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From Ireland and Rosen Number theory book(ch11.#11)

Consider the curve $y^2=x^{3}-Dx$ over $\mathbb{F}_{p}$, where $D \not= 0$. Call this curve $C_{1}$. It can be shown that the substitution $x=\frac{1}{2}(u+v^{2})$ and $y=\frac{1}{2}v(u+v^{2})$ transforms $C_{1}$ into the curve $C_{2}$ given by $u^{2}-v^{4}=4D$.

My question:

How can we show that in any finite field the number of finite points on $C_{1}$ is one more than the number of points on $C_{2}$?

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1 Answer 1

up vote 2 down vote accepted

There are rational functions $U$ and $V$ of $(x,y)$ that "solve for $u$ and $v$ given $x$ and $y$". They are defined for $(x,y) \neq (0,0)$ and, thought of as a pair $(U,V)$ defining a point on $C_2$, invert the polynomial map that "calculates $(x,y)$ from $(u,v)$". Evaluation of these functions defines a bijection between finite points of $C_2$ and finite points of $C_1 - (0,0)$.

The characteristic of the finite field should be different from 2.

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