
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR
January–June
2013
Thematic Program on Torsors, Nonassociative Algebras
and Cohomological Invariants
Coxeter
Lecture Series,
May 2123
Raman
Parimala,
Emory University
Lecture on Arithmetic of linear algebraic groups over
two dimensional fields
May 21, 2013 at 3:30 p.m. Bahen
Centre, Rm 1180
(map)
Lecture on Quadratic forms and Galois cohomology
May 22, 2013 at 3:30 p.m. Bahen
Centre, Rm 1190
(map)
Lecture
on A Hasse principle over function fields
May
23, 2013 at 3:30 p.m. Bahen
Centre, Rm 1190
(map)



A conjecture of Serre (Conjecture
II) states that every principal homogeneous space under
a semisimple simply connected linear algebraic group, defined
over a field of cohomological dimension two, has a rational
point. This conjecture for totally imaginary number fields,
known as the Hasse principle conjecture, is settled by KneserHarderChernousov.
A major breakthrough for general fields is a theorem of
MerkurjevSuslin which settles Conjecture II for groups
of inner type An. I will describe
some approaches to resolving this conjecture, each yielding
a positive answer to Conjecture II for special classes of
groups or fields.
Lecture II: Quadratic
forms and Galois cohomology
(presentation slides)
The classical invariants of quadratic
forms, the dimension, the discriminant and the Clifford
invariant classify quadratic forms over totally imaginary
number fields. Milnor proposed successive higher invariants
for quadratic forms with values in mod 2 Galois cohomology
groups, extending the classical invariants, and he conjectured
that these invariants classify quadratic forms up to isomorphism.
Milnor’s conjecture is a theorem due to Voevodsky,
Orlov and Vishik. Using this theorem, bounds on the generation
of Galois cohomology groups lead to arithmetic consequences.
Among them is the finiteness of the uinvariant,
equivalently, that quadratic forms in sufficiently many
variables represent zero nontrivially. I will discuss some
progress towards the determination of the uinvariant
of function fields of curves over fields of arithmetic interest.
Lecture III: A Hasse
principle over function fields (presentation
slides)
A localglobal principle for the
existence of nontrivial zeros of quadratic forms over function
fields, with respect to completions at discrete valuations,
has interesting consequences. For function fields of curves
over finite fields, it gives the classical theorem of HasseMinkowski.
One could look for a more general Hasse principle for the
existence of rational points on homogeneous spaces under
connected linear algebraic groups defined over function
fields of curves over a local or a global field. I will
explain some positive results in this direction for certain
classes of groups which include split simply connected groups
over function fields of curves over local fields. There
are recent examples of the failure of the Hasse principle
for rational points on principal homogeneous spaces under
nonrational tori defined over function fields of padic
curves. This leads to the question whether the Hasse principle
holds for all semisimple simply connected groups over these
fields.

Speakers in the Distinguished Lecture Series (DLS) have made
outstanding contributions to their field of mathematics. The
DLS consists of a series of three onehour lectures.
Index of Fields
Distinguished and Coxeter Lectures
Thematic
Year Home page


