MATH Seminar
| Title: A local-global principle for adjoint groups over function fields of p-adic curves |
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| Defense: Dissertation |
| Speaker: Jack Barlow of Emory University |
| Contact: Jack Barlow, jack.barlow@emory.edu |
| Date: 2023-03-23 at 2:30PM |
| Venue: MSC E406 |
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| Abstract: Let $k$ be a number field and $G$ a semisimple simply connected linear algebraic group over $k$. The Kneser conjecture states that the Hasse principle holds for principal homogeneous spaces under $G$. Kneser's conjecture is a theorem due to Kneser for all classical groups, Harder for exceptional groups other than $E_8$, and Chernousov for $E_8$. It has also been proved by Sansuc that if $G$ is an adjoint linear algebraic group over $k$, then the Hasse principle holds for principal\\ homogeneous spaces under $G$.\\ \par Now let $p\in\mathbb{N}$ be a prime with $p\neq 2$, and let $K$ be a $p$-adic field. Let $F$ be the function field of a curve over $K$. Let $\Omega_F$ be the set of all divisorial discrete valuations of $F$. It is a conjecture of Colliot-Thélène, Parimala and Suresh that if $G$ is a semisimple simply connected linear algebraic group over $F$, then the Hasse principle holds for principal homogeneous spaces under $G$. This conjecture has been proved for all groups of classical type. In this talk, we ask whether the Hasse principle holds for adjoint groups over $F$, motivated by the number field case. We give a positive answer to this question for a class of adjoint classical groups. |
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