Weierstrass 型椭圆曲线族中的不可能交点
数论
2016-07-19 v2 动力系统
摘要
考虑由非零 λ ∈ Q 2 ‾ \lambda\in\overline{\mathbb{Q}_2} λ ∈ Q 2 E λ : y 2 = x 3 + λ E_{\lambda}:y^2=x^3+\lambda E λ : y 2 = x 3 + λ P λ ( x ) = ( x , x 3 + λ ) ∈ E λ P_{\lambda}(x)=(x,\sqrt{x^3+\lambda})\in E_{\lambda} P λ ( x ) = ( x , x 3 + λ ) ∈ E λ α , β ∈ Q 2 ‾ \alpha,\beta\in\overline{\mathbb{Q}_2} α , β ∈ Q 2 α β ∈ Q \frac{\alpha}{\beta}\in\mathbb{Q} β α ∈ Q P λ ( α ) P_{\lambda}(\alpha) P λ ( α ) P λ ( β ) P_{\lambda}(\beta) P λ ( β ) E λ E_{\lambda} E λ λ \lambda λ α β ∈ { − 2 , − 1 2 } \frac{\alpha}{\beta}\in\{-2,-\frac{1}{2}\} β α ∈ { − 2 , − 2 1 } α β ∉ Q \frac{\alpha}{\beta}\notin\mathbb{Q} β α ∈ / Q α \alpha α β \beta β e ( Q 2 ( α β ) ∣ Q 2 ) e(\mathbb{Q}_2(\frac{\alpha}{\beta})\mid\mathbb{Q}_2) e ( Q 2 ( β α ) ∣ Q 2 ) E λ : y 2 = x ( x − 1 ) ( x − λ ) E_{\lambda}:y^2=x(x-1)(x-\lambda) E λ : y 2 = x ( x − 1 ) ( x − λ ) a , b a,b a , b { λ ∈ C ∖ { 0 , 1 } : ( a , a ( a − 1 ) ( a − λ ) ) , ( b , b ( b − 1 ) ( b − λ ) ) ∈ ( E λ ) tors } \{\lambda\in\mathbb{C}\setminus\{0,1\}~:~(a,\sqrt{a(a-1)(a-\lambda)}),(b,\sqrt{b(b-1)(b-\lambda)})\in (E_{\lambda})_{\text{tors}}\} { λ ∈ C ∖ { 0 , 1 } : ( a , a ( a − 1 ) ( a − λ ) ) , ( b , b ( b − 1 ) ( b − λ ) ) ∈ ( E λ ) tors }
引用
@article{arxiv.1507.07047,
title = {Impossible intersections in a Weierstrass family of elliptic curves},
author = {Niki Myrto Mavraki},
journal= {arXiv preprint arXiv:1507.07047},
year = {2016}
}
备注
16 pages