Number Theory
Stability, Non-Abelian Class Field Theory and
New Zeta Functions
Nagoya
University
Abstract
In this
talk, we use stability conditions to develop a non-abelian
class field theory for Riemann surfaces, and to introduce new non-abelian zeta functions for global fields.
More precisely, first, for Riemann surfaces, we show the Existence Theorem and establish the Reciprocity Law in non-abelian CFT for Riemann surfaces, based on the Narasimhan-Seshadri correspondence and theory of Tannakian categories.
Then we introduce new non-abelian zeta functions for curves defined over finite fields, by using the moduli spaces of semi-stable bundles, based on what we call the Harder-Narasimhan correspondence. Basic properties such as rationality and functional equation will be discussed as well.
Finally, we introduce our non-abelian zeta functions for number fields, based on our works on adelic moduli spaces of semi-stable lattices and new arithmetic cohomology.
While it is
not a fashionable one, our approach is by no means radical. (1) For Riemann
surfaces, it starts from where Weil left. (2) For curves over finite fields,
our new zeta functions, while are different from that of Weil, are indeed
natural generalizations of Artin zeta functions. And
(3) for number fields, our cohomology (resp.
stability, new zeta) is motivated by Tate's thesis (resp. Mumford's GIT, Iwasawa's work on Dedekind zeta functions).
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Date: |
December 28, 2001 (Friday) |
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Time: |
4:00 - 5:00pm |
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Place: |
Room 517, Meng Wah Complex |
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All are welcome |
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