Essential dimension of algebraic tori

Lötscher, Roland; MacDonald, Mark L.; Meyer, Aurel; Reichstein, Zinovy

doi:10.1515/crelle.2012.010

Cited by 24 publications

(32 citation statements)

References 28 publications

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“…The details (namely, the proofs of Propositions 4.2 and 4.3) will be supplied in the next two sections. The starting point of our argument is [LMMR,Theorem 3.1], which we reproduce as Theorem 4.1 below for the reader's convenience.…”

Section: Proof Of the Main Theorem: An Overviewmentioning

confidence: 99%

“…Assume the contrary. Then by [LMMR,Lemma 2.6] there exists a surjective homomorphism f : P → X(T ) of Gal(k sep /k)-lattices, where P is permutation and rank P = dim T . This implies that f has finite kernel and hence, is injective.…”

Section: Proof Of the Main Theorem: An Overviewmentioning

confidence: 99%

“…Recall that G 0 is a torus, which we denote by T and G/G 0 is a finite p-group which we denote by F . The case, where T = {1}, i.e., G = F is an arbitrary (possibly twisted) finite p-group, is established in the course of the proof of [LMMR,Theorem 7.1]. Our proof of Proposition 4.2 below is based on leveraging this case as follows.…”

Section: Dimensions Of Irreducible Representationsmentioning

confidence: 99%

“…In these cases Theorem 1.1 reduces to [LMMR,Theorems 1.1 and 7.1], respectively. In the case of constant finite p-groups this result is due to N. Karpenko and A. Merkurjev, whose work [KM] was the starting point for both [LMMR] and the present paper.…”

Section: Introductionmentioning

confidence: 99%

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Essentialp-dimension of algebraic groups whose connected component is a torus

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Algebra Number Theory

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Section: Proof Of the Main Theorem: An Overviewmentioning

confidence: 99%

Section: Proof Of the Main Theorem: An Overviewmentioning

confidence: 99%

Section: Dimensions Of Irreducible Representationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Essentialp-dimension of algebraic groups whose connected component is a torus

Lötscher

MacDonald

Meyer

et al. 2013

Algebra Number Theory

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“…Let γ ′ ∈ H 1 (F ′ , U Φ ) be the corresponding class of U Φ -torsors over F ′ under the isomorphism H 1 (F ′ , U Φ ) ≃ Br p s F ′ (Φ)/F ′ by (7). As γ is a generic U Φ -torsor, there exists an F ′ -morphism v : Spec F ′ → V Φ such that v * (γ) = γ ′ and Im(v) ⊂ W (see Section 2.3).…”

mentioning

confidence: 99%

Essential dimension of central simple algebras

Baek

Merkurjev

2012

Acta Math.

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Abstract. Let p be a prime integer, 1 ≤ s ≤ r integers and F a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed p (Alg p r ,p s ) of the class Alg p r ,p s of central simple algebras of degree p r and exponent dividing p s . In particular, we show that ed 2 (Alg 8,2 ) = 8 and ed p (Alg p 2 ,p ) = p 2 + p for p odd. IntroductionLet F : Fields/F → Sets be a functor from the category Fields/F of field extensions over F to the category Sets of sets. Let E ∈ Fields/F and K ⊂ E a subfield over F . An element α ∈ F (E) is said to be defined over K (and K is called a field of definition of α) if there exists an element β ∈ F (K) such that α is the image of β under the mapwhere the supremum is taken over all fields E ∈ Fields/F and all α ∈ F (E) (see [3, Def. 1.2] or [8, Sec.1]). Informally, the essential dimension of F is the smallest number of algebraically independent parameters required to define F and may be thought of as a measure of complexity of F . Let p be a prime integer. The essential p-dimension of α, denoted ed F p (α), is defined as the minimum of ed F (α E ′ ), where E ′ ranges over all field extensions of E of degree prime to p. The essential p-dimension of F is ed p (F ) = sup{ed F p (α)}, where the supremum ranges over all fields E ∈ Fields/F and all α ∈ F (E). By definition, ed(F ) ≥ ed p (F ) for all p.For every integer n ≥ 1, a divisor m of n and any field extension E/F , let Alg n,m (E) denote the set of isomorphism classes of central simple E-algebras of degree n and exponent dividing m. Equivalently, Alg n,m (E) is the subset of the m-torsion part Br m (E) of the Brauer group of E consisting of all elements a such that ind(a) divides n. In particular, if n = m, then Alg n (E) := Alg n,n (E)

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Essential dimension of algebraic groups, including bad characteristic

Garibaldi

Guralnick

2016

Arch. Math.

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We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group $G$ of rank at least two is at most $\mathrm{dim}(G) - 2(\mathrm{rank}(G)) - 1$. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra.Comment: v2 is a substantial revisio

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Essential dimension of algebraic tori

Abstract: Abstract. The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group.

Cited by 24 publications

References 28 publications

Essentialp-dimension of algebraic groups whose connected component is a torus

Essentialp-dimension of algebraic groups whose connected component is a torus

Essential dimension of central simple algebras

Essential dimension of algebraic groups, including bad characteristic

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