Van Wamelen [Math. Comp. 68 (1999) no. 225, 307-320] lists 19 curves of genus two over Q with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over Q, and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.We extend Van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest 'generic' examples of CM curves of genus two.We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike Van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of Lauter and Viray for Igusa class polynomials.
The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain (Z/2Z) 4 action. In particular, we describe families whose general member contains 8, 16, 24 or 32 lines as well as the 320 conics found by Eklund [Ekl10] (some of which degenerate into the mentioned lines). The second half of this paper is dedicated to finding the Picard group of a general member of each of these families, and describing it as a lattice. It turns out that for each family the Picard group of a very general surface is generated by the lines and conics lying on said surface.
In [Ekl10], Eklund showed that a general (Z/2Z) 4 -invariant quartic K3 surface contains at least 320 conics. In this paper we analyse the field of definition of those conics as well as their Monodromy group. As a result, we prove that the moduli space (Z/2Z) 4 -invariant quartic K3 surface with a marked conic has 10 irreducible components.
Let P denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z, and w; let X be the generic element of the family of surfaces in P given byThe surface X is a K3 surface over the function field Q(t). In this paper, we explicitly compute the geometric Picard lattice of X , together with its Galois module structure, as well as derive more results on the arithmetic of X and other elements of the family X.
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