IMSc Webinar

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

Jayanth Guhan

CMI

Let $G$ be an algebraic group over a field $F$. Let $X$ be a homogeneous space under $G$ over $F$. Let $\Omega$ be the set of all places of $F$. For $u \in \Omega$, let $F_{u}$ be the completion of $F$ at $u$. The Hasse Principle is said to hold for $X$ if

$$\prod X(F_{u}) eq \emptyset \implies X(F) \eq \emptyset.$$

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 $A \in {}_2Br(F)$ be a central simple algebra with an involution $\sigma$ of any kind and $F_0 =F^{\sigma}$. Let $h$ be an hermitian space over $(A, \sigma)$ and $G = SU(A, \sigma, h)$ if $\sigma$ is of first kind and $G = U(A, \sigma, h)$ if $\sigma$ is of second kind.
Suppose that char$(k)
\eq 2$ and ind$(A)\leq 4$. Then we prove that the Hasse principle holds for projective homogeneous spaces under $G$ over $F_0$. 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.
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 ∈ 2 Br(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) 6= 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.