Essential dimension, spinor groups, and quadratic forms

Brosnan, Patrick; Reichstein, Zinovy; Vistoli, Angelo

doi:10.4007/annals.2010.171.533

Cited by 34 publications

(63 citation statements)

References 17 publications

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“…Some of these have already appeared in print. In particular, we used Theorem 1.3 to study the essential dimension of spinor groups in [BRV10], N. Karpenko and A. Merkurjev [KM08] used it to study the essential dimension of finite p-groups, and A. Dhillon and N. Lemire [DL] used it, in combination with the Genericity Theorem 6.1, to give an upper bound for the essential dimension of the moduli stack of SL n -bundles over a projective curve. In this paper Theorem 1.3 (in combination with Theorem 6.1) is also used to study the essential dimension of the stacks of hyperelliptic curves (Theorem 7.2) and, in the appendix written by Najmuddin Fakhruddin, of principally polarized abelian varieties.…”

Section: Introductionmentioning

confidence: 99%

Essential dimension of moduli of curves and other algebraic stacks

Brosnan¹,

Reichstein²,

Vistoli³

2011

J. Eur. Math. Soc.

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Abstract. In this paper we consider questions of the following type. Let k be a base field and K/k be a field extension. Given a geometric object X over a field K (e.g. a smooth curve of genus g), what is the least transcendence degree of a field of definition of X over the base field k? In other words, how many independent parameters are needed to define X? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to the question above when the geometric objects X are smooth, stable or hyperelliptic curves. The appendix, written by Najmuddin Fakhruddin, answers this question in the case of abelian varieties.

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Section: Introductionmentioning

confidence: 99%

Essential dimension of moduli of curves and other algebraic stacks

Brosnan¹,

Reichstein²,

Vistoli³

2011

J. Eur. Math. Soc.

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“…This improves on the long-standing edÔPGL n Õ char F n 2 2 2 -5 2 -6 2 -3 2 -15 2 -64 2 -28 2 -8 [MR09,§1].) The gap between the upper and lower bounds for PGL n stands in interesting constrast with the situtation for Spin n : edÔSpin n Õ is both OÔ 2 n Õ and ΩÔ 2 n Õ, i.e., is asymptotically bounded both above and below by constants times 2 n , by [BRV10].…”

Section: ô¡ Gõmentioning

confidence: 99%

Open problems on central simple algebras

Auel

Brussel

Garibaldi

et al. 2011

Transformation Groups

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Abstract. We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners. Motivated by these and other developments, we now present an updated list of open problems. Some of these problems are-or have become-special cases of much more general problems for algebraic groups. To keep our task manageable, we (mostly) restrict our attention to central simple algebras and PGL n . The following is an idiosyncratic list that originated during the Brauer group conference held at Kibbutz Ketura in January 2010. We thank the participants in that problem session for their contributions to this text. We especially thank Darrel Haile,

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“…We believe it to be the case that for n ≥ 15 we have {BSpin n } = {Spin n } −1 . This would provide the first example of a connected group G for which {BG} = {G} −1 , and should be at least morally related to other odd behaviour of spin groups, regarding for example essential dimension [BRV10] and (conjectural) failure of stable rationality of fields of invariants [Mer,Conjecture 4.5]. We plan to return to this point in future work.…”

Section: Introductionmentioning

confidence: 91%

“…Note that ∆ n ⊂ Pin n is also isomorphic to the preimage of the diagonal matrices of SO n+1 in Spin n+1 , by sending the element e i ∈ C n to e i e n+1 ∈ C n+1 (recall that C n denotes the Clifford algebra). This finite subgroup is quite well-understood, and is responsible in particular for the exponential growth of the essential dimension of spin groups (see [BRV10]).…”

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confidence: 99%

On the Motivic Class of the Classifying Stack of $G_2$ and the Spin Groups

Pirisi

Talpo

2017

International Mathematics Research Notices

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We compute the class of the classifying stack of the exceptional algebraic group G 2 and of the spin groups Spin 7 and Spin 8 in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks BSpin n can be reduced to the computation of the classes of B∆n, where ∆n ⊂ Pinn is the "extraspecial 2-group", the preimage of the diagonal matrices under the projection Pinn → On to the orthogonal group.

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Essential dimension, spinor groups, and quadratic forms

Abstract: We prove that the essential dimension of the spinor group Spin n grows exponentially with n and use this result to show that quadratic forms with trivial discriminant and Hasse-Witt invariant are more complex, in high dimensions, than previously expected.

Cited by 34 publications

References 17 publications

Essential dimension of moduli of curves and other algebraic stacks

Essential dimension of moduli of curves and other algebraic stacks

Open problems on central simple algebras

On the Motivic Class of the Classifying Stack of $G_2$ and the Spin Groups

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