§ Hidden symmetries of alg varieties

• Given equations in $A$, can find solutions in any $B$ such that we have $\phi: A \to B$
• Can translate topological ideas to geometry.
• Fundamental theorem of riemann: fundamental group with finitely many covering becomes algebraic (?!)
• So we can look at finite quotients of the fundamental group.
• As variety, we take line minus one point. This can be made by considering $xy - 1 = 0$ in $R[x, y]$ and then projecting solutions to $R[x]$.
• If we look at complex solutions, then we get $\mathbb C - \{0 \} = C^\times$.
• The largest covering space is $\mathbb C \xrightarrow{\exp} \mathbb C^\times$. The fiber above $1 \in C^\times$ (which is the basepont) is $2 \pi i$.
• Finite coverings are $C^\times \xrightarrow{z \mapsto z^n} C^\times$. The subsitute for the fundamental group is the projective (inverse) limit of these groups.
• The symmetry of $Gal(\overline{\mathbb Q} / \mathbb Q)$ acts on this fundamental group.
• One can get not just fundamental group, but any finite coefficients!
• Category of coverings is equivalent to category of sets with action of fundamental group.