Abelian surfaces with an automorphism and quaternionic multiplication (Q2786450)

The paper in question aims at investigating abelian surfaces, with a non-principal polarization, carrying an automorphism of order 3 or 4, that also have quaternionic multiplication (QM) from an order in some rational quaternion algebra. It begins with an explicit description of abelian surfaces with a polarization of type \((1,d)\) for some \(d\in\mathbb{N}\), and then continues to show how they can be obtained as deformations of a self-product of an elliptic curve with CM (with a particular isomorphism of that self-product). It is then shown, again in terms of explicit matrices, what QM structure must be carried by elements of these deformations. An overview of the relevant congruent subgroups associated with polarizations of type \((2,4)\) and the theta embedding of the associated moduli spaces into \(\mathbb{P}^{7}\) is also given, based on the more general presentation from the book [Complex abelian varieties. Berlin: Springer-Verlag (1992; Zbl 0779.14012)]; 2nd augmented ed. Berlin: Springer (2004; Zbl 1056.14063)] by \textit{C. Birkenhake} and \textit{H. Lange}. The Heisenberg group and its Schrödinger representation (both defined more generally in [loc. cit.]) are also introduced, with a brief description of their properties.NEWLINENEWLINEIt is known that in general abelian surfaces with QM are parametrized by 1-dimensional moduli spaces, called Shimura curves, that are typically compact. The later parts of the paper in question deal with algebraic geometry (in particular the projective embeddings) of the Shimura curves forming the moduli spaces of the abelian surfaces described above, with \(d=2\) (hence with QM from a maximal order \(\mathcal{O}_{6}\) in the rational quaternion algebra of discriminant 6). A result of the reference [\textit{W. Barth}, Adv. Stud. Pure Math. 10, 41--84 (1987; Zbl 0639.14023)] implies that the theta embedding puts the associated moduli space \(M_{2,4}\) into a copy of \(\mathbb{P}^{5}\), which forms the main part of the fixed points of a certain involution on \(\mathbb{P}^{7}\). The paper shows that the Shimura curve in question is then the image of an embedding of \(\mathbb{P}^{1}\) (denoted \(\mathbb{P}^{1}_{QM}\)) into \(\mathbb{P}^{5}\), and relations with the action of the discriminant group \(T(2,4)\) of order 64 associated with \((2,4)\)-polarizations are also investigated. This Shimura curve is acted on naturally by \(S_{4}\), with 3 special orbits (what are also described explicitly) that form the branch points of the map from \(\mathbb{P}^{1}_{QM}\) onto its \(S_{4}\)-quotient.NEWLINENEWLINEThe paper proceeds with obtaining a good, symmetric \((1,2)\) polarization from the principal polarization on the Jacobians a certain genus 2 curve \(C\) studied in the reference [\textit{K.-I. Hashimoto} and \textit{N. Murabayashi}, Tohoku Math. J. (2) 47, No. 2, 271--296 (1995; Zbl 0838.11044)]. The map from \(C\) into its Jacobian can thus be manipulated to present \(C\) as a cover of a rational curve of degree 4 in \(\mathbb{P}^{5}\), passing through 6 nodes of the Kummer surfaces associated with the Jacobian of \(C\) as a subvariety of that \(\mathbb{P}^{5}\). Examining reducible hyperplane sections of these Kummer surfaces, one obtains relations with certain invariants of \(C\) (both, in fact, varying inside a family paramterized by the Shimura curve \(\mathbb{P}^{1}_{QM}\)) and the description of \(C\) in [loc. cit.]. The paper concludes with a study of the 3 special points, and with an application to the the Humbert surface \(S_{2}\) parametrizing abelian varieties with multiplication from \(\mathbb{Z}[\sqrt{-2}]\) (which is a \(K3\) surface).