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We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having as field of moduli, that admit models over every completion of , but have no model over .
Let be an elliptic surface defined over a number field , let be a section, and let be a rational prime. We bound the number of points of low algebraic degree in the -division hull of at the fibre . Specifically, for with such that is nonsingular, we obtain a bound on the number of such that , and such that for some . This bound depends on , , , , and , but is independent of .
For a complex abelian surface with endomorphism ring isomorphic to the maximal order in a quartic CM field , the Igusa invariants generate an unramified abelian extension of the reflex field of . In this paper we give an explicit geometric description of the Galois action of the class group of this reflex field on . Our description can be expressed by maps between various Siegel modular varieties, and we can explicitly compute the action for ideals of small norm. We use the Galois action to modify the CRT method for computing Igusa class polynomials, and our run time analysis shows that this yields a significant improvement. Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby implying that the ‘isogeny volcano’ algorithm to compute endomorphism rings of ordinary elliptic curves over finite fields does not have a straightforward generalization to computing endomorphism rings of abelian surfaces over finite fields.
We study the cluster category of a marked surface without punctures. We explicitly describe the objects in as direct sums of homotopy classes of curves in and one-parameter families related to noncontractible closed curves in . Moreover, we describe the Auslander–Reiten structure of the category in geometric terms and show that the objects without self-extensions in correspond to curves in without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.