Département de Mathématique, Orsay, 1979.
    With an English summary. MR 551495 (80m:12016)
  • [HPS89] Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske.
    The basis problem for modular forms on $ \Gamma _0(N)$.
    Mem. Amer. Math. Soc., 82(418):vi+159, 1989. MR 960090 (90d:11056)
  • [Li80] Wen Ch'ing Winnie Li.
    On the representations of $ {\rm GL}(2)$. I. $ \varepsilon $-factors and $ n$-closeness.
    J. Reine Angew. Math., 313:27-42, 1980. MR 552460 (81h:10042a)
  • [LW10] David Loeffler and Jared Weinstein.
    On the computation of local components of a newform.
    Math. Comp., 81(278):1179-1200 (2012). MR 2869056 (2012k:11064)
  • [PS10] Ariel Pacetti and Nicolás Sirolli.
    Computing ideal class representatives in quaternion algebras.
    Submitted, arXiv:1007.2821, 2010.
  • [Rio06] Anna Rio.
    Dyadic exercises for octahedral extensions. II.
    J. Number Theory, 118(2):172-188, 2006. MR 2223979 (2007c:11132)
  • [Roh93] David E. Rohrlich.
    Variation of the root number in families of elliptic curves.
    Compositio Math., 87(2):119-151, 1993. MR 1219633 (94d:11045)
  • [sag] SAGE mathematics software, version 4.5.2.
    http://www.sagemath.org/.
  • [Tat79] J. Tate.
    Number theoretic background.
    In Automorphic forms, representations and $ L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 3-26. Amer. Math. Soc., Providence, R.I., 1979. MR 546607 (80m:12009)
  • [Tor04] Gonzalo Tornaría.
    Data about the central values of the l-series of (imaginary and real) quadratic twists of elliptic curves, http://www.ma.utexas.edu/users/tornaria/cnt/.
    2004.
  • [Wei74] André Weil.
    Exercices dyadiques.
    Invent. Math., 27:1-22, 1974. MR 0379445 (52:350)

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Additional Information

Ariel Pacetti
Affiliation: Departamento de Matemática, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, CP 1428, Buenos Aires, Argentina
Email: apacetti@dm.uba.ar

DOI: https://doi.org/10.1090/S0002-9939-2013-11532-7
Keywords: Local factors, twisting epsilon factors
Received by editor(s): October 20, 2010
Received by editor(s) in revised form: November 8, 2011
Published electronically: April 24, 2013
Additional Notes: The first author was partially supported by PIP 2010-2012 GI and UBACyT X113
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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