Why do we say always "$1$-dimensional formal group law"?
What is the meaning of $1$-dimensional in this phrase?
2026-02-23 15:30:49.1771860649
Why do we say always "$1$-dimensional formal group law"?
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I do not know much about formal group laws, but I think I can answer that (without giving too many details).
What you call a $1$-dimensional formal group law is a formal power series $F$ in two indeterminates $x$ and $y$ over a commutative ring $R$ that fullfills some conditions. More generally, one can define $n$-dimensional formal group laws which are given by $n$ formal power series $F_1,\dots,F_n$ in $2n$ indeterminates $x_1,\dots,x_n,y_1,\dots,y_n$ over a commutative ring $R$ that fullfill some conditions.
Why $n$-dimensional?
I believe that comes from the fact that there is a natural way to associate a $n$-dimensional formal group law to any $n$-dimensional algebraic group in terms of the group structure. A prominent example being the $1$-dimensional formal group law associated to an elliptic curve or the additive formal group law coming from the additive group $\mathbb{G}_a$.