Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Congruence formula for certain dihedral twists

Authors: Sudhanshu Shekhar and R. Sujatha
Journal: Trans. Amer. Math. Soc. 367 (2015), 3579-3598
MSC (2010): Primary 14H52, 11F80, 11F11, 11F33
Published electronically: November 4, 2014
MathSciNet review: 3314817
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this article we prove a congruence formula for the special values of certain dihedral twists of two primitive modular forms of weight two with isomorphic residual Galois representation at a prime $ p$.

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

  • [B] Thanasis Bouganis, Special values of $ L$-functions and false Tate curve extensions, J. Lond. Math. Soc. (2) 82 (2010), no. 3, 596-620. With an appendix by Vladimir Dokchitser. MR 2739058 (2012g:11120),
  • [CE] Robert F. Coleman and Bas Edixhoven, On the semi-simplicity of the $ U_p$-operator on modular forms, Math. Ann. 310 (1998), no. 1, 119-127. MR 1600034 (99b:11043),
  • [Cr] J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151 (93m:11053)
  • [D] Fred Diamond, Congruences between modular forms: raising the level and dropping Euler factors, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11143-11146. Elliptic curves and modular forms (Washington, DC, 1996). MR 1491976 (98m:11033),
  • [DD] T. Dokchitser and V. Dokchitser, Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3) 94 (2007), no. 1, 211-272. With an appendix by J. Coates and R. Sujatha. MR 2294995 (2008g:11106),
  • [DDT] Henri Darmon, Fred Diamond, and Richard Taylor, Fermat's last theorem, Current developments in mathematics, 1995 (Cambridge, MA), Int. Press, Cambridge, MA, 1994, pp. 1-154. MR 1474977 (99d:11067a)
  • [DFG] Fred Diamond, Matthias Flach, and Li Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 663-727 (English, with English and French summaries). MR 2103471 (2006e:11089),
  • [EK] E. Kani, Binary theta series and modular forms with complex multiplication, Int. J. Number Theory 10 (2014), no. 4, 1025-1042. MR 3208873
  • [GV] Ralph Greenberg and Vinayak Vatsal, On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), no. 1, 17-63. MR 1784796 (2001g:11169),
  • [H] Haruzo Hida, A $ p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 3, 1-83. MR 976685 (89k:11120)
  • [H1] Haruzo Hida, A $ p$-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. Math. 79 (1985), no. 1, 159-195. MR 774534 (86m:11097),
  • [H2] Haruzo Hida, Congruence of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), no. 2, 225-261. MR 610538 (82g:10044),
  • [RS] K. Rubin and A. Silverberg, Families of elliptic curves with constant mod $ p$ representations, Elliptic curves, modular forms, and Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 148-161. MR 1363500 (96j:11078)
  • [SS] Sudhanshu Shekhar and R. Sujatha, Euler characteristic and congruences of elliptic curves, Münster J. of Math. 7 (2014), 327-343.
  • [V] V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), no. 2, 397-419. MR 1695203 (2000g:11032),
  • [Wi] Andrew Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035 (96d:11071),
  • [WS] W. Stein, Modular forms database,

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14H52, 11F80, 11F11, 11F33

Retrieve articles in all journals with MSC (2010): 14H52, 11F80, 11F11, 11F33

Additional Information

Sudhanshu Shekhar
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai-400005, India

R. Sujatha
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Keywords: Elliptic curve, Galois representation, modular forms, congruences for modular forms, special values of $L$-series, periods of modular forms
Received by editor(s): October 27, 2012
Received by editor(s) in revised form: July 2, 2013
Published electronically: November 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society