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Let E be an elliptic curve over a field K of characteristic 6= 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(More)
The main objective of this paper is to analyze the geometry of the modular diagonal quotient surface ZN,ε = ∆ε\(X(N) × X(N)) which classifies pairs of elliptic curves E1 and E2 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 minimal(More)
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, i.e. H = ∆α =(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)
AIM To describe the characteristics of patients attending a psychologist-led cognitive behavioural therapy (CBT) service for individuals with dental phobia and the outcomes of treatment. METHOD Analysis of routinely collected assessment and outcome data from 130 patients attending a single secondary service providing CBT for dental phobia. FINDINGS The(More)
The main aim of this paper, which is the sequel to [Ka], is to prove the existence of curves C of genus 2 admitting morphisms to two given elliptic curves E and E ′. More precisely, we are interested in the following problem. Question. Given two elliptic curves E and E ′ over an algebraically closed field K and an integer N ≥ 2, does there exist a curve C(More)
Let C be a curve of genus 2 defined over an algebraically closed field K, and suppose that C admits a non-constant morphism f : C → E to an elliptic curve E. If f does not factor over an isogeny of E, then we say that f is an elliptic subcover of C. Note that this last condition imposes no essential restriction since every nonconstant f : C → E factors over(More)
where Πg(C) is the geometric (profinite) fundamental group of C×Spec(Ks) (i.e. Πg(C) is equal to the Galois group of the maximal unramified extension of F (C)⊗Ks). This sequence induces a homomorphism ρC from GK to Out(Πg(C)) which is the group of automorphisms modulo inner automorphisms of Πg(C). It is well known that ρC is an important tool for studying(More)