Summary: Contemporary Mathematics
Results of Cohen-Lenstra type for quadratic function fields
Jeffrey D. Achter
Abstract. Consider hyperelliptic curves C of fixed genus over a finite field F.
Let L be a finite abelian group of exponent dividng N. We give an asymptotic
formula in |F|, with explicit error term, for the proportion of C for which
Jac(C)[N](F) = L.
Let C be a smooth, proper curve of positive genus g over a finite field F. 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)(F). Varying the coefficients of C yields a family of groups.
On the other hand, since Jac(C)(F) 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)(F) 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, b Z/p×
, find e such that ae