Automorphism groups of pseudoreal Riemann surfaces

A smooth complex projective curve is called pseudoreal if it is isomorphic to its conjugate but is not definable over the reals. Such curves, together with real Riemann surfaces, form the real locus of the moduli space $\mathcal M_g$. This paper deals with the classification of pseudoreal curves according to the structure of their automorphism group. We follow two different approaches existing in the literature: one coming from number theory, dealing more generally with fields of moduli of projective curves, and the other from complex geometry, through the theory of NEC groups. Using the first approach, we prove that the automorphism group of a pseudoreal Riemann surface $X$ is abelian if $X/Z({\rm Aut}(X))$ has genus zero, where $Z({\rm Aut}(X))$ is the center of ${\rm Aut}(X)$. This includes the case of $p$-gonal Riemann surfaces, already known by results of Huggins and Kontogeorgis. By means of the second approach and of elementary properties of group extensions, we show that $X$ is not pseudoreal if the center of $G={\rm Aut}(X)$ is trivial and either ${\rm Out}(G)$ contains no involutions or ${\rm Inn}(G)$ has a group complement in ${\rm Aut}(G)$. This extends and gives an elementary proof (over $\mathbb C$) of a result by D\`ebes and Emsalem. Finally, we provide an algorithm, implemented in MAGMA, which classifies the automorphism groups of pseudoreal Riemann surfaces of genus $g\geq 2$, once a list of all groups acting for such genus, with their signature and generating vectors, are given. This program, together with the database provided by J. Paulhus in \cite{Pau15}, allowed us to classifiy pseudoreal Riemann surfaces up to genus $10$, extending previous results by Bujalance, Conder and Costa.