A Hasse principle for quadratic forms over function fields

Parimala, Raman

doi:10.1090/s0273-0979-2014-01443-0

Cited by 8 publications

(5 citation statements)

References 19 publications

(6 reference statements)

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“…We will also use this result to prove a finiteness statement for spinor groups, as well as some unitary groups and groups of type , with good reduction over a class of fields that are not finitely generated. This class includes the function fields of -adic curves that have received a great deal of attention in recent years (we refer the reader to [Bru10, CPS12, Par14, PS98] and references therein for various results involving division algebras, quadratic forms and algebraic groups over those fields), but is in fact much larger. We will formulate our results using a generalization of Serre’s condition (see [Ser97, ch.…”

Section: Introductionmentioning

confidence: 99%

Spinor groups with good reduction

Chernousov¹,

Rapinchuk²,

Rapinchuk³

2019

Compositio Math.

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Let K be a 2-dimensional global field of characteristic = 2, and let V be a divisorial set of places of K. We show that for a given n 5, the set of K-isomorphism classes of spinor groups G = Spin n (q) of nondegenerate n-dimensional quadratic forms over K that have good reduction at all v ∈ V , is finite. This result yields some other finiteness properties, such as the finiteness of the genus gen K (G) and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups H i (K, µ2)V for i 1 established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type G2.

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Section: Introductionmentioning

confidence: 99%

Spinor groups with good reduction

Chernousov¹,

Rapinchuk²,

Rapinchuk³

2019

Compositio Math.

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“…In particular, X d (M j ) = ∅. Since the residue fields are either local or global (see [Par14,8.1]), [k j : k(v)] is odd and per(D ⊗ F F v )|2, by lemma 5.6 and lemma 5.7, conditions in lemma 5.1 are satisfied. By Morita equivalence and lemma 5.1, X d (F v ) = ∅ for all v. Finally by the Hasse principle theorem 4.5, X d (F ) = ∅, so h is isotropic.…”

Section: Application: Odd Degree Extensionsmentioning

confidence: 99%

Hasse principle for hermitian spaces over semi-global fields

2016

Journal of Algebra

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Abstract. In a recent paper, Colliot-Thélène, Parimala and Suresh conjectured that a local-global principle holds for projective homogeneous spaces under connected linear algebraic groups over function fields of p-adic curves. In this paper, we show that the conjecture is true for any linear algebraic group whose almost simple factors of its semisimple part are isogenous to unitary groups or special unitary groups of hermitian or skew-hermitian spaces over central simple algebras with involutions. 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.

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“…Here we just mention that these conjectures deal with such classical aspects of the theory as the local-global principle (formulated in terms of properties of the global-to-local map in Galois cohomology -cf. [119] and the recent survey [86]) as well as ways of extending the theorem on the finiteness of class numbers for groups over number fields to more general situations. It should also be pointed out that, if proven, the finiteness conjecture for groups with good reduction would have numerous applications: we will discuss the genus problem for absolutely almost simple algebraic groups and weakly commensurable Zariski-dense subgroups of these that play a crucial role in the analysis of length-commensurable locally symmetric spaces, particularly those that are not arithmetically defined (such as, for example, nonarithmetic Riemann surfaces).…”

Section: Introductionmentioning

confidence: 99%

Linear algebraic groups with good reduction

Rapinchuk¹,

Rapinchuk²

2020

Preprint

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This article is a survey of conjectures and results on reductive algebraic groups having good reduction at a suitable set of discrete valuations of the base field. Until recently, this subject has received relatively little attention, but now it appears to be developing into one of the central topics in the emerging arithmetic theory of (linear) algebraic groups over higher-dimensional fields. The focus of this article is on the Main Conjecture (Conjecture 5.7) asserting the finiteness of the number of isomorphism classes of forms of a given reductive group over a finitely generated field that have good reduction at a divisorial set of places of the field. Various connections between this conjecture and other problems in the theory of algebraic groups (such as the analysis of the global-to-local map in Galois cohomology, the genus problem, etc.) are discussed in detail. The article also includes a brief review of the required facts about discrete valuations, forms of algebraic groups, and Galois cohomology.

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A Hasse principle for quadratic forms over function fields

Cited by 8 publications

References 19 publications

Spinor groups with good reduction

Spinor groups with good reduction

Hasse principle for hermitian spaces over semi-global fields

Linear algebraic groups with good reduction

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