Romyar Sharifi (McMaster)  <BR><BR>


Investigations in the arithmetic of cyclotomic fields and modular curves
<BR><BR>

Class groups of cyclotomic fields have interested number theorists  
since the 19th century.  In the final quarter of the 20th century,  
Ribet and Mazur-Wiles demonstrated that aspects of their structure  
can be ascertained from the study of modular representations.  The  
goal of this talk is to give a modular interpretation of particular  
elements in these class groups that arise as values of a cup product  
pairing on cyclotomic units.  These pairing values yield information  
on a wealth of algebraic objects, but any analytic interpretation of  
them had been heretofore quite mysterious.  We will describe how,  
conjecturally, modular representations can be used to relate the  
pairing values to p-adic L-values of cusp forms.