In this note we consider a certain class of closed Riemann surfaces which are a natural generalization of the so called classical Humbert curves. They are given by closed Riemann surfaces S admitting H ∼ = Z k 2 as a group of conformal automorphisms so that S/H is an orbifold of signature (0, k + 1; 2, . . . , 2). The classical ones are given by k = 4. Mainly, we describe some of its generalities and provide Fuchsian, algebraic and Schottky descriptions.
We build a database of genus 2 curves defined over Q which contains all curves with minimal absolute height h ≤ 5, all curves with moduli height h ≤ 20, and all curves with extra automorphisms in standard form y 2 = f (x 2 ) defined over Q with height h ≤ 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space M 2 for which the field of moduli is a field of definition is discussed and some open problems are presented.
The known examples of explicit equations for Riemann surfaces, whose field of moduli is different from their field of definition, are all hyperelliptic. In this paper we construct a family of equations for non-hyperelliptic Riemann surfaces, each of them is isomorphic to its conjugate Riemann surface, but none of them admit an anticonformal automorphism of order 2; that is, each of them has its field of moduli, but not a field of definition, contained in R. These appear to be the first explicit such examples in the non-hyperelliptic case.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.