K-theory and L-functions

By Fernando Rodriguez Villegas

These are notes (taken by M. Lalín) for a course on $K$-theory and $L$-functions I gave at Harvard University in the Spring of 2002. Notes for the first few lectures were taken by S. Valverde.

The main goal of the course was to explain how to frame conceptually the numerical examples of D. Boyd relating the Mahler measure of certain two variable polynomials $A(x,y)$ and $L(E,2)$ for $E$ the elliptic curve defined by the zeros of $A$. This amounts to a refinement of the original work of Deninger, who was asked to explain Smyth’s identity of 1981

$$ m(1+x+y)=L'(\chi_{-3},-1). $$

In the introductory part of the course we followed J. Milnor’s algebraic $K$-theory.