Ramified primes in the field of definition for the Mordell-Weil group of an elliptic surface

Author:
Masato Kuwata

Journal:
Proc. Amer. Math. Soc. **116** (1992), 955-959

MSC:
Primary 11G35; Secondary 14D10, 14G05, 14J27

DOI:
https://doi.org/10.1090/S0002-9939-1992-1116264-0

MathSciNet review:
1116264

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an elliptic surface defined over a number field . We consider the field in which all the sections are defined. Assuming that the -invariant is nonconstant, is again a number field. We describe the primes of possible ramification of the extension in terms of the configuration of the points of bad fibers in . Aside from few possible exceptions, is unramified outside of the primes of bad reduction of and the primes for which two or more points of bad fibers become identical modulo .

**[B]**Sybilla Beckmann,*Ramified primes in the field of moduli of branched coverings of curves*, J. Algebra**125**(1989), no. 1, 236–255. MR**1012673**, https://doi.org/10.1016/0021-8693(89)90303-7**[K]**Masato Kuwata,*The field of definition of the Mordell-Weil group of an elliptic curve over a function field*, Compositio Math.**76**(1990), no. 3, 399–406. MR**1080009****[Mil1]**J. S. Milne,*Abelian varieties*, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103–150. MR**861974****[Mil2]**J. S. Milne,*Jacobian varieties*, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR**861976****[Mir]**Rick Miranda,*The basic theory of elliptic surfaces*, Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research], ETS Editrice, Pisa, 1989. MR**1078016****[Sil]**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****[Sh1]**Tetsuji Shioda,*The Galois representation of type 𝐸₈ arising from certain Mordell-Weil groups*, Proc. Japan Acad. Ser. A Math. Sci.**65**(1989), no. 6, 195–197. MR**1011870****[Sh2]**Tetsuji Shioda,*Mordell-Weil lattices and Galois representation. I*, Proc. Japan Acad. Ser. A Math. Sci.**65**(1989), no. 7, 268–271. MR**1030197**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
11G35,
14D10,
14G05,
14J27

Retrieve articles in all journals with MSC: 11G35, 14D10, 14G05, 14J27

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1116264-0

Keywords:
Elliptic surfaces

Article copyright:
© Copyright 1992
American Mathematical Society