The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians

  title={The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians},
  author={Nils Bruin and Kevin Doerksen},
  journal={Canadian Journal of Mathematics},
  pages={992 - 1024}
  • N. Bruin, Kevin Doerksen
  • Published 19 February 2009
  • Mathematics, Computer Science
  • Canadian Journal of Mathematics
Abstract In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general… 
Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians
  • T. Shaska
  • Mathematics, Computer Science
  • 2015
This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split and is determined in terms of the absolute invariants t1 ... t6.
Congruences of Elliptic Curves Arising from Non-Surjective Mod $N$ Galois Representations
We study N-congruences between quadratic twists of elliptic curves. If N has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In
Decomposition of Some Jacobian Varieties of Dimension 3
This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split.
Hilbert modular surfaces for square discriminants and elliptic subfields of genus 2 function fields
We compute explicit rational models for some Hilbert modular surfaces corresponding to square discriminants, by connecting them to moduli spaces of elliptic K3 surfaces. Since they parametrize
Quickly constructing curves of genus 4 with many points
The "defect" of a curve over a finite field is the difference between the number of rational points on the curve and the Weil-Serre bound for the curve. We present a construction for producing
Decomposing Jacobians Via Galois covers
Let φ : X → Y be a (possibly ramied) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of φ, up to isogeny, as the Jacobian of a quotient curve
On pairs of 17-congruent elliptic curves
We compute explicit equations for the surfaces Z(17, 1) and Z(17, 3) parametrising pairs of 17-congruent elliptic curves. We find that each is a double cover of the same elliptic K3-surface. We use
On the Prym variety of genus 3 covers of genus 1 curves
Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, this construction can be seen as a degenerate case of a result by Nils Bruin.
Computing the Cassels-Tate Pairing in the Case of a Richelot Isogeny
An explicit formula as well as a practical algorithm to compute the Cassels-Tate pairing on Sel( Ĵ)×Sel(Ĵ) where φ̂ is the dual isogeny of φ and the formula is given under the simplifying assumption that all two torsion points on J are defined overK.
The Intersection of Two Fermat Hypersurfaces in P^3 via Computation of Quotient Curves
We study the intersection of two particular Fermat hypersurfaces in $\mathbb{P}^3$ over a finite field. Using the Kani-Rosen decomposition we study arithmetic properties of this curve in terms of its


Genus 2 curves that admit a degree 5 map to an elliptic curve
Abstract We continue our study of genus 2 curves C that admit a cover C → E to a genus 1 curve E of prime degree n. These curves C form an irreducible 2-dimensional subvariety ℒ n of the moduli space
Genus 2 Curves with (3, 3)-Split Jacobian and Large Automorphism Group
There are exactly six C-isomorphism classes of genus two curves C with Aut(C) isomorphic to D8 (resp., D12) and with (3, 3)-split Jacobian and it is shown that exactly four of these classes with group D8 have representatives defined over Q.
The Hessian of a genus one curve
The development of the invariant theory of genus one curves is continued, and explicit formulae and algorithms for computing the Hessian are given, which leads to a practical algorithm for computing equations for visible elements of order n in the Tate—Shafarevich group of an elliptic curve.
The moduli space of curves of genus two covering elliptic curves
We consider the moduli spaceSn of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an
The arithmetic-geometric mean and isogenies for curves of higher genus
Computation of Gauss’s arithmetic-geometric mean involves iteration of a simple step, whose algebro-geometric interpretation is the construction of an elliptic curve isogenous to a given one,
Exhibiting SHA[2] on hyperelliptic Jacobians
Curves of genus 2 with split Jacobian
We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If X is a curve of genus 2, and f: X t E a map from X to an elliptic curve, then X has
Lectures on elliptic curves
Introduction 1. Curves of genus: introduction 2. p-adic numbers 3. The local-global principle for conics 4. Geometry of numbers 5. Local-global principle: conclusion of proof 6. Cubic curves 7.
Visualizing Elements of Sha[3] in Genus 2 Jacobians
Any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve and how to get explicit models of the genus 2 curves involved is described.
Explicit Families of Elliptic Curves with Prescribed Mod N Representations
In Part 1 we explain how to construct families of elliptic curves with the same mod 3, 4, or 5 representation as that of a given elliptic curve over Q. In §4 we give equations for the families in the