137 research outputs found
The distribution of class groups of function fields
Using equidistribution results of Katz and a computation in finite symplectic
groups, we give an explicit asymptotic formula for the proportion of curves C
over a finite field for which the l-torsion of Jac(C) is isomorphic to a given
abelian l-group. In doing so, we prove a conjecture of Friedman and WashingtonComment: To appear, JPA
Bounds of the rank of the Mordell-Weil group of jacobians of hyperelliptic curves
In this article we extend work of Shanks and Washington on cyclic extensions,
and elliptic curves associated to the simplest cubic fields. In particular, we
give families of examples of hyperelliptic curves defined over
, with of degree , where is a Sophie Germain prime,
such that the rank of the Mordell--Weil group of the jacobian of
is bounded by the genus of and the -rank of the class group of the
(cyclic) field defined by , and exhibit examples where this bound is
sharp.Comment: 22 pages, To appear in J. Th\'eor. Nombres Bordeau
A Generic Approach to Searching for Jacobians
We consider the problem of finding cryptographically suitable Jacobians. By
applying a probabilistic generic algorithm to compute the zeta functions of low
genus curves drawn from an arbitrary family, we can search for Jacobians
containing a large subgroup of prime order. For a suitable distribution of
curves, the complexity is subexponential in genus 2, and O(N^{1/12}) in genus
3. We give examples of genus 2 and genus 3 hyperelliptic curves over prime
fields with group orders over 180 bits in size, improving previous results. Our
approach is particularly effective over low-degree extension fields, where in
genus 2 we find Jacobians over F_{p^2) and trace zero varieties over F_{p^3}
with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average
time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio
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