Scarcity and abundance of trivial zeros in division towers
Author:
David E. Rohrlich
Journal:
J. Algebraic Geom. 17 (2008), 643-675
DOI:
https://doi.org/10.1090/S1056-3911-08-00462-1
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
- John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, and Otmar Venjakob, The $\rm GL_2$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208. MR 2217048, DOI https://doi.org/10.1007/s10240-004-0029-3
- 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, DOI https://doi.org/10.1007/s00208-004-0609-z
- J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
- 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
- Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
- Tom Fisher, Some examples of 5 and 7 descent for elliptic curves over $\bf Q$, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 169–201. MR 1831874, DOI https://doi.org/10.1007/s100970100030
- P. X. Gallagher, Determinants of representations of finite groups, Abh. Math. Sem. Univ. Hamburg 28 (1965), 162–167. MR 185017, DOI https://doi.org/10.1007/BF02993247
- 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
- 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, DOI https://doi.org/10.4153/CJM-1997-037-7
- 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, DOI https://doi.org/10.1103/physrevlett.73.918
- David E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compositio Math. 100 (1996), no. 3, 311–349. MR 1387669
- David E. Rohrlich, Root numbers of semistable elliptic curves in division towers, Math. Res. Lett. 13 (2006), no. 2-3, 359–376. MR 2231124, DOI https://doi.org/10.4310/MRL.2006.v13.n3.a3
- Jean-Pierre Serre, Abelian $l$-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute. MR 0263823
- Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI https://doi.org/10.1007/BF01405086
- 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
- Jean-Pierre Serre and John Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. MR 236190, DOI https://doi.org/10.2307/1970722
- M. Shuter, Rational points of elliptic curves in $p$-division fields (to appear).
- 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
References
- J. Coates, T. Fukaya, K. Kato, R. Sujatha, and O. Venjakob, The $GL_{2}$ main conjecture for elliptic curves without complex multiplication, Publ. Math. IHES 101 (2005), 163–208. MR 2217048 (2007b:11172)
- J. Coates and R. Sujatha, Fine Selmer groups of elliptic curves over $p$-adic Lie extensions, Math. Ann. 331 (2005), 809–839. MR 2148798 (2006g:11219)
- J. E. Cremona, Algorithms for elliptic curves, Cambridge University Press, 1992. MR 1201151 (93m:11053)
- P. Deligne, Les constantes des équations fonctionelles des fonctions $L$, Modular Functions of One Variable, II, SLN 349, Springer-Verlag, New York, 1973, pp. 501–595. MR 0349635 (50:2128)
- W. Feit, Characters of Finite Groups, W. A. Benjamin, New York, 1967. MR 0219636 (36:2715)
- T. Fisher, Some examples of $5$ and $7$ descents for elliptic curves over $\mathbb {Q}$, Jour. Eur. Math. Soc. 3 (2001), 169–201. MR 1831874 (2002m:11045)
- P. X. Gallagher, Determinants of representations of finite groups, Abh. Math. Sem. Univ. Hamburg 28 (1965), 162–167. MR 0185017 (32:2487)
- R. Greenberg, Non-vanishing of certain values of $L$-functions, Analytic Number Theory and Diophantine Problems, Prog. in Math. 70, Birkhäuser, Boston, 1987, pp. 223–235. MR 1018378 (90j:11124)
- L. Howe, Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups, Can. J. Math. 49 (1997), 749–771. MR 1471055 (98h:11087)
- D. E. Rohrlich, Elliptic curves and the Weil-Deligne group, Elliptic Curves and Related Topics, CRM Proceedings & Lecture Notes, Vol. 4, Amer. Math. Society, Providence, 1994, pp. 125–157. MR 1260960 (95a:11054)
- D. E. Rohrlich, Galois theory, elliptic curves, and root numbers, Compos. Math. 100 (1996), 311–349. MR 1387669 (97m:11075)
- D. E. Rohrlich, Root numbers of semistable elliptic curves in division towers, Math. Research Letters 13 (2006), 359–376. MR 2231124 (2007c:11072)
- J.-P. Serre, Abelian $l$-adic Representations and Elliptic Curves, written with the collaboration of W. Kuyk and J. Labute, W. A. Benjamin, New York, 1968. MR 0263823 (41:8422)
- J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331. MR 0387283 (52:8126)
- J.-P. Serre, Linear Representations of Finite Groups, translated from the French by L. S. Scott, Springer-Verlag, New York, 1977. MR 0450380 (56:8675)
- J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. Math. 88 (1968), 492–517. MR 0236190 (38:4488)
- M. Shuter, Rational points of elliptic curves in $p$-division fields (to appear).
- J. Tate, Number theoretic background, Automorphic Forms, Representations, and $L$-Functions, Proc. Sympos. Pure Math. Vol. 33 – Part 2, Amer. Math. Society, Providence, 1979, pp. 3–26. MR 546607 (80m:12009)
Additional Information
David E. Rohrlich
Affiliation:
Department of Mathematics and Statistics, Boston University, Boston, Massachusetts 02215
MR Author ID:
149885
Email:
rohrlich@math.bu.edu
Received by editor(s):
July 4, 2006
Published electronically:
February 7, 2008
