an:01998527
Zbl 1053.11087
Bauer, Mark L.
The arithmetic of certain cubic function fields
EN
Math. Comput. 73, No. 245, 387-413 (2004).
00101220
2004
j
11R58 94A60 14H05 11Y40
Summary: We discuss the properties of curves of the form \(y^3=f(x)\) over a given field \(K\) of characteristic different from 3. If \(f(x)\) satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the Jacobian of such curves. From a purely mathematical perspective, this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory. At the same time, it provides another source of groups which are suitable for Diffie-Hellman type protocols in cryptographic applications.