Use of frobenius map of an elliptic curve

Question

I was reading about elliptic curves from https://www.math.brown.edu/~jhs/Presentations/WyomingEllipticCurve.pdf. Page No. 44 defines Frobenius map. It defines the frobenius map as $f(x,y)=(x^p,y^p) \mod p$. Isn't it just an identity map by fermat's little theorem? What's the use of this map in elliptic curves?

Answer 1

In the notation of the notes you link to $E(K)$ stands for the points $(x,y)\in K^2$ that satisfy the equation $y^2=x^3 + Ax + b$ that defines the elliptic curve together with the point at infinity. Here $K$ can be any field in which the coefficients $A,B$ are contained. On $E(\mathbb{F}_p)$ the Frobenius map $(x,y)\mapsto (x^p,y^p)$ is indeed the identity. But the Frobenius map is defined on $E(\overline{\mathbb{F}_p})$.

So to understand it you first need to understand what $\overline{\mathbb{F}_p}$ is. It is an algebraic closure of $\mathbb{F}_p$, i.e. an algebraic extension of $\mathbb{F}_p$ in which every non-constant polynomial has a root. This is an infinite field that contains $\mathbb{F}_p$.

In fact $\mathbb{F}_p$ is the subset of elements in $\overline{\mathbb{F}_p}$ that are fixed by the map $x\mapsto x^p$. Hence $E(\mathbb{F}_p)$ is the subset of elements of $E(\overline{\mathbb{F}_p})$ that are fixed by the Frobenius map.