The discriminant of an algebraic torus

Bitan, Rony A.

doi:10.1016/j.jnt.2011.03.001

Cited by 8 publications

(7 citation statements)

References 14 publications

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“…Hence t ∞ (G) = |H 1 (I ∞ , F ∞ (K s ∞ )) σ∞ |. More explicitly, the Kottwitz epimorphism together with Galois descent, yields an epimorphism [Bit,Corollary 3.2]). We get the following exact diagram on which the lower row can be also obtained by the following steps: applying the contravariant left-exact functor Hom(−, Z) on the exact sequence of character g ∞ -modules…”

Section: On the Cohomology Of O {∞} -Schemes And Relative Local Covolmentioning

confidence: 98%

“…The I ∞ -coinvariants functor is in general only right exact, but here as T sc ∞ is connected, X * (T sc ∞ ) I∞ is free (see [Bit,Formula (3.1)]) and embedded into X * (T ∞ ) I∞ . Thus applying this functor on…”

Section: On the Cohomology Of O {∞} -Schemes And Relative Local Covolmentioning

confidence: 99%

“…Over anyÔ p , the norm torus is SpecÔ ∞ [a, b]/(a 2 − pb 2 − 1). Its reduction provides at each place p, e p connected components, where e p stands for the ramification index there (see [Bit,Example 3.3]),…”

Section: Application and Examplesmentioning

confidence: 99%

“…Its preimage in G sc is the Weil torus T sc = R L/K (G m,L ), fitting into the exact sequence 1 → F → T sc π −→ T → 1.Over any Ôp , the norm torus is Spec Ô∞ [a, b]/(a 2 − pb 2 − 1). Its reduction provides at each place p, e p connected components, where e p stands for the ramification index there (see[Bit, Example 3.3]),i.e. [T p ( Ôp ) : T 0 p ( Ôp )] = e p .…”

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confidence: 99%

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A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields

Bitan¹,

Köhl²

2015

Funct. Approx. Comment. Math.

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Let G be a semisimple group defined over a global function field K of a rational curve, not anistropic of type An. We express the (relative) Tamagawa number of G in terms of local data including the number t∞(G) of types in one orbit of a special vertex in the Bruhat-Tits building of G∞(K∞) for some place ∞ and the class number h∞(G) of G at ∞. denote suitable Cartan subgroups of G sc and G respectively; cf. Proposition 7.8. These concrete computations allow us to also provide a wealth of examples in Section 6 for which we compute the relative Tamagawa numbers. We also demonstrate the result in a case of a split group defined over the function field of an elliptic curve (Remark 7.11).This article is organized as follows: In the preliminary Section 2 we fix the relevant notions from Bruhat-Tits theory. In Section 3 we compute volumes of parahoric subgroups over local fields, their maximal unramified extensions, and their valuation rings. In Section 4 we revise the definition of the Tamagawa number of semisimple K-groups and establish a decomposition of G(A)/G(K) enabling us to express τ (G) in terms of a global invariant and a local one. In Section 5 we compute cohomology groups over rings of S-integers with |S| = 1, use Bruhat-Tits theory and Serre's formula ([Ser1, p. 84], [BL, Corollary 1.6]) in order to derive the above-mentioned formula (13) for computing the Tamagawa number. In Section 6 we express the number t ∞ (G) of types in the orbit of a special point in terms of F ∞ , accomplishing the proof of our Main Theorem. The final Section 7 addresses the above-mentioned application and examples.

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Section: On the Cohomology Of O {∞} -Schemes And Relative Local Covolmentioning

confidence: 98%

Section: On the Cohomology Of O {∞} -Schemes And Relative Local Covolmentioning

confidence: 99%

Section: Application and Examplesmentioning

confidence: 99%

mentioning

confidence: 99%

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A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields

Bitan¹,

Köhl²

2015

Funct. Approx. Comment. Math.

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“…Accordingly, by Theorem 4.5, the kernel of Tr Gr R (T ) is H 2 (π 0 (T )k,Q × ) W . By [6,Equation (3.1)], the special fibre of the component group scheme for T is given by…”

Section: Quasicharacter Sheaves For P-adic Torimentioning

confidence: 99%

From the Function-Sheaf Dictionary to Quasicharacters of -Adic Tori

Cunningham¹,

Roe²

2015

J. Inst. Math. Jussieu

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Abstract. We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme G over a finite field k and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on G and show that it is an extension of the group of characters of G(k) by a cohomology group determined by the component group scheme of G. We also classify all morphisms in the category character sheaves on G. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of p-adic tori.

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Some Analytic and Arithmetic Properties of Integral Models of Algebraic Tori

Bagdasaryan

2023

Trends in Mathematics

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The discriminant of an algebraic torus

Cited by 8 publications

References 14 publications

A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields

A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields

From the Function-Sheaf Dictionary to Quasicharacters of -Adic Tori

Some Analytic and Arithmetic Properties of Integral Models of Algebraic Tori

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