# The Hasse principle for bilinear symmetric forms over the ring of integers of a global function field

Seminar
Speaker
Dr. Rony Bitan (Bar-Ilan University)
Date
29/04/2015 - 11:30 - 10:30Add to Calendar 2015-04-29 10:30:00 2015-04-29 11:30:00 The Hasse principle for bilinear symmetric forms over the ring of integers of a global function field 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 - המחלקה למתמטיקה 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.

תאריך עדכון אחרון : 19/04/2015