On zeta functions and Iwasawa modules

Arithmetic incarnations of zeta in Iwasawa theory

We give a brief exposition of the Iwasawa theory of cyclotomic extensions, so as to discuss its relationship with p-adic zeta functions. We give an overview of these connections and the arithmetic significance of the theory, leading up to a statement of the main conjecture.

Filtrations of smooth principal series and Iwasawa modules

Let $G$ be a reductive $p$-adic group‎. ‎We consider the general question‎ ‎of whether the reducibility of an induced representation can be detected in a‎ ‎``co-rank one‎" ‎situation‎. ‎For smooth complex representations induced from supercuspidal‎ ‎representations‎, ‎we show that a sufficient condition is the existence of a subquotient‎ ‎that does not appear as a subrepresentation‎. ‎An import...

Iwasawa-Tate on ζ-functions and L-functions

After a too-brief introduction to adeles and ideles, we sketch proof of analytic continuation and functional equation of Riemann’s zeta, in the modern form due independently to Iwasawa and Tate about 1950. The sketch is repeated for Dedekind zeta functions of number fields, noting some additional complications. The sketch is repeated again for Hecke’s (größencharakter) L-functions, noting furth...

Galois modules, Iwasawa theory and leading term conjectures

Iwasawa Theory and Motivic L-functions

We illustrate the use of Iwasawa theory in proving cases of the (equivariant) Tamagawa number conjecture.