9 research outputs found
On the invariants of the quotients of the Jacobian of a curve of genus 2
The original publication is available at www.springerlink.comInternational audienceLet C be a curve of genus 2 that admits a non-hyperelliptic involution. We show that there are at most 2 isomorphism classes of elliptic curves that are quotients of degree 2 of the Jacobian of C. Our proof is constructive, and we present explicit formulae, classified according to the involutions of C, that give the minimal polynomial of the j-invariant of these curves in terms of the moduli of C. The coefficients of these minimal polynomials are given as rational functions of the moduli
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
Rational points in the moduli space of genus two
We build a database of genus 2 curves defined over which contains
all curves with minimal absolute height , all curves with moduli
height , and all curves with extra automorphisms in
standard form defined over with height .
For each isomorphism class in the database, an equation over its minimal field
of definition is provided, the automorphism group of the curve, Clebsch and
Igusa invariants. The distribution of rational points in the moduli space
for which the field of moduli is a field of definition is
discussed and some open problems are presented
Four-dimensional GLV via the Weil restriction
The Gallant-Lambert-Vanstone (GLV) algorithm uses efficiently computable endomorphisms to accelerate the computation of scalar multiplication of points on an abelian variety. Freeman and Satoh proposed for cryptographic use two families of genus 2 curves defined over \F_{p} which have the property that the corresponding Jacobians are -isogenous over an extension field to a product of elliptic curves defined over \F_{p^2}. We exploit the relationship between the endomorphism rings of isogenous abelian varieties to exhibit efficiently computable endomorphisms on both the genus 2 Jacobian and the elliptic curve. This leads to a four dimensional GLV method on Freeman and Satoh\u27s Jacobians and on two new families of elliptic curves defined over \F_{p^2}
Sato-Tate distributions and Galois endomorphism modules in genus 2
For an abelian surface A over a number field k, we study the limiting
distribution of the normalized Euler factors of the L-function of A. This
distribution is expected to correspond to taking characteristic polynomials of
a uniform random matrix in some closed subgroup of USp(4); this Sato-Tate group
may be obtained from the Galois action on any Tate module of A. We show that
the Sato-Tate group is limited to a particular list of 55 groups up to
conjugacy. We then classify A according to the Galois module structure on the
R-algebra generated by endomorphisms of A_Qbar (the Galois type), and establish
a matching with the classification of Sato-Tate groups; this shows that there
are at most 52 groups up to conjugacy which occur as Sato-Tate groups for
suitable A and k, of which 34 can occur for k = Q. Finally, we exhibit examples
of Jacobians of hyperelliptic curves exhibiting each Galois type (over Q
whenever possible), and observe numerical agreement with the expected Sato-Tate
distribution by comparing moment statistics.Comment: 59 pages, 2 figures, minor edits, to appear in Compositio Mathematic
Sato-Tate groups of abelian threefolds
Given an abelian variety over a number field, its Sato-Tate group is a
compact Lie group which conjecturally controls the distribution of Euler
factors of the L-function of the abelian variety. It was previously shown by
Fit\'e, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to
conjugation) that occur as Sato-Tate groups of abelian surfaces over number
fields; we show here that for abelian threefolds, there are 410 possible
Sato-Tate groups, of which 33 are maximal with respect to inclusions of finite
index. We enumerate candidate groups using the Hodge-theoretic construction of
Sato-Tate groups, the classification of degree-3 finite linear groups by
Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura's theory of
CM types that rules out 23 candidate groups; we cross-check this using
extensive computations in Gap, SageMath, and Magma. To show that these 410
groups all occur, we exhibit explicit examples of abelian threefolds realizing
each of the 33 maximal groups; we also compute moments of the corresponding
distributions and numerically confirm that they are consistent with the
statistics of the associated L-functions.Comment: Simplified a calculation in Section 6.4; 87 page