School of Mathematics and Statistics   Number Theory Seminar

# Steve Ward

## Thursday 1 June, 4:05-4:55PM, Carslaw 375

### Class Field Theory and the Kronecker Weber Theorem

 The Main Theorem of Complex Multiplication for abelian varieties is a generalisation of the Kronecker-Weber theorem which states that every abelian extension of $\Q$ is contained in a cyclotomic field, and also the result of Weber and Takagi which says that every abelian extension of an imaginary quadratic field is generated by singular moduli, i.e. the j-invariant of a lattice [1, \tau] with complex multiplication. This week I will lay the groundwork from class field theory necessary to state and prove the main theorem seen in Shimura's book. As an application, we will already be able to prove the Kronecker-Weber theorem. Next week I will explain the (general) main theorem and sketch a proof.