Local-global Galois theory of arithmetic function fields

Galois deformation theory for norm fields and its arithmetic applications

Chebyshev’s bias in Galois extensions of global function fields

Article history: Received 18 November 2009 Revised 6 January 2011 Accepted 15 March 2011 Available online xxxx Communicated by David Goss MSC: primary 11N05 secondary 11M38, 11G05

Fields and Galois Theory

These notes give a concise exposition of the theory of fields, including the Galois theory of finite and infinite extensions and the theory of transcendental extensions. They are an expansion of those handed out during a course taught to first-year graduate students. v2.01 (August 21, 1996). First version on the web. v2.02 (May 27, 1998). Fixed about 40 minor errors; 57 pages. v3.00 (April 3, 2...

Finiteness properties of soluble arithmetic groups over global function fields

Let G be a Chevalley group scheme and B ≤ G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K , and OS be the corresponding S–arithmetic ring. Then, the S– arithmetic group B(OS) is of type F |S|−1 but not of type FP |S| . Moreover one can derive lower and upper bounds for the geometric invariants Σ(B(OS)). These are sha...

Global Ideal Theory of Meromorphic Function Fields

It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved. Introduction. Throughout this paper let X and Y denote noncompact (connected) Riemann surfaces. Let A(X) (or A for short), denote the ring of all analytic functions on X, and let F(X) (or ...