Buchmann's Algorithm

Buchmann's algorithm to compute class groups and units of an
algebraic number fields has been implemented in Pari, Lidia and 
Magma. We explain the algorithm  from the point of view of Arakelov
class groups.


Odlyzko's Discriminant bounds

The discriminant bounds for number fields   
derived by Andrew Odlyzko in the 1970's
are analogous to the bounds for the number of rational points
on curves over finite fields derived by Drinfeld and Vladut in 
the 1980's. We derive the bounds and explain the analogy. 

See:

http://www.dtc.umn.edu/~odlyzko/unpublished/
http://www.mat.uniroma2.it/~eal/poitou-bb.pdf