R. Parimala, R. Preeti: Hasse principle for Classical groups over function fields of curves over number fields

parimala@math.tifr.res.in, preeti@math.tifr.res.in

Submission: 2002, Sep 16

In ([CT]), Colliot-Th\'{e}l\{e}ne conjectures the following: \\ Let $F$ be a function field in one variable over a number field, with field of constants $k$ and $G$ be a semisimple simply connected linear algebraic group defined over $F$. Then the map $H^1(F, G) \to \prod_{v \in \Omega_k} H^1( F_v, G)$ has trivial kernel, $\Omega_k$ denoting the set of places of $k$. \smallskip The conjecture is true if $G$ is of type ${\,}^1A^*$, i.e., isomorphic to $SL_1 (A)$ for a central simple algebra $A$ over $F$ of square free index, as pointed out by Colliot-Th\'{e}l\{e}ne, being an immediate consequence of the theorems of Merkurjev-Suslin ([S1]) and Kato ([K]). Gille ([G]) proves the conjecture if $G$ is defined over $k$ and $F = k(t)$, the rational function field in one variable over $k$. We prove that the conjecture is true for groups $G$ defined over $k$ of the types ${\,}^2A^*$, $B_n$, $C_n$, $D_n$ ($D_4$ nontrialitarian), $G_2$ or $F_4$; a group is said to be of type ${\,}^2A^*$, if it is isomorphic to $SU(B, \tau)$ for a central simple algebra $B$ of square free index over a quadratic extension $k'$ of $k$ with a unitary $k'|k$ involution $\tau$.

2000 Mathematics Subject Classification:

Keywords and Phrases:

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