Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Scarcity and abundance of trivial zeros in division towers

Author: David E. Rohrlich
Journal: J. Algebraic Geom. 17 (2008), 643-675
Published electronically: February 7, 2008
MathSciNet review: 2424923
Full-text PDF

Abstract | References | Additional Information

Abstract: Explicit formulas and asymptotic estimates are derived for twisted root numbers of elliptic curves in division towers. The key assumption on the elliptic curves considered is that the image of the Galois representation afforded by the first layer of the division tower is contained in a Borel subgroup. In [Math. Research Letters 13 (2006), 359-376], by contrast, the Galois representation was assumed to be surjective.

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

  • 1. John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, and Otmar Venjakob, The 𝐺𝐿₂ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208. MR 2217048, 10.1007/s10240-004-0029-3
  • 2. J. Coates and R. Sujatha, Fine Selmer groups of elliptic curves over 𝑝-adic Lie extensions, Math. Ann. 331 (2005), no. 4, 809–839. MR 2148798, 10.1007/s00208-004-0609-z
  • 3. J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
  • 4. P. Deligne, Les constantes des équations fonctionnelles des fonctions 𝐿, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1973, pp. 501–597. Lecture Notes in Math., Vol. 349 (French). MR 0349635
  • 5. Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
  • 6. Tom Fisher, Some examples of 5 and 7 descent for elliptic curves over 𝑄, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 169–201. MR 1831874, 10.1007/s100970100030
  • 7. P. X. Gallagher, Determinants of representations of finite groups, Abh. Math. Sem. Univ. Hamburg 28 (1965), 162–167. MR 0185017
  • 8. Ralph Greenberg, Nonvanishing of certain values of 𝐿-functions, Analytic number theory and Diophantine problems (Stillwater, OK, 1984), Progr. Math., vol. 70, Birkhäuser Boston, Boston, MA, 1987, pp. 223–235. MR 1018378
  • 9. Lawrence Howe, Twisted Hasse-Weil 𝐿-functions and the rank of Mordell-Weil groups, Canad. J. Math. 49 (1997), no. 4, 749–771. MR 1471055, 10.4153/CJM-1997-037-7
  • 10. David E. Rohrlich, Elliptic curves and the Weil-Deligne group, Elliptic curves and related topics, CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 125–157. MR 1260960
  • 11. David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. MR 1387669
  • 12. David E. Rohrlich, Root numbers of semistable elliptic curves in division towers, Math. Res. Lett. 13 (2006), no. 2-3, 359–376. MR 2231124, 10.4310/MRL.2006.v13.n3.a3
  • 13. Jean-Pierre Serre, Abelian 𝑙-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0263823
  • 14. Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 0387283
  • 15. Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380
  • 16. Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 0236190
  • 17. M. Shuter, Rational points of elliptic curves in $ p$-division fields (to appear).
  • 18. J. Tate, Number theoretic background, Automorphic forms, representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607

Additional Information

David E. Rohrlich
Affiliation: Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215

Received by editor(s): July 4, 2006
Published electronically: February 7, 2008

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
AMS Website