Résumé
The aim of this paper is to study the birational geometry of certain moduli spaces of abelian surfaces with some additional structures. In particular, we inquire moduli of abelian surfaces with a symmetric theta structure and an odd theta characteristic. More precisely, on one hand, for a $(d_1,d_2)$-polarized abelian surfaces, we study how the parity of $d_1$ and $d_2$ influence the relation between the datum of a canonical level structure and that of a symmetric theta structure. On the other, for certain values of $d_1$ and $d_2$, the datum of a theta characteristic, seen as a quadratic form on the points of $2$-torsion induced by a symmetric line bundle, is necessary in order to well-define Theta-null maps. Finally we use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geometry and the Kodaira dimension these moduli spaces.