MATH Seminar

Title: Reduced Unitary Whitehead Groups over Function Fields of $p$-adic Curves
Defense: Dissertation
Speaker: Zitong Pei of Emory Unviersity
Contact: Zitong Pei,
Date: 2024-04-22 at 11:00AM
Venue: MSC E406
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The study of the Whitehead group of semi-simple simply connected groups is classical with an abundance of new open questions concerning the triviality of these groups. The Kneser-Tits conjecture on the triviality of these groups was answered in the negative by Platanov for general fields. There is a relation between reduced Whitehead groups and $R$-equivalence classes in algebraic groups.\\ \\ Let $G$ be an algebraic group over a field $F$. The $R$-equivalence, defined by Manin, is the equivalence relation on $G(F)$ defined by $x\sim y$ for $x, y \in G(F)$ if there exists a $F$-rational morphism ${\mathbb A}^1_K \cdots \to G$ defined at $0$ and $1$ and sending 0 to $x$ and 1 to $y$. Let $RG(F)$ be the equivalence class of the identity element in $G(F)$. Then $RG(F)$ is a normal subgroup of $G(F)$ and the quotient $G(F)/RG(F)$ is called the group of $R$-equivalence classes of $G(F)$. It is well known that for the semi-simple simply connected isotropic group $G$ over $F$, $W(G, F)$ is isomorphic to the group of $R$-equivalence classes. Thus the group of $R$-equivalence classes can be thought as Whitehead groups for general algebraic groups. The group of $R$-equivalence classes, is very useful while studying the rationality problem for algebraic groups, the problem to determine whether the variety of an algebraic group is rational or stably rational.\\ \\ Suppose that $D_0$ is a central division $F_0$-algebra. If the group $G(F_0)$ of rational points is given by $SL_n(D)$ for some $n>1$, then $W(G, F_0) $ is the reduced Whitehead group of $D_0$. Let $F$ be a quadratic field extension of $F_0$ and $D$ be a central division $F$-algebra. Suppose that $D$ has an involution of second kind $\tau$ such that $F^{\tau}=F_0$. If the hermitian form $h_{\tau}$ induced by $\tau$ is isotropic and the group $G(F_0)$ is given by $SU(h_\tau, D)$, then $W(G, F_0)$ is isomorphic to the reduced unitary Whitehead group of $D.$\\ \\ We start from the fundamental facts on reduced unitary Whitehead groups of central simple algebras, then introduce the patching techniques. Finally, let $F/F_0$ be a quadratic field extension of the function field of a $p$-adic curve. Let $A$ be a central simple algebra over $F$. Assume that the period of $A$ is two and $A$ has a unitary $F/F_0$ involution. We provide a proof for the triviality of the reduced unitary Whitehead group of $A$.

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