Variation of the root number in families of elliptic curves
Compositio Mathematica, Volume 87 (1993) no. 2, p. 119-151
@article{CM_1993__87_2_119_0,
     author = {Rohrlich, David E.},
     title = {Variation of the root number in families of elliptic curves},
     journal = {Compositio Mathematica},
     publisher = {Kluwer Academic Publishers},
     volume = {87},
     number = {2},
     year = {1993},
     pages = {119-151},
     zbl = {0791.11026},
     mrnumber = {1219633},
     language = {en},
     url = {http://www.numdam.org/item/CM_1993__87_2_119_0}
}
Rohrlich, David E. Variation of the root number in families of elliptic curves. Compositio Mathematica, Volume 87 (1993) no. 2, pp. 119-151. http://www.numdam.org/item/CM_1993__87_2_119_0/

1 B.J. Birch and N.M. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5 (1966), 295-299. | MR 201379 | Zbl 0146.42401

2 J.W.S. Cassels, Arithmetic on curves of genus 1. VIII, J. reine angew. Math. 217 (1965), 180-199. | MR 179169 | Zbl 0241.14017

3 J.W.S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193-291. | MR 199150 | Zbl 0138.27002

4 J.W.S. Cassels and A. Schinzel, Selmer's conjecture and families of elliptic curves, Bull. London Math. Soc. 14 (1982), 345-348. | MR 663485 | Zbl 0474.14010

5 I. Connell, Good reduction of elliptic curves in abelian extensions, J. reine angew. Math. (to appear). | MR 1207284 | Zbl 0759.14025

6 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 349635 | Zbl 0271.14011

7 B. Edixhoven, A. De Groot, and J. Top, Elliptic curves over the rationals with bad reduction at only one prime, Math. of Comp. 54 (1990), 413-419. | MR 995209 | Zbl 0756.14030

8 A. Fröhlich and J. Queyrut, On the functional equation of the Artin L-function for characters of real representations, Invent. Math. 20 (1973), 125-138. | MR 321888 | Zbl 0256.12010

9 P.X. Gallagher, Determinants of representations of finite groups, Abh. Math. Sem. Univ. Hamburg 28 (1965), 162-167. | MR 185017 | Zbl 0132.27306

10 F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4 (1991), 1-23. | MR 1080648 | Zbl 0725.11027

11 L.D. Olson, Torsion points on elliptic curves with given j-invariant, Manuscripta Math. 16 (1975), 145-150. | MR 371898 | Zbl 0314.14006

12 D.E. Rohrlich, Nonvanishing of L-functions and structure of Mordell-Weil groups, J. reine angew. Math. 417 (1991), 1-26. | MR 1103904 | Zbl 0731.11030

13 E.S. Selmer, A conjecture concerning rational points on cubic curves, Math. Scand. 2 (1954), 49-54. | MR 62767 | Zbl 0055.27107

14 J-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. | MR 387283 | Zbl 0235.14012

15 J-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. Math. 68 (1968), 492-517. | MR 236190 | Zbl 0172.46101

16 T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. Sancti Pauli 39 (1990), 211-240. | MR 1081832 | Zbl 0725.14017

17 J.H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1985. | MR 817210 | Zbl 0585.14026

18 J. Tate, Algorithm for finding the type of a singular fiber in an elliptic pencil, Modular Functions of One Variable IV, Lect. Notes Math. 476, Springer-Verlag, 1975, pp. 33-52. | MR 393039

19 J. Tate, Number theoretic background, Automorphic Forms, Representations, and L-Functions, Proc. Symp. Pure Math. Vol. 33-Part 2, Amer. Math. Soc., Providence, 1979, pp. 3-26. | MR 546607 | Zbl 0422.12007

20 J. Tunnell, On the local Langlands conjecture for GL(2), Invent. Math. 46(1978), 179-200. | MR 476703 | Zbl 0385.12006

21 J. Tunnell, Report on the local Langlands conjecture for GL(2), Automorphic Forms, Representations, and L-Functions, Proc. Symp. Pure Math. Vol. 33-Part 2, Amer. Math. Soc., Providence, 1979. | MR 546614 | Zbl 0409.12026

22 J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219-307. | MR 1103429 | Zbl 0724.11026