Contemporary Mathematics
Volume 463, 2008
Results of Cohen-Lenstra type for quadratic function fields
Jeffrey D. Achter
ABSTRACT.
Consider hyperelliptic curves C of fixed genus over a finite field lF.
Let
L
be a finite abelian group of exponent dividng
N.
We give an asymptotic
formula in
IJFI,
with explicit error term, for the proportion of C for which
Jac(C)[NJ(lF)
~
L.
1. Introduction
Let
C
be a smooth, proper curve of positive genus
g
over a finite field lF. Its
Jacobian Jac(C) is a g-dimensional abelian variety. On one hand, if an explicit
model of
C
is chosen, then there are efficient methods for computing in the finite
abelian group Jac(C)(JF). Varying the coefficients of C yields a family of groups.
On the other hand, since Jac(C)(JF) is isomorphic to the class group of the function
field of C, studying these groups is tantamount to analyzing the class groups of
certain families of global fields, an endeavor with a rich history of its own.
The groups Jac(C)(JF) are extremely useful in public-key cryptography and
computational number theory. For instance, the security of ElGamal's encryption
scheme relies on the difficulty of the discrete logarithm problem in
Z/p:
given
a,
bE
Z/px,
find
e
such that
ae
=
b
mod
p.
There is an an analogous problem in
the abelian group Jac(C)(JF): given
A, B
E
Jac(C)(JF), find
e
such that
eA
=
B.
One can use to this to create an encryption scheme based on Jacobian varieties.
Understanding the security of such a system relies on understanding the expected
structure of the group Jac(C)(JF). Such groups also arise in primality testing
[3]
and integer factorization
[19, 20];
again, results on expected divisibility properties
of IJac(C)(JF)I as
C
varies are crucial to estimates of efficiency.
The Cohen-Lenstra heuristics conjecturally describe the frequency with which
a given abelian group occurs as the class group of a quadratic imaginary number
field
[9].
Although these heuristics remain unproven, they have inspired detailed
studies of class groups of function fields. Friedman and Washington conjecturally
[10]
describe the probability with which a given abelian £-group occurs as the
£-Sylow part of the class group of a function field drawn randomly from all, or
even all hyperelliptic, function fields. (Since the function field of a hyperelliptic
curve admits a presentation as JF(x)[y]/(y2
-
f(x))
for some polynomial
f(x)
E
lF[x], such fields are a good analogue for quadratic number fields.) A variety of
2000 Mathematics Subject Classification. 11G20; 11R58, 52B30.
@2008 American Mathematical Society
http://dx.doi.org/10.1090/conm/463/09041
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