1,309 research outputs found
Exhibiting Sha[2] on hyperelliptic jacobians
We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Sharevich-Tate group. Finally we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32
Families of explicitly isogenous Jacobians of variable-separated curves
We construct six infinite series of families of pairs of curves (X,Y) of
arbitrarily high genus, defined over number fields, together with an explicit
isogeny from the Jacobian of X to the Jacobian of Y splitting multiplication by
2, 3, or 4. For each family, we compute the isomorphism type of the isogeny
kernel and the dimension of the image of the family in the appropriate moduli
space. The families are derived from Cassou--Nogu\`es and Couveignes' explicit
classification of pairs (f,g) of polynomials such that f(x_1) - g(x_2) is
reducible
Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
In this article, we show that in each of four standard families of
hyperelliptic curves, there is a density- subset of members with the
property that their Jacobians have adelic Galois representation with image as
large as possible. This result constitutes an explicit application of a general
theorem on arbitrary rational families of abelian varieties to the case of
families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit
examples of hyperelliptic curves of genus and over whose
Jacobians have such maximal adelic Galois representations.Comment: 24 page
On Lagrangian fibrations by Jacobians, II
Let Y → Pn be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian X = Jd (Y/Pn) is a Lagrangian fibration. We prove that X is a Beauville-Mukai integrable system if n = 3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville-Mukai system if n = 3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curve
Constructing genus 3 hyperelliptic Jacobians with CM
Given a sextic CM field , we give an explicit method for finding all genus
3 hyperelliptic curves defined over whose Jacobians are simple and
have complex multiplication by the maximal order of this field, via an
approximation of their Rosenhain invariants. Building on the work of Weng, we
give an algorithm which works in complete generality, for any CM sextic field
, and computes minimal polynomials of the Rosenhain invariants for any
period matrix of the Jacobian. This algorithm can be used to generate genus 3
hyperelliptic curves over a finite field with a given zeta
function by finding roots of the Rosenhain minimal polynomials modulo .Comment: 20 pages; to appear in ANTS XI
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