no. 3
The field of definition of the Mordell-Weil group of an elliptic curve over a function field
Compositio Mathematica, Tome 76 (1990) no. 3, pp. 399-406. 
@article{CM_1990__76_3_399_0,
     author = {Kuwata, Masato},
     title = {The field of definition of the Mordell-Weil group of an elliptic curve over a function field},
     journal = {Compositio Mathematica},
     pages = {399--406},
     publisher = {Kluwer Academic Publishers},
     volume = {76},
     number = {3},
     year = {1990},
     zbl = {0731.14010},
     mrnumber = {1080009},
     language = {en},
     url = {archive.numdam.org/item/CM_1990__76_3_399_0/}
}
Kuwata, Masato. The field of definition of the Mordell-Weil group of an elliptic curve over a function field. Compositio Mathematica, Tome 76 (1990) no. 3, pp. 399-406. http://archive.numdam.org/item/CM_1990__76_3_399_0/

[C] M.R. Christie, Positive definite rational functions of two variables which are not the sum of three squares, Journal of Number Theory, 8 (1976), 224-232. | MR 412162 | Zbl 0331.14017

[K] Masato Kuwata, The Canonical Height and Elliptic Surfaces, (to appear from Journal of Number Theory). | MR 1072465 | Zbl 0741.14008

[Sil] J.H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986. | MR 817210 | Zbl 0585.14026

[S-D] H.P.F. Swinnerton-Dyer, The field of definition of the Néron-Severi group, Studies in Pure Mathematics, 719-731. | MR 820265 | Zbl 0575.14018