What group does $\mathbb{G}_m$ denote?

Question

What group does $\mathbb{G}_m$ denote? I saw it used here.

Answer 1

$\mathbb G_m$ is an algebraic group: the group of multiplication (in a field). That is: For any field $F$, you obtain a group $\mathbb G_m(F)$, where the elements are tuples of elements of $F$ that satisfy certain polynomial conditions (with polynomials not depending on $F$) and the group operation is given by polynomials (that do not depend on $F$). In fact, you can take $\mathbb G_m(F)=\{(x,y)\in F^2\mid x y-1=0\}$ and define $(x,y)\cdot (x',y')=(xx',yy')$. It seems tha tthe author prefers to describe the group rather as a subgroup of $GL_2$ (or in fact $SL_2$).