Geodesic
A finiteness criterion for the number of rational points for twisted elliptic Weil curves
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987), pp. 41-53 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Weil elliptic curve $E/\mathbb{Q}$ and let $L(E,s)=\sum^\infty_{n=1}a(n)n^{-s}$ be its canonical $L$-series. Admitting the Birch–Swinnerton–Dyer conjecture and fixing the curve $E$, a criterion is given for the finiteness of the group $E_D(\mathbb{Q})$ for twisted elliptic curves $E_D$, defined by the condition $$ L(E_D,s)=\sum^\infty_{n=1}\chi(n)a(n)n^{-s}, $$ where $D$ is the discriminant of the quadratic field and $\chi(D)$ is its quadratic character. 
@article{ZNSL_1987_160_a4,
     author = {P. I. Guerzhoy and A. A. Panchishkin},
     title = {A finiteness criterion for the number of rational points for twisted elliptic {Weil} curves},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {41--53},
     year = {1987},
     volume = {160},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_160_a4/}
}
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P. I. Guerzhoy; A. A. Panchishkin. A finiteness criterion for the number of rational points for twisted elliptic Weil curves. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 8, Tome 160 (1987), pp. 41-53. http://geodesic.mathdoc.fr/item/ZNSL_1987_160_a4/