Local global theorems for diagonal forms of higher degree.
Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
Let be a 2-dimensional normal excellent henselian local domain in which is invertible and let and be its fraction field and residue field respectively. Let be the set of rank 1 discrete valuations of corresponding to codimension 1 points of regular proper models of . We prove that a quadratic form over satisfies the local-global principle with respect to in the following two cases: (1) has rank 3 or 4; (2) has rank and , where is a complete discrete valuation ring with...
Local-global principle for Witt equivalence of function fields over global fields
We examine the conditions for two algebraic function fields over global fields to be Witt equivalent. We develop a criterion solving the problem which is analogous to the local-global principle for Witt equivalence of global fields obtained by R. Perlis, K. Szymiczek, P. E. Conner and R. Litherland [12]. Subsequently, we derive some immediate consequences of this result. In particular we show that Witt equivalence of algebraic function fields (that have rational places) over global fields implies...
Maximal Subgroups of the Finite Orthogonal and Unitary Groups Stabilizing Anisotropic Subspaces.
Milnor’s conjecture on quadratic forms and motivic complexes
Minimal forms with respect to function fields of conics.
Motivic cohomology and unramified cohomology of quadrics
This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension and . Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics....
Motivic equivalence of quadratic forms.
O jedenáctém a dvanáctém Hilbertově problému
On 14-dimensional quadratic forms in , 8-dimensional forms in , and the common value property.
On a conjecture of Izhboldin on similarity of quadratic forms.
On elementary equivalence, isomorphism and isogeny
Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...
On indefinite binary quadratic forms and quadratic ideals.
On near-neighbor quadratic forms. (Autour des formes quadratiques quasi-voisines.)
On octahedral extensions of and quadratic -curves
We give a necessary condition for a surjective representation Gal to arise from the -torsion of a -curve. We pay a special attention to the case of quadratic -curves.
On orthogonal decomposition of homogeneous polynomials
On quadratic forms of height 3 and degree . (Sur les formes quadratiques de hauteur 3 et de degré au plus 2.)
On Rank 4 Quadratic Spaces with Given Arf and Witt Invariants.
On Real Local-Global Principles.
On the annihilating ideal for trace forms
We give several examples of classes of trace forms for which the ideal of annihilating polynomials is principal. We prove, that in general, the annihilating ideal is not a principal ideal.