Finite subgroups of maximal order of the Cremona group over the rationals

Abstract: Let $\Cr_\Q(2)$ be the Cremona group of rank $2$ over rational numbers. We give a classification of large finite subgroups $G$ of $\Cr_\Q(2)$ and give a new sharp bound smaller (but not multiplicative) than $M(\Q)=120960 = 2^7\cdot3^3\cdot5\cdot7$; the one given in \cite{MR2567402}. In particular, we prove that any finite subgroup $G \subset\Cr_\Q(2)$ has order $\mid G\mid \le 432$ and Lemma \ref{lemm-25} provides a group of order $432$. We use the modern approach of minimal $G-$surfaces, given a (smooth) rational surface $S\subset\p^2$ defined over $\Q$, we study the finite subgroups $G \subset \Aut_{\Q}(S)$ of automorphisms of $S$. We give the best bound for the order of $G\subset\Aut(S)$ for surfaces with a conic bundle structure invariant by $G$. We also give the best bound for the order of $G\subset \Aut_\Q(S)$ for all rational Del Pezzo surfaces of some given degree. In addition, we give descriptions of the finite subgroups of automorphisms of conic bundles and Del Pezzo surfaces of maximal size.