Let E be an elliptic curve over a field K of characteristic = 2 and let N > 1 be an integer prime to char(K). The purpose of this paper is to study the family of genus 2 covers of E of fixed degree N , i.e. those covers f : C → E for which C/K is a curve of genus 2 and deg(f) = N. Since we can (without loss of generality) restrict our attention those covers… (More)
The main objective of this paper is to analyze the geometry of the modular diagonal quotient surface Z N,ε = ∆ ε \(X(N) × X(N)) which classifies pairs of elliptic curves E 1 and E 2 together with an isomorphism of " determinant ε " between their associated modular representations mod N. In particular, we calculate some of the numerical invariants of its… (More)
Introduction Let X be a smooth projective curve over C which admits a finite group G of automorphisms. Then G × G acts on the product surface Y = X × X, and so for each subgroup H ≤ G × G, the quotient Z = H\Y is a normal algebraic surface. Here we shall restrict attention to the case that the subgroup H is the graph of a group automorphism α ∈ Aut(G) of G,… (More)
Let K be a finitely generated field with separable closure K s and absolute Galois group G K. By a curve C/K we always understand a smooth geometrically irreducible projective curve. Let F (C) be its function field and let Π(C) be the Galois group of the maximal unramified extension of F (C). We have the exact sequence where Π g (C) is the geometric… (More)
The purpose of this paper is to present two theorems which give an overview of the set of elliptic curves lying on an abelian surface and to discuss several applications. One of these applications is a classical theorem of Biermann (1883) and Humbert (1893) on the characterization of abelian surfaces containing elliptic curves in terms of the " singular… (More)
1 Motivation In the whole paper K is a field of finite type over its prime field K 0 of characteristic p ≥ 0 and not equal to 2. As typical case we can take K as number field or as function field of one variable over a finite field. With K s we denote its separable closure. With n we denote an odd number larger than 5 and prime to p. We begin by stating… (More)
The purpose of this article is to give a partly historical survey of the main results on Euler's idoneal numbers and to study several generalizations. Some of these generalizations were suggested by the study of genus 2 curves.
The main aim of this paper is to determine the number c N,D of genus 2 covers of an elliptic curve E of fixed degree N ≥ 1 and fixed discriminant divisor D ∈ Div(E). In the case that D is reduced, this formula is due to Dijkgraaf. The basic technique here for determining c N,D is to exploit the geometry of a certain compactification C = C E,N of the… (More)
1 Introduction Let C/K be a smooth, geometrically connected curve over a finitely generated field K, and let F = κ(C) denote its function field.