CAPITAL REGION ALGEBRA/NUMBER THEORY SEMINAR 


Speaker:       Allison Pacelli, Williams College 

Time:          Wednesday, December 15, at 4:30pm       

Place:         Bailey Hall 100, Union College 

Title:         Class Groups of Global Function Fields        


Abstract:

The parallelism between algebraic number fields and global
function fields is both beautiful and useful.  Frequently 
the analogy is used to provide insight into the number field 
realm by formulating suitable analogues of results previously 
known for function fields. The analogy is useful in the opposite 
direction as well.  For example, less is known about the class 
group in the function field situation than in the number field case.

In this talk, we'll discuss  class groups of  global  
function fields.  Let  q  be a power of an odd prime, and
F_q  the finite field with  q  elements.  It is known that
for any natural numbers  m  and  n, infinitely many function fields
K  of degree  m  over the rational function field  F_q(T) 
have class numbers divisible by  n.  This is a consequence of
stronger results about the structure of the class group, specifically
the rank of a subgroup of the class group.  We will see that the rank
of a subgroup of the class group is closely related to the rank of the
unit group and the decomposition of the prime at infinity.  In
particular,  for any integers  m  and  n, and  1 < g < m,  there
are infinitely many function fields  K  of degree  m  over  F_q(T)  
such that the prime at infinity splits into  g  primes in  K  and 
the ideal class group of  K  contains a subgroup isomorphic 
to  ( Z/nZ )^{m-g} .