The Power Operation in the Galois Cohomology of a Reductive Group Over a Global Field

Let K be a field, Br(K) its Brauer group. If L/K is a field extension, then the relative Brauer group Br(L/K) is the kernel of the restriction map resL/K : Br(K)→ Br(L). Relative Brauer groups have been studied by Fein and Schacher. Every subgroup of Br(K) is a relative Brauer group Br(L/K) for some extension L/K, and the question arises as to which subgroups of Br(K) are algebraic relative Brauer groups, i.e. of the form Br(L/K) with L/K an algebraic extension. For example if L/K is a finite extension of number fields, then Br(L/K) is infinite, so no finite subgroup of Br(K) is an algebraic relative Brauer group. In [1] the question was raised as to whether or not the n-torsion subgroup Brn(K) of the Brauer group Br(K) of a field K is an algebraic relative Brauer group. For example, if K is a (p-adic) local field, then Br(K) ∼= Q/Z, so Brn(K) is an algebraic relative Brauer group for all n. A counterexample was given in [1] for n = 2 and K a formal power series field over a local field. For global fields K, the problem is a purely arithmetic one, because of the fundamental local-global description of the Brauer group of a global field. In particular, for a Galois extension L/K of global fields, if the local degree of L/K at every finite prime is equal to n, and is equal to 2 at the real primes for n even, then Br(L/K) = Brn(K). In [1], it was proved that Brn(Q) is an algebraic relative Brauer group for all squarefree n. In [2], the arithmetic criterion above was verified for any number field K Galois over Q and any n prime to the class number of K, so in particular, Brn(Q) is an algebraic relative Brauer group for all n. In [3], Popescu proved that for a global function field K of characteristic p, the arithmetic criterion holds for n prime to the order of the non-p part of the Picard group of K. In this paper we settle the question completely, by verifying the arithmetic criterion for all n and all global fields K. In particular, the n-torsion subgroup of the Brauer group of K is an algebraic relative Brauer group for all n and all global fields K. The proof, an extension of the ideas in [2], reduces to the case n a prime power `. We first carry out the proof for number fields K. The proof for the function field case when ` 6= char(K) is essentially the same as the proof in the number field case. The proof for ` = char(K) appears in [3].

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets H1(K,G) for all reductive K-groups G in a functorial way.

Статья посвящена памяти Олега Николаевича Введенского (1937 – 1981 гг.). О. Н. Введенский был учеником академика И. Р. Шафаревича. Исследования О. Н. и полученные им результаты связаны с двойственностью в эллиптических кривых и с соответствующими когомологиями Галуа над локальными полями, со спариванием Шафаревича-Тэйта и с другими спариваниями, с локальной и квази-локальной теорией полей классов эллиптических кривых, с теорией абелевых многообразий размерности больше 1, с теорией коммутативных формальных групп над локальными полями. Представлены как результаты, полученные О. Н. Введенским, так и новые избранные результаты, развивающие исследования в направлениях фундаментальных групп схем, главных однородных пространств (торсеров) и двойственности. Первая часть статьи, представлення здесь, является введением как в результаты, полученные О. Н. Введенским в направлении двойственности абелевых многообразий и формальных групп, так и в новые избранные результаты, развивающие исследования в направлениях фундаментальных групп схем, главных однородных пространств (торсеров) и двойственности. Во Введении приведены предварительные сведения и представлено содержание статьи. В первом разделе дан краткий обзор избранных результатов по теории алгебраических, квазиалгебраические и проалгебраические группы и групповых схем. Далее, в разделе 2 преставлены избранные результаты по фундаментальным группам алгебраических многообразий, по фундаментальным группам схем, а в разделе 3 - избранные результаты о главных однородных пространствах (торсерах), развивающие исследования О. Н. и других авторов. Термин торсер мы используем как перевод на русский язык в редакции И.Р. Шафаревича английского термина torsor. В разделе 4 даны сведения о двойственности, а в разделе 5 представлены результаты О. Н. по арифметической теории формальных групп и их развитие. Результаты, этого раздела, представленные над локальными и квази-локальными полями K, над их кольцами целых, и над их полями вычетов k, связанны (1) с формальной структурой абелевых многообразий, (2) с коммутативными формальными группами, (3) с соответствующими гомоморфизмами и изогениями. В статье алгебраические многообразия, абелевы схемы и коммутативные формальные групповые схемы определены, как правило, над локальными и квази-локальными полями, над их кольцами целых, и над их полями вычетов. Но кратко рассматриваются эти объектыи и над глобальными полями, так как О. Н. интересовала тематика алгебраических многообразий над глобальными полями и он проводил соответствующие исследования. Предполагается, что характеристика полей вычетов больше 3, если не оговаривается иное.Я признателен В.Н. Чубарикову за предложение опубликовать статью в сборнике.Особая признательность Н. М. Добровольскому за помощь и поддержку в процессе подготовки статьи к печати.

On the Tits indices of absolutely almost simple algebraic groups over local and global fields

On analogs of Cassels–Tate's exact sequence for connected reductive groups and Brauer-Manin obstruction for homogeneous spaces over global function fields

For arithmetical schemes $X$, K. Kato \[J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)] introduced certain complexes $C^{r,s}(X)$ of Gersten-Bloch-Ogus type whose components involve Galois cohomology groups of all the residue fields of $X$. For specific $(r,s)$, he stated some conjectures on their homology generalizing the fundamental isomorphisms and exact sequences for Brauer groups of local and global fields. We prove some of these conjectures in small degrees and give applications to the class field theory of smooth projective varieties over local fields, and finiteness questions for some motivic cohomology groups over local and global fields.

Witt equivalent fields can be understood to be fields having the same symmetric bilinear form theory. Witt equivalence of finite fields, local fields and global fields is well understood. Witt equivalence of function fields of curves defined over archimedean local fields is also well understood. In the present paper, Witt equivalence of general function fields over global fields is studied. It is proved that for any two such fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K comma upper L"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K,L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , any Witt equivalence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K tilde upper L"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo> ∼ </mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">K \sim L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induces a cannonical bijection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v left-right-arrow w"> <mml:semantics> <mml:mrow> <mml:mi>v</mml:mi> <mml:mo stretchy="false"> ↔ </mml:mo> <mml:mi>w</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">v \leftrightarrow w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> between Abhyankar valuations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v"> <mml:semantics> <mml:mi>v</mml:mi> <mml:annotation encoding="application/x-tex">v</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having residue field not finite of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and Abhyankar valuations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w"> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding="application/x-tex">w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having residue field not finite of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main tool used in the proof is a method for constructing valuations due to Arason, Elman and Jacob [J. Algebra 110 (1987), 449–467]. The method of proof does not extend to non-Abhyankar valuations. The result is applied to study Witt equivalence of function fields over number fields. It is proved, for example, that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are number fields and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis tilde script l left-parenthesis x 1 comma ellipsis comma x Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∼ </mml:mo> <mml:mi> ℓ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">k(x_1,\dots ,x_n) \sim \ell (x_1,\dots ,x_n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>1</mml:mn>

Embeddings of maximal tori in classical groups over fields of characteristic not 2 are the subject matter of several recent papers. The aim of the present paper is to give necessary and sufficient conditions for such an embedding to exist, when the base field is a local field, or the field of real numbers. This completes the results of [3], where a complete criterion is given for the Hasse principle to hold when the base field is a global field.

So far we have been working with the polynomial ring A inside the rational function field k = F(T). In this section we extend our considerations to more general function fields of transcendence degree one over a general constant field. This process is somewhat like passing from elementary number theory to algebraic number theory. The Riemann-Roch theorem is the fundamental result needed to accomplish this generalization. We will give a proof of this fundamental result in Chapter 6. In this chapter we give the basic definitions, state the theorem, and derive a number of important corollaries. After this is accomplished, attention will be shifted to function fields over a finite constant field. Such fields are called global function fields. The other class of global fields are algebraic number fields. All global fields share a great number of common features. We introduce the zeta function of a global function field and explore its properties. The Riemann hypothesis for such zeta functions will be explained in some detail, and we will derive several very important consequences, among others an analogue for the prime number theorem for arbitrary global function fields. A proof of the Riemann hypothesis will be given in the appendix. In this chapter we will prove a weak version. This is enough to yield the analogue of the prime number theorem, albeit with a poor error term. In later chapters we will also explore L-functions associated to global function fields - both Hecke L-functions (generalizations of Dirichlet L-functions) and Artin L-functions.KeywordsZeta FunctionFunction FieldGlobal FunctionAlgebraic FunctionRiemann HypothesisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let $K$ be a field, $G$ a finite group, and $\rho : G \to \mathbf {GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant. II. If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on $G$ or $K$ under which two orthogonal, symplectic or unitary representations of $G$ are equivalent if and only if they are equivalent as linear representations and their underlying forms are “isotypically” equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) $KG$-module components of their spaces are equivalent. We assume throughout that the characteristic of $K$ does not divide $2|G|$. Solutions to I and II are given when $K$ is a finite or local field, or when $K$ is a global field and the representation is “split”. The results for III are strongest when the degrees of the absolutely irreducible representations of $G$ are odd – for example if $G$ has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index – and, in the case that $K$ is a local or global field, when the representations are split.

There is reason to believe that there is a close relation between the irreducible representations, in the sense of harmonic analysis, of the group of rational points on a reductive algebraic group over a local field and the representations of the Weil group of the local field in a certain associated complex group. There should also be a relation, although it will not be so close, between the representations of the global Weil group in the associated complex group and the representations of the adele group that occur in the space of automorphic forms. The nature of these relations will be explained elsewhere. For now all I want to do is explain and prove the relations when the group is abelian. I should point out that this case is not typical. For example, in general there will be representations of the algebraic group not associated to representations of the Weil group. The proofs themselves are merely exercises in class field theory. I am writing them down because it is desirable to confirm immediately the general principle, which is very striking, in a few simple cases. Moreover, it is probably impossible to attack the problem in general without having first solved it for abelian groups. If the proofs seem clumsy and too insistent on simple things remember that the author, to borrow a metaphor, has not cocycled before and has only minimum control of his vehicle. It is well known that there is a one-to-one correspondence between isomorphism classes of algebraic tori defined over a field F and split over the Galois extension K of F and equivalence classes of lattices on which G(K/F ) acts. If T corresponds to L then TK , the group of K-rational points on T , may, and shall, be identified as a G(K/F )-module with Hom(L,K∗). If K is a global field and A(K) is the adele ring of K the group TA(K)/TK may be identified with Hom(L,CK) if CK is the idele class group of K. If K is a local field CK will be the multiplicative group of K. Suppose L is the lattice Hom(L,Z). If C∗ is the multiplicative group of nonzero complex numbers and Cu the group of complex numbers of absolute value 1 we set T = Hom(L,C∗) and Tu = Hom(L,Cu). There are natural actions of G(K/F ) on L, T , and Tu. The semidirect product T oG(K/F ) is a complex Lie group with Tu oG(K/F ) as a real subgroup. If F is a local or global field the Weil group WK/F is an extension

On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I

On absolute Galois splitting fields of central simple algebras

In analogy with the study of representations of GL 2 n ⁡ ( F ) \operatorname {GL}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) , where F F is a local field, we study representations of U 2 n ⁡ ( F ) \operatorname {U}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) in this paper. (Only quasisplit unitary groups are considered in this paper since they are the only ones which contain Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) .) We prove that there are no cuspidal representations of U 2 n ⁡ ( F ) \operatorname {U}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) for F F a nonarchimedean local field. We also prove the corresponding global theorem that there are no cuspidal automorphic representations of U 2 n ⁡ ( A k ) \operatorname {U}_{2n}(\mathbb {A}_k) with nonzero period integral on Sp 2 n ⁡ ( k ) ∖ Sp 2 n ⁡ ( A k ) \operatorname {Sp}_{2n}(k) \backslash \operatorname {Sp}_{2n}(\mathbb {A}_k) for k k any number field or a function field. We completely classify representations of quasisplit unitary groups in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasisplit unitary group distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) .

Louis Solomon introduced the notion of a zeta function ${\zeta _\Theta }(s)$ of an order $\Theta$ in a finite-dimensional central simple K-algebra A, with K a number field or its completion ${K_P}$ (P a non-Archimedean prime in K). In several papers, C. J. Bushnell and I. Reiner have developed the theory of zeta functions and they gave explicit formulae in some special cases. One important property of these zeta functions is the Euler product, which implies that in order to calculate ${\zeta _\Theta }(s)$, it is sufficient to consider the zeta function of local orders ${\Theta _P}$. However, since these local orders ${\Theta _P}$ are in general not principal ideal domains, their zeta function is a finite sum of so-called ’partial zeta functions’. The most complicated term is the ’genus zeta function’, ${Z_{{\Theta _P}}}(s)$, which is related to the free ${\Theta _P}$-ideals. I. Reiner and C. J. Bushnell calculated the genus zeta function for hereditary orders in quaternion algebras (i.e., $[A:K] = 4$). The authors mention the general case but they remark that the calculations are cumbersome. In this paper we derive an explicit method to calculate the genus zeta function ${Z_{{\Theta _P}}}(s)$ of any local hereditary order ${\Theta _P}$ in a central simple algebra over a local field. We obtain ${Z_{{\Theta _P}}}(s)$ as a finite sum of explicit terms which can be calculated with a computer. We make some remarks on the programming of the formula and give a short list of examples. The genus zeta function of the minimal hereditary orders (corresponding to the partition (1, 1, ... , 1) of n) seems to have a surprising property. In all examples, the nominator of this zeta function is a generating function for the q-Eulerian polynomials. We conclude with some remarks on a conjectured identity.