Higher dimensional class field theory: The variety case / Linda M. Gruendken.

Format:

Book

Manuscript

Thesis/Dissertation

Author/Creator:

Gruendken, Linda M.

Contributor:

Pop, Florian, 1952- advisor.

University of Pennsylvania.

Language:

English

Subjects (All):

Penn dissertations--Mathematics.

Mathematics--Penn dissertations.

Local Subjects:

Penn dissertations--Mathematics.

Mathematics--Penn dissertations.

Physical Description:

vi, 134 pages ; 29 cm

Production:

2011.

Summary:

Let k be a finite field, and suppose that the arithmetical variety X &sub; Pnk is an open subset in projective space. Suppose that CX is the Wiesend idele class group of X, pab1 (X) the abelianised fundamental group, and rho X : CX&rarrr;pab 1 (X) the Wiesend reciprocity map. We use the Artin-Schreier-Witt and Kummer Theory of affine k-algebras to prove a full reciprocity law for X. We find necessary and sufficent conditions for a subgroup H < CX to be a norm subgroup: H is a norm subgroup if and only if it is open and its induced covering datum is geometrically bounded. We show that rhoX is injective and has dense image. We obtain a one-to-one correspondence of open geometrically bounded subgroups of CX with open subgroups of pab1 (X). Furthermore, we show that for an etale cover X'' &rarr; X with maximal abelian subcover X' &rarr; X, the reciprocity morphism induces an isomorphism CX/NCX' ' &sime; Gal(X'/X).

Notes:

Adviser: Florian Pop.

Thesis (Ph.D. in Mathematics) -- University of Pennsylvania, 2011.

Includes bibliographical references.