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Quadratic and Hermitian forms
... We present the notation and main tools used in this paper and refer to the standard references [12], [13], [14] and [24] as well as to [2], [3] and [4] for the details. ...
... Then there is a finite field extension L of F such that L ⊆ F P and A ⊗ F L M n P (L). Since an ordering Q ∈ X F extends to L if and only if sign Q ((Tr L/F ) * 1 ) > 0 (see [24,Chapter 3,Theorem 4.4]), the set U := {Q ∈ X F | Q extends to L} is clopen in X F and contains P. Then for Q ∈ U we have L ⊆ F Q and thus ...
... In particular, and using that α 1 , . . . , α 2 k belong to every R ∈ X N /P, we obtain, using the quadratic Knebusch trace formula [24,Chapter 3,Theorem 4.5], that sign P Tr * ( α 1 , . . . , α 2 k ) = ...
We introduce positive cones on algebras with involution. These allow us to prove analogues of Artin's solution to Hilbert's 17th problem, the Artin-Schreier theorem characterizing formally real fields, and to define signatures with respect to positive cones. We consider the space of positive cones of an algebra with involution and investigate its topological properties, showing in particular that it is a spectral space. As an application we solve the problem of the existence of positive involutions.
... For B ∈ H nd n (o), we put D B = (−4) [n/2] det B. D B (or its image in F × /F ×2 ) is often called the signed determinant ( [8]) or the discriminant ( [10]) of 2B. Here, F ×2 = {x 2 | x ∈ F × }. ...
... Definition 0.4. The Clifford invariant (see Scharlau [10], p. 333) of B ∈ H nd n (o) is the Hasse invariant of the Clifford algebra (resp. the even Clifford algebra) of B if n is even (resp. ...
... (See Scharlau [10] pp. 80-81.) ...
Let B be a half-integral symmetric matrix of size n defined over . The Gross-Keating invariant of B was defined by Gross and Keating, and has important applications to arithmetic geometry. But the nature of the Gross-Keating invariant was not understood very well for . In this paper, we establish basic properties of the Gross-Keating invariant of a half-integral symmetric matrix of general size over an arbitrary non-archimedean local field of characteristic zero.
... Thus, the generalized quadratic spaces over Q R are the (−1, R)-unitary spaces over Q R defined by Knus in [9, I(5.2)] or the regular (−1, R)-quadratic spaces over Q R defined by Scharlau in [14,Ch. 7,§3]. ...
... For this, pick an element ε ♯ ∈ I such that a u (ε ♯ , ε) = 1. The elements ε, ε ♯ then form an F -base of I, and (14) ξ = ε a u (ε ♯ , ξ) − ε ♯ a u (ε, ξ) for all ξ ∈ I. ...
We provide a new proof of the analogue of the Artin-Springer theorem for groups of type that can be represented by similitudes over an algebra of Schur index 2: an anisotropic generalized quadratic form over a quaternion algebra Q remains anisotropic after generic splitting of Q, hence also under odd degree field extensions of the base field. Our proof is characteristic free and does not use the excellence property.
... (p. 259) of [6], there is an isomorphism i : A → M d (K) and a diagonal matrix α ∈ M d (K) such that the isomorphism i takes the involution x → x † to the involution η of M d (K) defined by η(x) = αx * α −1 . Now suppose we are given a totally positive b in K. Let β ∈ M d (K) be the diagonal matrix with b in the upper left corner and 1's elsewhere. ...
... (p. 259) of [6], there is an isomorphism i : A → M d (D) and a diagonal matrix α ∈ M d (D) with α * = −α such that the isomorphism i takes the involution x → x † to the involution η of M d (D) defined by η(x) = αx * α −1 ; furthermore, as is argued on pp. 194-195 of [3], the entries of the diagonal matrix α 2 are totally negative elements of K. Let α 1 , . . . ...
We provide a simple method of constructing isogeny classes of abelian varieties over certain fields k such that no variety in the isogeny class has a principal polarization. In particular, given a field k, a Galois extension of k of odd prime degree p, and an elliptic curve E over k that has no complex multiplication over k and that has no k-defined p-isogenies to another elliptic curve, we construct a simple -dimensional abelian variety X over k such that every polarization of every abelian variety isogenous to X has degree divisible by . We note that for every odd prime p and every number field k, there exist and E as above. We also provide a general framework for determining which finite group schemes occur as kernels of polarizations of abelian varieties in a given isogeny class. Our construction was inspired by a similar construction of Silverberg and Zarhin; their construction requires that the base field k have positive characteristic and that there be a Galois extension of k with a certain non-abelian Galois group. Note: Theorem 3.2 in this paper is incorrect. The current version of the paper includes comments explaining the mistake.
... , x n ) with real coefficients that has nonnegative values, Artin proved that f can be written as a sum of squares of rational functions over R. By a famous theorem of Pfister [13], 2 n squares are always sufficient to represent f . For n ≤ 2 this bound is known to be best possible, whereas for n ≥ 3 it is only known that the best general bound lies between n + 2 and 2 n (see [14], for instance). ...
... except for (n, d, s) = (3, 2, 5), (4,2,9) and (5,2,14), where max has to be replaced by min. ...
For let p(n,2d) denote the smallest number p such that every sum of squares of forms of degree d in is a sum of p squares. We establish lower bounds for these numbers that are considerably stronger than the bounds known so far. Combined with known upper bounds they give in the ternary case. Assuming a conjecture of Iarrobino-Kanev on dimensions of tangent spaces to catalecticant varieties, we show that for and all . For ternary sextics and quaternary quartics we determine the exact value of the invariant, showing p(3,6)=4 and p(4,4)=5.
... Witt's theorem [51,Thm. 5.3] asserts that if E, F ∈ Gr k (V ) and T : E → F is a linear map such that T * Q| F = Q| E then one can findT ∈ O(Q) withT | E = T . ...
... Denote by X k a,b ⊂ Gr k (V ) the subset consisting of those subspaces E for which the restriction of Q to E has signature (a, b). It follows by Witt's theorem [51,Thm. 5.3.] that whenever X k a,b is non-empty, it is a G-orbit in Gr k (V ). ...
Let denote the identity connected component of the real orthogonal group with signature (p,q). We give a complete description of the spaces of continuous and generalized translation- and -invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invariant valuations. As a result of independent interest, we identify within the space of translation-invariant valuations the class of Klain-Schneider continuous valuations, which strictly contains all continuous translation-invariant valuations. The operations of pull-back and push-forward by a linear map extend naturally to this class.
... Let K be a field of characteristic not equal to 2. Let W (K) denote the Witt group of quadratic forms over K and I(K) the fundamental ideal of W (K) consisting of classes of even dimensional forms (cf. [25,Ch. 2]). ...
... For a 1 , · · · , a n ∈ F * , let << a 1 , · · · , a n >> denote the n-fold Pfister form < 1, −a 1 > ⊗ · · · ⊗ < 1, −a n > (cf. [25,Ch. 4]. ...
Let F be the function field of a curve over totally imaginary number field. Let p be a prime. If F contains a primitive p th root of unity, then every element in the third Galois cohomology group of F with values in the group of p th roots of unity, is a symbol.
... this happens exactly when its trace form q h (which is a quadratic form over Q in 4(n + 1) variables) is isotropic. Then G is isotropic by [32,Cor. 5.7.3 (iii)]. ...
... Recall that by definition the hermitian module (L R , h) is regular (or nonsingular) if the map φ h : x → h(x, ·) induces an isomorphism of O D,R -modules from L R onto its dual module (L R ) * , seen as a right module via f α = αf (see [32,Sect. 7.1]). When R is a field this is equivalent to (L R , h) being nondegenerate, i.e., (L R ) ⊥ = 0. ...
For any n>1 we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group Sp(n,1). We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all n>1.
... . A more detailed exposition of this material may be found e.g. in [15]. ...
... If the real Schur index of χ is 2, then W is uniform and [18,Theorem B] also gives the Clifford invariant of (W, F ): [15,Theorem 10 ...
The rational invariants of the SL_2(q)-invariant quadratic forms on the real irreducible representations are determined. There is still one open question (see Remark 6.5) if q is an even square.
... Now suppose that k is a totally real number field, and δ v ą 0 for all real places v. The Hasse principle (see [54] Theorem 6.6.6) ensures that q is uniquely determined, up to equivalence, by all its localizations q v " q b k k v , where v ranges over all places of k. ...
... (see [54] Theorem 6.6.10). We give more details in §2.2 and §2.3, depending on the parity of d. ...
Let G be the group of rational points of a quasi-split p-adic special orthogonal, symplectic or unitary group for some odd prime number p. FollowingArthur and Mok, there are a positive integer N, a p-adic field E and a local functorial transfer from isomorphism classes of irreducible smooth complex representations of G to those of GL(N,E). By fixing a prime number l different from p and an isomorphism between the field of complex numbers and an algebraic closure of the field of l-adic numbers, we obtain a transfer map between representations with l-adic coefficients. Now consider a cuspidal irreducible l-adic representation pi of G: we can define its reduction mod l, which is a semi-simple smooth representation of G of finite length, with coefficients in a field of characteristic l. Let pi' be a cuspidal irreducible l-adic representation of G whose reduction mod l is isomorphic to that of pi. We prove that the transfers of pi and pi' have reductions mod l which may not be isomorphic, but which have isomorphic supercuspidal supports. When G is not the split special orthogonal group SO(2), we further prove that the reductions mod l of the transfers of pi and pi' share a unique common generic component.
... Notation for bilinear forms in GW( ) will follow standard rules, as in [34] or [60]. In particular, the rank 1 symmetric bilinear form associated with a unit ∈ is denoted ⟨ ⟩. ...
We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof,, we obtain a motivic version of mod kk Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety (NGLrT)∖GLr which turns out to be not stably A1‐connected. We also show that the higher motivic stable stems are of bounded torsion.
... Note that the number of such extensions equals Sign P (T r * 1 ), cf.[31,§3, Theorem 4.5]). ...
We investigate the real cycle class map for singular varieties. We introduce an analog of Borel–Moore homology for algebraic varieties over the real numbers, which is defined via the hypercohomology of the Gersten–Witt complex associated with schemes possessing a dualizing complex. We show that the hypercohomology of this complex is isomorphic to the classical Borel–Moore homology for quasi-projective varieties over the real numbers.
... We will limit ourselves to the case of Milnor's exact sequence. In a forthcoming paper, we will extend this sequence to an analogue of the Milnor-Scharlau exact sequence that uses transfer maps [19,Theorem 3.5,page 215]. In order to state our main result, we need to recall some definitions and results related to quadratic forms and Kato-Milne cohomology. ...
Let F be a field of characteristic 2. In this paper we determine the Kato-Milne cohomology of the rational function field F(x) in one variable x. This will be done by proving an analogue of the Milnor exact sequence [4] in the setting of Kato-Milne cohomology. As an application, we answer the open case of the norm theorem for Kato-Milne cohomology that concerns separable irreducible polynomials in many variables. This completes a result of Mukhija [17, Theorem A.3] that gives the norm theorem for inseparable polynomials.
... 4a. Hermitian forms on R-modules. A convenient reference is Scharlau [53] (see also [60,Section 6.2]). Let R be a ring with involution and let L be an R-module. ...
Kreck and Schafer produced the first examples of stably diffeomorphic closed smooth 4-manifolds which are not homotopy equivalent. They were constructed by applying the doubling construction to 2-complexes over certain finite abelian groups of odd order. By extending their methods, we formulate a new homotopy invariant on the class of 4-manifolds arising as doubles of 2-complexes with finite fundamental group. As an application we show that, for any , there exist a family of k closed smooth 4-manifolds which are all stably diffeomorphic but are pairwise not homotopy equivalent.
... Remark 2.8. Using the Cartan-Dieudonné theorem on the generation of the orthogonal group of a quadratic space over a field by reflections (see [Sc85,Theorem I.5.4]), one can show by similar means that the natural projection SO(V, q) → SO(V , q) on special orthogonal groups is surjective. This observation is due to M. Knebusch (see [Kn69]). ...
Let R be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over R remain anisotropic after base change to any odd-degree finite \'{e}tale extension of R. This generalization of the classical Artin-Springer theorem (concerning the situation where R is a field) was previously established in the case where all residue fields of R are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic to the case where the ring has at least one residue field which is finite.
... . By Witt's extension theorem [42], the representation ...
We study the O(p,q)-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an O(p,q)-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the homogeneous case of O(p,p), yielding a Crofton formula for the centro-affine surface area when . We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all O(p,2)-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the family of distributions on the space of symmetric matrices, which we use to construct a family of O(p,q)-invariant Crofton distributions. We conjecture there are no others, which we then prove for O(p,2) with p even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.
... (v) Witt-GrothendieckŴ (F ) and Witt rings W (F ) of quadratic forms over fields F (see [6] or [11] for further details on quadratic forms and Witt rings). Let F be a field of characteristic not 2. Consider the group ring Z[F * /F * 2 ] and the canonical ring homomorphism ...
In ``New Proofs of the structure theorems for Witt Rings'', Lewis shows how the standard ring-theoretic results on the Witt ring can be deduced in a quick and elementary way from the fact that the Witt ring of a field is integral and from the specific nature of the explicit annihilating polynomials he provides. We will show in the present article that the same structure results hold for larger classes of commutative rings and not only for Witt rings. We will construct annihilating polynomials for these rings.
... [18,Chap. I] or [32,Chap. 7]). We identify hermitian forms with their Witt class in W (A, σ), unless indicated otherwise. ...
We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.
... Proof. See Scharlau [14] Ch. 9, Remark 2.12, p333. ...
It is well-known that the Fourier coefficients of Siegel-Eisenstein series can be expressed in terms of the Siegel series. The functional equation of the Siegel series of a quadratic form over was first proved by Katsurada. In this paper, we prove the functional equation of the Siegel series over a non-archimedean local field by using the representation theoretic argument by Kudla and Sweet.
... . It is known that dim C(V, q) = 2 n , see for example [28,§9.2,Corollary 2.7]. ...
We begin the study of PBW deformations of graded algebras relevant to the theory of Hopf algebras. One of our examples is the Fomin-Kirillov algebra FK3. Another one appeared in a paper of Garc\'ia Iglesias and Vay. As a consequence of our methods, we determine when the deformations are semisimple and we are able to produce PBW bases and polynomial identities for these deformations.
... It is well-known that two definite hermitian spaces V and W are isometric if and only if dim(V ) = dim(W ) and dV = dW (cf. [Sch85] Ch. 10, ex. 1.6). ...
In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occuring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is "no" by giving a counter example, while the answer to the second question is "yes" by proving a mass formula for partial masses. Finally, we determine a list of all single-class partial genera and show that a partial genus consists of only one class if and only if the whole genus consists of only one class.
... We assume familiarity with the theory of quaternion algebras, for which we refer to [5,6,10]. ...
We study two transitivity properties for group actions on buildings, called Weyl transitivity and strong transitivity. Following hints by Tits, we give examples involving anisotropic algebraic groups to show that strong transitivity is strictly stronger than Weyl transitivity. A surprising feature of the examples is that strong transitivity holds more often than expected.
... The Clifford algebra of ϕ splits if and only if ϕ is a multiple of a 3-fold Pfister form, by [16,Ch. 2,Th. ...
We study central simple algebras with involution of the first kind that become hyperbolic over the function field of the conic associated to a given quaternion algebra Q. We classify these algebras in degree~4 and give an example of such a division algebra with orthogonal involution of degree~8 that does not contain Q with its canonical involution, even though it contains Q and is totally decomposable into a tensor product of quaternion algebras with involution.
... for some integer m, and the H occurring in the description of h d is the usual unitary hyperbolic plane as described in [Sch85,7.7.3]. ...
For G an almost simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map R_G: H^1(F, G) --> H^3(F, Q/Z(2)). This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for Albert algebras as special cases. We show that R_G has trivial kernel if G is quasi-split of type E_6 or E_7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.
... The algebraic knot concordance group C has a description within a Witt group of quadratic forms (see for example [42], mainly §5). Levine's approach is to consider the Witt group of isometric structures of the symmetrized Seifert form over Q. (Working over Z is much harder, and the arising invariants were given later by Stoltzfus [49].) ...
It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the Rudolph-Bennequin inequality can be used indirectly to prove that some knots are not slice.
... In view of the previous theorem, to prove it, we just need to show that any simply connected absolutely almost simple k-group G of outer type A n for n 2 is k-isotropic. Since there does not exist a noncommutative finite dimensional division algebra with center a quadratic Galois extension of k which admits an involution of the second kind with fixed field k (see [Sch,Ch. 10,Thm. ...
The purpose of the paper is to present an alternative approach to unramified descent in Bruhat-Tits theory of reductive groups over nonarchimedean local fields. This approach appears to be conceptually simpler than the approach in the papers of Bruhat and Tits.
... Both norm principles were proved for nondegenerate quadratic forms over fields of characteristic not 2 (cf. [Sch,II.8.6] or [L, p. 205, p. 206]). For finite extensions of semi-local regular rings containing a field of characteristic 0, Knebusch's norm principle (for quadratic forms) was proved in [Z] and for finiteétale extensions of semi-local Noetherian domains with infinite residue fields of characteristic different from 2 in [O-P-Z]. ...
Let F be a field of characteristic 0 or greater than d. Scharlau's norm principle holds for separable field extensions K over F, for certain forms of degree d over F which permit composition.
... Here s p (V ) is the Hasse-Witt invariant at p (as normalized in [28]) and γ p depends only on det(L) · (Q × p ) 2 , in particular it is one, if det(L) is a square. There are obvious generalizations of this, if L is of more general level. ...
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.
We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current paper we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.
Our aim in this paper is to prove in the setting of Kato-Milne cohomology in characteristic 2 an exact sequence which is analogue to the Milnor-Scharlau sequence [8, Theorem 6.2]. This is an extension of the Milnor exact sequence proved in [6].
Let q be an odd prime power, and the finite symplectic group. We give an expression for the total Stiefel-Whitney Classes (SWCs) for orthogonal representations of G, in terms of character values of at elements of order 2. We give "universal formulas'' for the fourth and eighth SWCs. For n=2, we compute the subring of the mod 2 cohomology generated by the SWCs .
Let K be a complete discretely valued field with residue field k with . Assuming that the norm principle holds for spinor groups for every regular skew-hermitian form over every quaternion algebra (with respect to the canonical involution on ) defined over any finite extension of k, we show that the norm principle holds for spinor groups Spin(h) for every regular skew-hermitian form h over every quaternion algebra D (with respect to the canonical involution on D) defined over K.
Let O(p,q) be the orthogonal groups of signature (p,q) over the reals. It is shown that an element of the commutator subgroup of O(p,q) is bireflectional (product of 2 involutions in ) if and only if it is reversible (conjugate to its inverse). Moreover, the bireflectional elements of are classified.
We define degree two cohomological invariants for G-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self--dual normal basis. In some cases, we show that these conditions are also sufficient.
We offer some elementary characterisations of group and round quadratic forms. These characterisations are applied to establish new (and recover existing) characterisations of Pfister forms. We establish "going-up" results for group and anisotropic round forms with respect to iterated Laurent series fields, which contrast with the established results with respect to rational function field extensions. For forms of two-power dimension, we determine when there exists a field extension over which the form becomes an anisotropic group form that is not round.
Let k be a field of characteristic not 2, let q be a quadratic space over k and let f be an irreducible polynomial with coefficients in k. In 1969, Milnor raised the following question : how can we decide whether q has an isometry with minimal polynomial f ? We give an answer to this question in the case of global fields. A more general version of the question is also considered.
This paper gives an introduction to Kuga-Satake varieties and discusses some aspects of the Hodge conjecture related to them. Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We give a detailed account of the construction of Kuga-Satake varieties and of their decomposition in simple subvarieties. We recall the Hodge conjecture and we point out a connection between the Hodge conjecture for abelian fourfolds and Kuga-Satake varieties. We discuss the implications of the Hodge conjecture on the geometry of surfaces whose second cohomology group has a Kuga-Satake variety. We conclude with some recent results on Kuga-Satake varieties of Hodge structures on which an imaginary quadratic field acts.
The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in characteristic 2 and investigate the relationships between them. We also investigate these invariants in the case of a quaternion algebra and in particular when this quaternion algebra is the unique quaternion division algebra over a field.
We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map eta over fields of cohomological dimension at most 2 with characteristic different from 2 (this includes the p-adic fields and the finite fields of odd characteristic) and the field of rational numbers; the ring structure is also determined.
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.
We generalize the Shimura-Waldspurger correspondence, which describes the generic part of the automorphic discrete spectrum of the metaplectic group , to the metaplectic group of higher rank. To establish this, we transport Arthur's endoscopic classification of representations of the odd special orthogonal group with by using a result of J. S. Li on global theta lifts in the stable range.
If M is a manifold with an action of a group G, then the homology group H_1(M,Q) is naturally a Q[G]-module, where Q[G] denotes the rational group ring. We prove that for every finite group G, and for every Q[G]-module V, there exists a closed hyperbolic 3-manifold M with a free G-action such that the Q[G]-module H_1(M,Q) is isomorphic to V. We give an application to spectral geometry: for every finite set P of prime numbers, there exist hyperbolic 3-manifolds N and N' that are strongly isospectral such that for all p in P, the p-power torsion subgroups of H_1(N,Z) and of H_1(N',Z) have different orders. We also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a G-action "knows" nothing about the fixed point structure under G, in contrast to the 2-dimensional case. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger-M\"uller formula, but we also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.
Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. Both old and new results on these invariants are collected.
We study the birational properties of geometrically rational surfaces from a derived categorical point of view. In particular, we give a criterion for the rationality of a del Pezzo surface over an arbitrary field, namely, that its derived category decomposes into zero-dimensional components. For del Pezzo surfaces of degree at least 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of the surface from Brauer classes and Chern classes associated to these vector bundles.
We classify the bireflections (products of 2 involutions) in the commutator subgroup G an orthogonal group O(V) over a finite field GF(q) of characteristic not 2. We show that every element of G is a bireflection if it is reversible (conjugate to its inverse in G), except when and V is hyperbolic. We also classify the reversible elements of G.
