759 research outputs found
Genus 3 curves whose Jacobians have endomorphisms by Q(ζ7 + ζ-7)
In this work we consider constructions of genus 3 curves X such that End(Jac(X))⊗Q contains the totally real cubic number field Q(ζ7+ζ-7). We construct explicit two-dimensional families defined over Q(s,t) whose generic member is a nonhyperelliptic genus 3 curve with this property. The case when X is hyperelliptic was studied in Hoffman and Wang (2013). We calculate the zeta function of one of these curves. Conjecturally this zeta function is described by a modular form
On Shimura curves in the Schottky locus
We show that a given rational Shimura curve Y with strictly maximal Higgs
field in the moduli space of g-dimensional abelian varieties does not
generically intersect the Schottky locus for large g.
We achieve this by using a result of Viehweg and Zuo which says that if Y
parameterizes a family of curves of genus g, then the corresponding family of
Jacobians is isogenous over Y to the g-fold product of a modular family of
elliptic curves. After reducing the situation from the field of complex numbers
to a finite field, we will see, combining the Weil and Sato-Tate conjectures,
that this is impossible for large genus g.Comment: 23 pages, shortened version of my PhD thesi
Abelian surfaces of GL2-type as Jacobians of curves
We study the set of isomorphism classes of principal polarizations on abelian
varieties of GL2-type. As applications of our results, we construct examples of
curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share
isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a
method to obtain genus two curves over \Q whose Jacobian varieties are
isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.Comment: To appear in Acta Arithmetic
Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians
The first step in elliptic curve scalar multiplication algorithms based on
scalar decompositions using efficient endomorphisms-including
Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as
well as higher-dimensional and higher-genus constructions-is to produce a short
basis of a certain integer lattice involving the eigenvalues of the
endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar
coefficients, and the faster the resulting scalar multiplication. Typically,
knowledge of the eigenvalues allows us to write down a long basis, which we
then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more
specialized algorithm. In this work, we use elementary facts about quadratic
rings to immediately write down a short basis of the lattice for the GLV, GLS,
GLV+GLS, and Q-curve constructions on elliptic curves, and for genus 2 real
multiplication constructions. We do not pretend that this represents a
significant optimization in scalar multiplication, since the lattice reduction
step is always an offline precomputation---but it does give a better insight
into the structure of scalar decompositions. In any case, it is always more
convenient to use a ready-made short basis than it is to compute a new one
Distortion maps for genus two curves
Distortion maps are a useful tool for pairing based cryptography. Compared
with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more
complicated since the full torsion subgroup has rank 2g. In this paper we prove
that distortion maps always exist for supersingular curves of genus g>1 and we
construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).Comment: 16 page
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