The Bloch-Nevanlinna conjecture revisited

Bloch-Kato Conjecture: Baby Version

In this talk K be a global field and GK := Gal(K/K) the absolute Galois group over K. We let V be a p-adic representation of GK (finite-dimensional over Qp), and Σ a finite set of places of K containing p and ∞, outside which V is unramified. So we can and will view V as a representation of GK,Σ for technical reasons, it is at times important to work with that rather than the full GK ! We are g...

Composition Operators from Nevanlinna Type Spaces to Bloch Type Spaces

Let X and Y be complete metric spaces of analytic functions over the unit disk in the complex plane. A linear operator T : X → Y is a bounded operator with respect to metric balls if T takes every metric ball in X into a metric ball in Y . We also say that T is metrically compact if it takes every metric ball in X into a relatively compact subset in Y . In this paper we will consider these prop...

The Nevanlinna Theorem of the Classical Theory of Moments Revisited

The canonical solutions of the truncated Hamburger moment problem (both in the classical and degenerate cases) are found. The Nevanlinna theorem which provides the noncanonical solutions of the truncated Hamburger problem is also rederived in the framework of the operator approach.

The Mertens Conjecture Revisited

The Bloch-Kato Tamagawa Number Conjecture

Here, τ(G) is roughly the volume of G(AK)/G(K) with respect to Haar measures on G(Kv) for all places v, where we have to use L-functions to make the product measure converge, and also have to restrict to measuring some “compact part” of G(Kv), by taking the kernel of all |χ|v, χ : G→ Gm. Now, let’s formulate a version of this for abelian varieties. For simplicity, assume that E is an elliptic c...