Seminar

Speaker

Dr. Rony Bitan (Bar-Ilan University)

Date

29/04/2015 - 11:30 - 10:30Add to Calendar

`The Hasse principle for bilinear symmetric forms over the ring of integers of a global function field``2015-04-29 10:30:00``2015-04-29 11:30:00``Let C be a smooth projective curve defined over a finite field F_q (q is odd), and let K=F_q(C) be its (global) function field. This field is considered as the geometric analogue of a number field. Removing one closed point from C results in an affine curve C^af. The ring of regular functions over C^af is an integral domain, over which we consider a non-degenerate bilinear and symmetric form f of any rank n. We express the number c(f) of isomorphism classes in the genus of f in cohomological terms and use it to present a sufficient and necessary condition depending only on C^af, under which f admits the Hasse local-global principle. We say that f admits the Hasse local-global principle if c(f)=1, namely, |Pic(C^\af)| is odd for any n other than 2 and equal to 1 for n=2. This result emphasizes the difference between Galois cohomology and etale cohomology. Examples are provided.``Third floor seminar room``אוניברסיטת בר-אילן - Department of Mathematics``mathoffice@math.biu.ac.il``Asia/Jerusalem``public`Place

Third floor seminar room

Abstract

Let C be a smooth projective curve defined over a finite field F_q (q is odd), and let K=F_q(C) be its (global) function field. This field is considered as the geometric analogue of a number field.

Removing one closed point from C results in an affine curve C^af. The ring of regular functions over C^af is an integral domain, over which we consider a non-degenerate bilinear and symmetric form f of any rank n.

We express the number c(f) of isomorphism classes in the genus of f in cohomological terms and use it to present a sufficient and necessary condition depending only on C^af, under which f admits the Hasse local-global principle. We say that f admits the Hasse local-global principle if c(f)=1, namely, |Pic(C^\af)| is odd for any n other than 2 and equal to 1 for n=2.

This result emphasizes the difference between Galois cohomology and etale cohomology. Examples are provided.

Last Updated Date : 19/04/2015