7 research outputs found
Primitive prime divisors in zero orbits of polynomials
Let be a sequence of integers. A primitive prime
divisor of a term is a prime which divides but does not divide any
of the previous terms of the sequence. A zero orbit of a polynomial is a
sequence of integers where the -th term is the -th iterate of
at 0. We consider primitive prime divisors of zero orbits of polynomials. In
this note, we show that for integers and , where and , every iterate in the zero orbit of contains a primitive
prime whenever zero has an infinite orbit. If , then every iterate
after the first contains a primitive prime.Comment: 6 page
The arithmetic of genus two curves with (4,4)-split Jacobians
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 description of Richelot isogenies
between Jacobians of genus 2 curves, where previously only Richelot isogenies
with kernels that are pointwise defined over the base field were considered.
Our main tool is a Galois theoretic characterization of genus 2 curves
admitting multiple Richelot isogenies.Comment: 30 page
On the arithmetic of genus 2 curves with (4,4)-split Jacobians
In this thesis, we study genus 2 curves whose Jacobians allow a decomposition into two elliptic curves. More specifically, we are interested in genus 2 curves C whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We restrict to base fields of characteristic distinct from 2 or 3, but we do not require them to be algebraically closed. In the first half of the thesis, we obtain a full classification of principally polarized abelian surfaces that can arise from gluing two elliptic curves together along their 4-torsion and we derive the relation which their absolute invariants must satisfy. In the process, we derive a description of Richelot isogenies between Jacobians of genus 2 curves. Previous literature only considered Richelot isogenies whose kernels are pointwise defined over the base field. We also obtain a Galois theoretic characterization of genus 2 curves which admit multiple Richelot isogenies on their Jacobians. As a corollary to this classification, we obtain a model for the universal elliptic curve over the modular curve of elliptic curves with 4-torsion anti-isometric to E[4]. The final chapter of the thesis considers elements of order m of the Shafarevich-Tate group of an elliptic curve E, denoted Sha(E/k)[m]. For a given elliptic curve, E, we consider the question of making Sha(E/k)[4] visible in the sense of Mazur. We show that the visibility argument for m=4 is less tractable than the arguments in the m=2 and m=3 cases. In the m=4 case, we encounter a challenge of trying to find rational points on a K3 surface. We also show that finding the appropriate twist of this surface is a non-trivial problem. Nevertheless, in particular cases, one can proceed with this construction and we conclude the thesis by working through a couple of examples in detail