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

@article{Bruin2011TheAO,
  title={The Arithmetic of Genus Two Curves with (4,4)-Split Jacobians},
  author={Nils Bruin and Kevin Doerksen},
  journal={Canadian Journal of Mathematics},
  year={2011},
  volume={63},
  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… 
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References

SHOWING 1-10 OF 60 REFERENCES
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
TLDR
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
TLDR
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
TLDR
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
...
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