Seminar Announcement Date: Tuesday, 31 October 2023 Time: 3:30  4:30 PM Venue: Lecture Hall 802 Localglobal principle for hermitian spaces over semiglobal fields Jayanth Guhan Chennai Mathematical Institute. 311023 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 HasseMinkowski theorem, which shows that that a quadratic form over a number field has a nontrivial zero if and only if it it has nontrivial zeroes over completions at all places of the field. b) the AlbertBrauerHasseNoether 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 Springertype theorem for isotropy of hermitian spaces over odd degree extensions of function fields of padic curves.
