Seminar Announcement
Date: Tuesday, 31 October 2023
Time: 3:30 - 4:30 PM
Venue: Lecture Hall 802
Local-global principle for hermitian spaces over semi-global fields
Jayanth Guhan
Chennai Mathematical Institute.
31-10-23

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Abstract

Let G be an algebraic group over a field F. Let X be a homogeneous space under G over F . Let Ω be the set of all places of F. For ν ∈ Ω, let Fν be the completion of F at ν. The Hasse Principle is said to hold for X if

∏ X(Fν ) ≠ ∅ ⇒ X(F ) 6 ≠ ∅.

Some well known examples of the Hasse principle include; a) the Hasse-Minkowski theorem, which shows that that a quadratic form over a number field has a non-trivial zero if and only if it it has non-trivial zeroes over completions at all places of the field. b) the Albert-Brauer-Hasse-Noether theorem which shows that a central simple algebra over a number field, which is a locally a matrix algebra, is a matrix algebra.

Let K be a complete discrete valued field with residue field k and F the function field of a curve over K. Let A ∈ 2Br(F) be a central simple algebra with an involution σ of any kind and F0 = F^σ. Let h be an hermitian space over (A, σ) and G = SU (A, σ, h) if σ is of first kind and G = U (A, σ, h) if σ is of second kind. Suppose that char(k) ≠ 2 and ind(A) ≤ 4. Then we prove that the Hasse principle holds for projective homogeneous spaces under G over F0. The proof implements patching techniques of Harbater, Hartmann and Krashen. As an application, we obtain a Springer-type theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves.