Residually Pseudo-Split Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Division Algebra ◽  
Residue Field ◽  
Quadratic Extension ◽  
Splitting Field ◽  
Quadratic Space ◽  
Descent Group ◽  
Tits Building ◽  
Quaternion Division Algebra

This chapter presents results about a residually pseudo-split Bruhat-Tits building Ξ‎L. It begins with a case for some quadratic space of type E⁶, E₇, and E₈ in order to identify an unramified extension such that the residue field is a pseudo-splitting field. It then considers a wild quaternion or octonion division algebra and the existence of an unramified quadratic extension L/K such that L is a splitting field of the quaternion division algebra. It also discusses the properties of an unramified extension L/K and shows that every exceptional Bruhat-Tits building is the fixed point building of a strictly semi-linear descent group of a residually pseudo-split building.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

Quadratic Forms of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
The Other ◽  
Quadratic Space ◽  
Division Algebras ◽  
Arbitrary Characteristic ◽  
Definition Of ◽  
Quaternion Division Algebra

This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.

Quadrangles of Type E6, E7 and E8: Summary

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Space ◽  
The Third ◽  
First Case ◽  
Root Group ◽  
Tits Building ◽  
Quaternion Division Algebra

This chapter summarizes the different cases about Moufang quadrangles of type E⁶, E₇ and E₈. The first case is that the building at infinity of the Bruhat-Tits building Ξ‎ is an unramified quadrangle; the second, a semi-ramified quadrangle; and the third, a ramified quadrangle. The chapter considers a theorem that takes into account two root group sequences, both of which are either indifferent or the various dimensions, types, etc., are as indicated in exactly one of twenty-three cases. It also presents a number of propositions relating to a quaternion division algebra and a quadratic space of type Eℓ for ℓ = 6, 7 or 8. Finally, it emphasizes the fact that the quadrangles of type F₄ could have been overlooked in the classification of Moufang polygons.

Semi-ramified Quadrangles of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Space ◽  
Basic Proposition ◽  
Tits Building ◽  
Quaternion Division Algebra

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang semi-ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a semi-ramified quadrangle if δ‎Λ‎ = 1 and δ‎Ψ‎ = 2 holds. The chapter first considers the theorem supposing that ℓ = 6, that δ‎Λ‎ = 1 and δ‎Ψ‎ = 2, and that the Moufang residues R0 and R1 are not both indifferent. This is followed by cases ℓ = 7 and ℓ = 8 as well as theorems concerning an anisotropic pseudo-quadratic space, a quaternion division algebra, standard involution, a proper involutory set, and isotropic and anisotropic quadratic spaces.

Totally Wild Quadratic Forms of Type E7

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Space ◽  
Trace Map ◽  
Quaternion Division Algebra

This chapter assumes that (K, L, q) is a totally wild quadratic space of type E₇. The goal is to prove the proposition that takes into account Λ‎ of type E₇, D as the quaternion division algebra over K whose image in Br(K) is the Clifford invariant of q, and the trace and trace map. The chapter also considers two other propositions: the first states that if the trace map is not equal to zero, then the Moufang residues R₀ and R₁ are not indifferent; the second states that if the trace map is equal to zero, then the Moufang residues R₀ and R₁ are both indifferent.

Linked Tori, II

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Integral Domain ◽  
Quadratic Space ◽  
Small Observation ◽  
Quaternion Division Algebra

This chapter proves several more results about weak isomorphisms between Moufang sets arising from quadratic forms and involutory sets. It first fixes a non-trivial anisotropic quadratic space Λ‎ = (K, L, q) before considering two proper anisotropic pseudo-quadratic spaces. It then describes a quaternion division algebra and its standard involution, a second quaternion division algebra and its standard involution, and an involutory set with a quaternion division algebra and its standard involution. It concludes with one more small observation regarding a pointed anisotropic quadratic space and shows that there is a unique multiplication on L that turns L into an integral domain with a multiplicative identity.

Ramified Quadrangles of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Basic Proposition ◽  
Tits Building ◽  
Quaternion Division Algebra

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a ramified quadrangle if δ‎Λ‎ = δ‎Ψ‎ = 1 holds. The chapter proves the theorem that if δ‎Ψ‎ = 1 and the Moufang residues R₀ and R₁ are not both indifferent, there exists an involutory set. It also discusses the cases ℓ = 6, ℓ = 7, and ℓ = 8, in which D is a quaternion division algebra.

Unramified Galois Involutions

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point ◽  
Complete System ◽  
Canonical Isomorphism ◽  
Descent Group ◽  
Tits Building

This chapter describes the fixed point building of an automorphism of a Bruhat-Tits building Ξ‎ which induces an unramified Galois involution on the building at infinity Ξ‎∞. An element of G (for example, a Galois involution of Δ‎) is unramified if the subgroup of G it generates is unramified. Before presenting the main result, the chapter presents the notation stating that Δ‎ = Ξ‎∞ is the building at infinity of Ξ‎ with respect to its complete system of apartments and G = Aut(Δ‎), followed by definitions. The central theorem shows how an unramified Galois involution of Δ‎ is obtained. Here Γ‎ := τ‎ is a descent group of both Δ‎ and Ξ‎, there is a canonical isomorphism from Δ‎Γ‎ to (Ξ‎Γ‎), where Ξ‎Γ‎ and Ξ‎Γ‎ are the fixed point buildings.

Linked Tori, I

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Space ◽  
Quaternion Division Algebra

This chapter investigates the consequences of the assumption that one Moufang set is weakly isomorphic to another. It first introduces some well-known facts about involutions which are assembled in a few lemmas, including those dealing with an involutory set, a biquaternion division algebra, and a quaternion division algebra with a standard involution. It then presents a notation for a non-trivial anisotropic quadratic space and another for an involutory set are presented, along with assumptions for a pointed anisotropic quadratic space and the standard involution of a quaternion. It also makes a number of propositions regarding the standard involution of a quaternion and a biquaternion. Results about weak isomorphisms between Moufang sets arising from involutory sets are given.

Indices for the Exceptional Bruhat-Tits Buildings

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Capital Letter ◽  
Coxeter Diagram ◽  
Root Group ◽  
Relative Type ◽  
Quaternion Division Algebra

This chapter considers the affine Tits indices for exceptional Bruhat-Tits buildings. It begins with a few small observations and some notations dealing with the relative type of the affine Tits indices, the canonical correspondence between the circles in a Tits index and the vertices of its relative Coxeter diagram, and Moufang sets. It then presents a proposition about an involutory set, a quaternion division algebra, a root group sequence, and standard involution. It also describes Θ‎-orbits in S which are disjoint from A and which correspond to the vertices of the Coxeter diagram of Ξ‎ and hence to the types of the panels of Ξ‎. Finally, it shows how it is possible in many cases to determine properties of the Moufang set and the Tits index for all exceptional Bruhat-Tits buildings of type other than Latin Capital Letter G with Tilde₂.

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