807 research outputs found
A New Method for Decomposition in the Jacobian of Small Genus Hyperelliptic Curves
Decomposing a divisor over a suitable factor basis in the Jacobian of a hyperelliptic curve is a crucial step in an
index calculus algorithm for the discrete log problem in the Jacobian. For small genus curves, in the year 2000, Gaudry had proposed
a suitable factor basis and a decomposition method. In this work, we provide a new method for decomposition over the same factor
basis. The advantage of the new method is that it admits a sieving technique which removes smoothness checking of polynomials
required in Gaudry\u27s method. Also, the total number of additions in the Jacobian required by the new method is less than
that required by Gaudry\u27s method. The new method itself is quite simple and we present some example decompositions and timing
results of our implementation of the method using Magma
Tautological classes on the moduli space of hyperelliptic curves with rational tails
We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. Our result gives a complete description of tautological relations. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology. (C) 2017 Elsevier B.V. All rights reserved11sci
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
Constructing hyperelliptic curves with surjective Galois representations
In this paper we show how to explicitly write down equations of hyperelliptic
curves over Q such that for all odd primes l the image of the mod l Galois
representation is the general symplectic group. The proof relies on
understanding the action of inertia groups on the l-torsion of the Jacobian,
including at primes where the Jacobian has non-semistable reduction. We also
give a framework for systematically dealing with primitivity of symplectic mod
l Galois representations.
The main result of the paper is the following. Suppose n=2g+2 is an even
integer that can be written as a sum of two primes in two different ways, with
none of the primes being the largest primes less than n (this hypothesis
appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is
an explicit integer N and an explicit monic polynomial of degree n, such that the Jacobian of every curve of the
form has for all odd primes l and
, whenever
is monic with and with no
roots of multiplicity greater than in for any p
not dividing N.Comment: 24 pages, minor correction
Complete moduli of cubic threefolds and their intermediate Jacobians
The intermediate Jacobian map, which associates to a smooth cubic threefold
its intermediate Jacobian, does not extend to the GIT compactification of the
space of cubic threefolds, not even as a map to the Satake compactification of
the moduli space of principally polarized abelian fivefolds. A much better
"wonderful" compactification of the space of cubic threefolds was constructed
by the first and fourth authors --- it has a modular interpretation, and
divisorial normal crossing boundary. We prove that the intermediate Jacobian
map extends to a morphism from the wonderful compactification to the second
Voronoi toroidal compactification of the moduli of principally polarized
abelian fivefolds --- the first and fourth author previously showed that it
extends to the Satake compactification. Since the second Voronoi
compactification has a modular interpretation, our extended intermediate
Jacobian map encodes all of the geometric information about the degenerations
of intermediate Jacobians, and allows for the study of the geometry of cubic
threefolds via degeneration techniques. As one application we give a complete
classification of all degenerations of intermediate Jacobians of cubic
threefolds of torus rank 1 and 2.Comment: 56 pages; v2: multiple updates and clarification in response to
detailed referee's comment
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