## On the rank of the elliptic curves with a rational point of order 4.(English)Zbl 1050.11058

The author proves the following two theorems: 1) There is an elliptic curve defined over $$\mathbb Q(t)$$ with a rational point of order $$4$$ and rank $$\geq 4$$. 2) There are infinitely many elliptic curves defined over $$\mathbb Q$$ with a rational point of order $$4$$ and rank $$\geq 5$$.

### MSC:

 11G05 Elliptic curves over global fields

### Keywords:

elliptic curve; rank
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### References:

 [1] Kihara, S.: On the rank of the elliptic curve $$y^2=x^3+kx$$. Proc. Japan Acad., 74A , 115-116 (1998). · Zbl 0919.11039 [2] Mestre, J.-F.: Rang de courbes elliptiques d’invariant donné. C. R. Acad. Sci. Paris Sér. I Math., 314 , 919-922 (1992). · Zbl 0766.14023 [3] Nagao, K.: On the rank of elliptic curve $$y^2=x^3-kx$$. Kobe J. Math., 11 , 205-210 (1994). · Zbl 0855.11026
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