65 research outputs found
Computing functions on Jacobians and their quotients
We show how to efficiently compute functions on jacobian varieties and their
quotients. We deduce a quasi-optimal algorithm to compute isogenies
between jacobians of genus two curves
Galois invariant smoothness basis
This text answers a question raised by Joux and the second author about the
computation of discrete logarithms in the multiplicative group of finite
fields. Given a finite residue field \bK, one looks for a smoothness basis
for \bK^* that is left invariant by automorphisms of \bK. For a broad class
of finite fields, we manage to construct models that allow such a smoothness
basis. This work aims at accelerating discrete logarithm computations in such
fields. We treat the cases of codimension one (the linear sieve) and
codimension two (the function field sieve)
The geometry of some parameterizations and encodings
We explore parameterizations by radicals of low genera algebraic curves. We
prove that for a prime power that is large enough and prime to , a fixed
positive proportion of all genus 2 curves over the field with elements can
be parameterized by -radicals. This results in the existence of a
deterministic encoding into these curves when is congruent to modulo
. We extend this construction to parameterizations by -radicals for
small odd integers , and make it explicit for
Approximate computations with modular curves
This article gives an introduction for mathematicians interested in numerical
computations in algebraic geometry and number theory to some recent progress in
algorithmic number theory, emphasising the key role of approximate computations
with modular curves and their Jacobians. These approximations are done in
polynomial time in the dimension and the required number of significant digits.
We explain the main ideas of how the approximations are done, illustrating them
with examples, and we sketch some applications in number theory
Explicit Riemann-Roch spaces in the Hilbert class field
Let be a finite field, and two curves over ,
and an unramified abelian cover with Galois group . Let
be a divisor on and its pullback on . Under mild conditions the
linear space associated with is a free -module. We study
the algorithmic aspects and applications of these modules
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