Alladi Ramakrishnan Hall

Local-global principle for hermitian spaces over semi-global fields

Jayanth Guhan

Emory University

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) ≠ ∅.
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 be a 2-torsion central simple F-algebra with an involution σ of any kind and F0 =Fσ. 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.