ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0.

Q4 Mathematics Integers Pub Date : 2021-01-01

Abhishek Juyal, Dustin Moody

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Abstract

We study various properties of the family of elliptic curves x+1/x+y+1/y+t = 0, which is isomorphic to the Weierstrass curve $E_{t} : Y^{2} = X (X^{2} + (\frac{t^{2}}{4} - 2) X + 1) .$ . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of $E_{t} (\bar{Q} (t))$ is 0 and the torsion subgroup of $E_{t} (Q (t))$ is isomorphic to $Z ∕ 4 Z$ . Over the rational field $Q$ we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of E_t with rank 5 and 6. We also determine all possible torsion subgroups of $E_{t} (Q)$ and conclude with some results regarding integral points in arithmetic progression on E_t .

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关于椭圆曲线族X + 1/X + Y + 1/Y + t = 0。

研究了与Weierstrass曲线E t: y 2 = x (x 2 + (t 2 4 - 2) x+1)同构的椭圆曲线族x+1/x+y+ t = 0的各种性质。这个方程源于对多项式的马勒测度的研究。证明了et (Q¯(t))的秩为0，并且et (Q (t))的扭转子群同构于Z∕4z。在有理域Q上，我们得到了秩至少为1和秩至少为2的无限子族，并找到秩为5和6的Et的具体实例。我们还确定了Et (Q)的所有可能的扭转子群，并得到了关于Et上等差数列积分点的一些结果。

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