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The homology groups h1(l/k), h2(l/k), and h3(l/k) of the Brauer complex for a triquadratic field extension l=k(a,b,c) are studied. In particular, given D∈APTARAMOPREFIX2 Br (k(a,b,c)/k), we find equivalent conditions for the image of D in h2(l/k) to be zero. We consider as well the second divided power operation γ2:APTARAMOPREFIX2 Br (l/k)→H4(k,Z/2Z), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h2⊗h1→H3, which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F/k such that the cohomology group h1(F(a,b,c)/F) of the Brauer complex contains the images of prescribed elements of k∗, provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L/F with nontrivial h3(L/F) are given. As a consequence we show that the homology group h3(L/F) can be arbitrarily big.

We show that Rost nilpotence holds for a geometrically integral threefold X over a field k of characteristic 0 if and only if αk(X)*∘N( CH 0(Xk(X)))=0 for some integer N>0 for all correspondences α of degree 0 which vanish over some field extension of k. As a corollary we get the Rost nilpotence property for three-dimensional smooth projective geometrically integral schemes over a field of characteristic zero, which are birationally isomorphic to a toric model.

Let F be a number field, and let F\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively correspond to the representations of the product of these polynomials by a norm from K to F. Combining the circle method with descent we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X., 25 pages

In this paper we give a new formula for adding$2$-coverings and$3$-coverings of elliptic curves that avoids the need for any field extensions. We show that the$6$-coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.

We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields. The classification of projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero will also be given., Comment: 11 pages, 1 figure

In this article we compute the ring of algebraic cobordisms of a Pfister quadric. This is a rare example of a non-cellular variety where such a computation is known. We consider the algebraic cobordisms Ω* of Levine and Morel, as well as the MGL 2*, * of Voevodsky. The methods of computation in these two cases are quite different. However, the results do agree (which supports the expectation that the two theories actually coincide). We show that the restriction homomorphism in our case is injective for any field extension E/F.

Let $F$ be a $p$-adic field and $K/F$ a finite, cyclic, tamely ramified field extension. We show, without any restriction, that base change from $F$ to K, in the sense of Arthur and Clozel, is compatible with the operation of $K/F$-lifting for simple characters defined by the authors in an earlier paper. This extends and completes the previous work, and represents a substantial step towards the goal of describing tamely ramified base change in terms of the structure theory of representations of Bushnell and Kutzko. We also give a consequence for the Langlands correspondence. The main step in the proof is to show that lifting of simple characters is compatible with automorphic induction, in the sense of Henniart and Herb, through an unramified extension. This is achieved via a new and independently interesting result concerning the conductor of a pair of supercuspidal representations. It gives a complete description of the minima of (a suitably normalized version of) the conductor when one of the representations is fixed. All results concerning the conductor are also valid for non-Archimedean local fields of positive characteristic.

This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.

Let φ be an anisotropic quadratic form over a field F of characteristic not 2. The splitting pattern of φ is defined to be the increasing sequence of nonnegative integers obtained by considering the Witt indices iW(φk) of φ over K where K ranges over all field extensions of F. Restating earlier results by HURRELBRINK and REHMANN, we show how the index of the Clifford algebra of φ influences the splitting pattern. In the case where F is formally real, we investigate how the signatures of φ influence the splitting behaviour. This enables us to construct certain splitting patterns which have been known to exist, but now over much “simpler” fields like formally real global fields or ℝ(t). We also give a full classification of splitting patterns of forms of dimension less than or equal to 9 in terms of properties of the determinant and Clifford invariant. Partial results for splitting patterns in dimensions 10 and 11 are also provided. Finally, we consider two anisotropic forms φ and φ of the same dimension m with φ ⟂ − φ ∈ InF and give some bounds on m depending on n which assure that they have the same splitting pattern.

Let K 0 (x) be a simple transcendental extension of a field K 0 , υ 0 be a valuation of K 0 with value group G 0 and residue field K 0 . Suppose G 0 ⊆G 1 ⊆G is an inclusion of totally ordered abelian groups with [G 1 : G 0 ] 1 and an infinite cyclic group. It is proved that there exists an (explicitly constructible) valuation υ of K 0 (x) extending υ 0 such that the value group of υ is G and its residue field is k, where k is a given finite extension of k 0 . This is analogous to a result of Matignon and Ohm [2, Corollary 3.2] for residually non-algebraic prolongations of υ 0 to K 0 (x).