Brauer Class Research Articles

Index Reduction for Brauer Classes via Stable Sheaves

We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field k, a k-variety X, and a class α ∈ Br(k), compute the index of the class α k(X) ∈ Br(X) obtained from α by extension of scalars to k(X). We give a general method for computing index reduction which refines classical results of Schofield and van den Bergh. When X is a curve of genus 1, we use Atiyah's theorem on the structure of stable vector bundles with integral slope to show that our formula simplifies dramatically, giving a complete solution to the index reduction problem in this case. Using the twisted Fourier-Mukai transform, we show that a similarly simple formula describes homogeneous index reduction on torsors under higher-dimensional abelian varieties.

Cyclic algebras over p-adic curves

In this paper we study division algebras over the function fields of curves over Q p . The first and main tool is to view these fields as function fields over nonsingular S which are projective of relative dimension 1 over the p adic ring Z p . A previous paper showed such division algebras had index bounded by n 2 assuming the exponent was n and n was prime to p. In this paper we consider algebras of prime degree (and hence exponent) q ≠ p and show these algebras are cyclic. We also find a geometric criterion for a Brauer class to have index q.

On the Brauer Class of Modular Endomorphism Algebras

In this paper we study the Brauer class of the endomorphism algebra Xf of the motive attached to a primitive elliptic modular cusp form f without complex multiplication (CM). Our study includes the case of forms of weight 2, where the associated motive is an abelian variety. It is a fundamental fact that Xf has a natural crossed product structure. This was proved by Ribet [8] and Momose [6] in the case of weight 2, and extended to forms of higher weight in [2] subject to an injectivity constraint, which we remove here. It follows that Xf is a central simple algebra over a subfield F of the Hecke field of f. Moreover Xf is 2-torsion when considered as an element of the Brauer group of F. Thus Xf is isomorphic to a matrix algebra over F, or a matrix algebra over a quaternion division algebra over F. Ribet has remarked in [8] that it seems difficult to distinguish these cases by pure thought. His remark pertains to the case of weight 2, but is equally relevant in higher weight. The chief motivation of this paper (and to a large extent [2]) is to give as complete a picture as possible of the Brauer class of Xf. In recent years the notion of slope has played an important role in the theory of elliptic modular forms. For instance this notion is fundamental in parameterizing families of elliptic modular cusp forms, as in the work of Hida (slope 0) and Gouvea, Mazur,

The extension of the reduced Clifford algebra and its Brauer class

The shape Clifford algebra C f of a binary form f of degree d is the k-algebra k{x,y}/I, where I is the ideal generated by {(αx+βy) d −f(α,β)∥α,βk}. C f has a natural homomorphic image A f , called the reduced Clifford algebra, which is a rank d 2 Azumaya algebra over its center. The center is isomorphic to the coordinate ring of the complement of an explicit Θ -divisor in Pic C/k d +g −1 , where C is the curve (w d −f(u,v)) and g is the genus of C ([9]). We show that the Brauer class of A f can be extended to a class in the Brauer group of Pic C/k d + g −1 . We also show that if d is odd, then the algebra A f is split if and only if the principal homogeneous space Pic C/k 1 of the jacobian of C has a k-rational point.

Hasse principles for the Brauer groups of algebraic function fields of genus zero over global fields

Let F be a global field with char(F) # 2 and K an algebraic function field in one variable of genus zero over F. In this paper, we investigate two kinds of Hasse principles for Brauer classes on K. If Br(K) is the Brauer group of K and Br(K)' is the subgroup of Br(K) whose elements have order relatively prime to char(F), then we precisely determine the kernels of the maps h 1 : Br(K)' → Π/p Br(F p K) and h 2 : Br(K) → Π/p Br(K P ), where p runs over the prime spots of F and P runs over the places of K which are trivial over F, and F p , K P are the completions at p, P respectively. To facilitate the determination of these kernels, we compute the kernel of the map h: Br(K) → Π P Br(KVp) where V P is the residue field with respect to P and show that the kernels of these three maps coincide. We then consider a more general version of the maps above by describing the 2-torsion subgroup of the kernel of h 1 when a finite number of prime spots in the product are omitted.

Relative Brauer Groups of Function Fields of Curves of Genus One

Let Kbe an algebraic function field in one variable over a constant field k. In this paper, we investigate the relative Brauer groups Br(K/k) of Kover kin various cases. When kis a global field, we focus on function fields K = k(C) of genus 1 where Cis the curve of the form y 2 = at 4 + bwith a, b ∈ k − {0}, and we describe the Brauer classes in Br(K/k). More precisely, we show that each algebra in Br(K/k) is a quaternion algebra which can be obtained by taking one of a finite number of the x-coordinates of k-rational points on the Jacobian of the curve C. In particular, for the field ℚ of rational numbers, we determine Br(K/ℚ) precisely in numerous cases and give examples.

Invariant fields of symplectic and orthogonal groups

The projective orthogonal and symplectic groups PO n ( F ) and PSp n ( F ) have a natural action on the F vector space V ′= M n ( F )⊕⋯⊕ M n ( F ). Here we assume F is an infinite field of characteristic not 2. If we assume there is more than one summand in V ′, then the invariant fields F ( V ′) PO n and F ( V ′) PSp n are natural objects. They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case 1, 2, or 4 are the highest powers of 2 that divide n . We derive rationality when n is odd, or when 2 is the highest power, and stable rationality when 4 is the highest power. In a companion paper joint with Tignol, we prove retract rationality when 8 is the highest power of 2 dividing n . Back in this paper, along the way, we consider two generic ways of forcing a Brauer class to be in the image of restriction.

Special forms of two symbols are division algebras

If F is a valued field of characteristic p certain tensor products of cyclic F-algebras are called special forms. It is known that if F is maximally complete then every Brauer class of exponent p is represented by a special form. It is shown here that special forms of two symbols are division algebras, but that special forms of three symbols need not be, and can represent indecomposible division algebras of index $p^2$.

A criterion of decomposability for degree 4 algebras with unitary involution

Let A be a degree 4 central simple algebra endowed with a unitary involution σ. We prove that (A,σ) is decomposable if and only if its discriminant (i.e. the Brauer class of its discriminant algebra) is trivial.

Rationality of moduli of vector bundles on curves

The moduli space M(r,d) of stable, rank r, degree d vector bundles on a smooth projective curve of genus g>1 is shown to be birational to M(h,0) x A, where h=hcf(r,d) and A is affine space of dimension (r^2-h^2)(g-1). The birational isomorphism is compatible with fixing determinants in M(r,d) and M(h,0) and we obtain as a corollary that the moduli space of bundles of rank r and fixed determinant of degree d is rational, when r and d are coprime. A key ingredient in the proof is the use of a naturally defined Brauer class for the function field of M(r,d).

R-Equivalence and Special Unitary Groups

A norm homomorphism for the group ofR-equivalence classes of all simply connected semisimple classical algebraic groups is constructed. The group ofR-equivalence classes for special unitary groupsSU(B,τ) is computed. It is proved that the variety ofSU(B,τ) is rational if ind(B)≤3 and the stable birational type ofSU(B,τ) depends only on the Brauer class ofBand does not depend on the involution τ.

La classe de Brauer de l'algèbre d'endomorphismes d'une variété abélienne modulaire

A formula is given for the Brauer class of the endomorphism algebra Q ⊗ End (A f) of the abelian variety A f/ Q attached by Shimura to a new form ƒ S 2 (N,∈) as a product of quaternion algebras and a term coming from the character ∈.

P-Algebras over an algebraic function field over a perfect field

Let K be a field of characteristic p > 0. Suppose that K is an algebraic function field of r variables over a perfect field. We shall consider the structure of p-algebras over K. When r = 1, Albert proved that every p-algebra is, in fact, a cyclic algebra and the exponent is equal to the index [a]. In this note we shall generalize Albert’s result to the following: Let K be an algebraic function field of r variables over a perfect field, A any p-algebra over K. Then there are cyclic division algebras D, , D2,..., D, so that A is similar to D, OK D, ok ... ok D,, the exponent of each Di is equal to its index, and the exponent of each Di is no greater than that of A. In fact a more general theorem can be established. I thank Professor Shuen Yuan and the referee for their very helpful suggestions about revising my original manuscripts. Throughout this note a cyclic algebra is denoted by (a, L/K, a) where 0 is a generator of Gal(L/K). The Brauer classes of a central simple K-algebra A and a cyclic algebra (a, L/K, a) will be denoted by [A] and [a, L/K, 01, respectively. o(A) is the exponent of A, i(A) its index and deg A = Jm its degree. The following easy lemma is from field theory. We omit its proof.

Brauer groups and class groups for Krull domains: a K-theoretic approach

Brauer groups and class groups for Krull domains: a K-theoretic approach

Simple components in group algebras

Let G be a finite group of exponent n, K a field of characteristic 0, and KG the group algebra of G over K. We write B(K) for the Brauer group of K, and C(K, G) for the collection of all simple summands of KG each of which has center K. If [A], E C(K, G) and {A} belongs to B(K), and we say that (A} is represented by C(K, G). This gives a natural multiplication to C(K, G), which may not be closed. This paper begins a study of C(K, G) with multiplication, to determine its closeness to a group structure. If G is an elementary group, we show that if {A ] is represented by C(K, G), then so is (A)’ for all r. If no restriction is put on G, then we can only show that r must be relatively prime to {A} in B(K). We shall write {A)‘= (A’}, where A r denotes the division algebra in the Brauer class (A )‘, and we say that b[d], E KG if

Brauer Groups, Class Groups and Maximal Orders for a Krull Scheme

In a previous paper [13] one of us considered Brauer groups Br(C) and class groups Cl(C) attached to certain monoidal categories C of divisorial R-lattices. That paper showed that the groups arising for a suitable pair of categories C1 ⊆ C2 could be related by a tidy exact sequenceIt was shown that this exact sequence specializes to a number of exact sequences which had formerly been handled separately. At the same time the conventional setting of noetherian normal domains was replaced by that of Krull domains, thus generalizing previous results while also simplifying the proofs. This work was carried out in an affine setting, and one aim of the present paper is to carry these results over to Krull schemes. This will enable us to recover the non-affine version of an exact sequence obtained by Auslander [1, p. 261], as well as to introduce a new, non-affine version of a different sequence derived by the same author [2, Theorem 1].

Remarks on a principle of localization

We prove that the Brauer class of a crossed product is a sum of symbols iff its “local” components are. Analogously we show that a solution of the “Goldie rank conjecture” would follow from the “local” statements; an extension of a result of Cliff-Sehgal is an easy corollary.

Brauer's class number relation

Brauer's class number relation

On Projective Modules and Automorphisms of Central Separable Algebras

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

Crossed product algebras attached to weight one forms

Abstract. We completely determine the Brauer class of the crossed product algebraattached to a non-dihedral cusp form of weight one. 1. IntroductionLet f be a holomorphic cuspidal newform of weight at least one. In weights twoand higher, there is a Grothendieck motive M f (abelian variety A f in weight two)attached to f. It is a fundamental fact, due to Momose and Ribet [8] in weight twoand [1], [4] in higher weights, that the full algebra of endomorphisms of this motivehas the structure of an explicit crossed product algebra X, when the form f is ofnon-dihedral type. Moreover, much, but not all, is known about the Brauer class ofthis algebra [9], [6], [4].In this paper, we completely determine the Brauer class of X for non-dihedral cuspforms of weight one. The Brauer class in this case is closely related to the secondWitt invariant of the trace form of a number field determined by the projective Artinrepresentation associated to the form.We show that, much as in higher weights, the Brauer class of X is completelydetermined at a prime of good reduction by the parity of a certain slope, when thisslope is finite (Theorem 2). An interpretation of this slope in terms of the adjointrepresentation allows us to compute it, showing that, in contrast to what happens inhigher weights, the Brauer class is essentially unramified at all primes of good reduction(Theorem 4, Corollary 7). At primes of bad reduction we show that the Brauer classis determined purely by the nebentypus of the form (Theorem 5, Corollary 7). Finally,as examples, we determine the Brauer class of X for all non-dihedral weight one formsof prime level (Corollary 9).2. The crossed product algebra XThe crossed product algebra X we wish to study can be attached to non-dihedralforms in all weights. We recall the definition for forms of weight one.Let f =Pa