The formal group law of height $ \ 2$ or $ \ 3 $ over $\mathbb{F}_p$

Question

I need to calculate formal group law over $\mathbb{F}_p$ of height $ \ 2$ or $ \ 3$ ?

I need the arithmetic calculation.

Please help me with a method

Answer 1

I’m sure that you know that there are infinitely many formal group laws of the same height over $\Bbb F_p$ that are not isomorphic over that field, but become isomorphic only over some infinite extension (usually an algebraic closure of $\Bbb F_p$). Here’s one way of getting one particular formal group of any height $h$. For typographical convenience, put $q=p^h$, and then first form the logarithm \begin{align} \log_\Phi(x)&=x+\frac{x^q}p+\frac{x^{q^2}}{p^2}+\cdots\\ &=\sum_{n=0}^\infty\frac{x^{q^n}}{p^n}\in\Bbb Q_p[[x]]\,. \end{align} Then calculate $\Phi(x,y)\in\Bbb Q_p[[x,y]]$ satisfying $$ \log_\Phi\bigl(\Phi(x,y)\bigr)=\log_\Phi(x)+\log_\Phi(y)\,, $$ in which $\Phi$ actually has its coefficients in $\Bbb Z_p$. Now just reduce modulo $p$.

There are other ways of getting formal groups of any height; I’ll be glad to go into more detail.