The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus L g of such genus-g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves H g . The authors present a birational parameterization of L g via dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ L g . They conjecture that for p ∈ H g with | Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈ L g such that the Klein 4-group is embedded in the reduced automorphism group of p. Further, for g = 3, they show that for every moduli point p ∈ H 3 such that | Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

We study genus 2 function fields with elliptic subfields of degree 2. The locus L2 of these fields is a 2-dimensional subvariety of the moduli space M2 of genus 2 fields. An equation for L2 is already in the work of Clebsch and Bolza. We use a birational parameterization of L2 by affine 2-space to study the relation between the j-invariants of the degree 2 elliptic subfields. This extends work of Geyer, Gaudry, Stichtenoth and others. We find a 1-dimensional family of genus 2 curves having exactly two isomorphic elliptic subfields of degree 2; this family is parameterized by the j-invariant of these subfields.

Abstract. In this paper we study genus 2 function fields K with degree 3 elliptic subfields. We show that the number of Aut(K)-classes of such subfields of fixed K is 0,1,2 or 4. Also we compute an equation for the locus of such K in the moduli space of genus 2 curves. IntroductionWe study genus two curves C whose function fields have a degree 3 subfield of genus 1. Such subfields we call elliptic subfields. Such curves C have already occurred in the work of Hermite, Goursat, Burkhardt, Brioschi, and Bolza, see Krazer [6] (p. 479). More generally, degree n elliptic subfields of genus 2 fields have been studied by Frey [1], Frey and Kani [2], Kuhn [8], Shaska [11], Shaska and Voelklein [13]. In the degree 3 case, explicit equations can be used to answer questions which remain open in the general case. This was the theme of the author's PhD thesis (see [12]) which also covered the case of degree 2 (which is subsumed in [13]). Equation (5) gives a normal form for pairs (K, E) where E is a degree 3 elliptic subfield of K. This normal form depends on two parameters a, b ∈ k. Isomorphism classes of pairs (K, E) are parameterized by parameters u, v where u := ab and v := b 3 . This yields an expression for the invariants (i 1 , i 2 , i 3 ) of K (see Igusa [9]) in terms of u and v. Our central object of study is the mapThis map is shown to have degree 2, which means that a general genus 2 field having elliptic subfields of degree 3 has exactly two such subfields. There are exactly four genus 2 fields which have four Aut(K)-classes and a 1-dimensional family of such fields with one Aut(K)-class of such subfields.Although the map θ is given by explicit equations (see (13)), it is a non-trivial computational task to show that it has degree 2 and to compute the action of its Galois group on u and v. The direct approach exceeds available computer power, so we introduce auxiliary parameters r 1 , r 2 that parameterize pairs of cubic polynomials. They arise from the fact that the subfield E induces a particular sextic defining K which splits naturally as a product of 2 cubic polynomials. We show that θ factorizes as (u, v) → (r 1 , r 2 ) → (i 1 , i 2 , i 3 )where the latter map is birational, and the former has degree 2. Thus r 1 , r 2 yield a birational parameterization of the locus L 3 of genus 2 fields having a degree 3 elliptic subfield. That L 3 is a rational variety follows also from the general theory of "diagonal modular surfaces", see Kani [4]. We also compute the equation in i 1 , i 2 , i 3 that defines L 3 as a sublocus of the moduli space M 2 of genus 2 curves. We further find relations between the j-invariants of the degree 3 elliptic subfields of K and classify all K in L 3 with extra automorphisms.All the computations were done using Maple [10].Acknowledgment: I would like to express my sincere gratitude to my PhD advisor Prof. H. Voelklein for all the time and effort spent in guiding me towards my dissertation (from which this paper originated). Genus Two Fields With Degree 3 Elliptic SubfieldsLet k be an alge...

In this note we discuss techniques for determining the automorphism group of a genus g hyperelliptic curve Xg defined over an algebraically closed field k of characteristic zero. The first technique uses the classical GL2(k)-invariants of binary forms. This is a practical method for curves of small genus, but has limitations as the genus increases, due to the fact that such invariants are not known for large genus.The second approach, which uses dihedral invariants of hyperelliptic curves, is a very convenient method and works well in all genera. First we define the normal decomposition of a hyperelliptic curve with extra automorphisms. Then dihedral invariants are defined in terms of the coefficients of this normal decomposition. We define such invariants independently of the automorphism group Aut(Xg). However, to compute such invariants the curve is required to be in its normal form. This requires solving a nonlinear system of equations.We find conditions in terms of classical invariants of binary forms for a curve to have reduced automorphism group A4, S4, A5. As far as we are aware, such results have not appeared before in the literature.

We study the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. We prove that the family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with H ⊕ E 7 (−1) ⊕ E 7 (−1) lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type IV . We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions. As a result, we obtain novel F-theory models, dual to non-geometric heterotic string compactifications in eight dimensions with two non-vanishing Wilson line parameters.