Multiples of Trace Forms and Algebras with Involution

Let k be a field of characteristic ≠ 2, and let G be a finite group. The aim of this article is to give a cohomological criterion for the isomorphism of multiples of trace forms of G-Galois algebras over k. The proof uses results concerning multiples of hermitian forms over division algebras with involution that are of independent interes

Cancellation of hyperbolic ε-hermitian forms and of simple knots

An n-knot will be a smooth, oriented submanifold Kn ⊂ Sn+2 such that Kn is homeomorphic to Sn. Given two knots and , we define their connected sum as in [13], p. 39. The cancellation problem for n-knots is the followin

Automorphisms of K3 surfaces, signatures, and isometries of lattices

Every Salem numbers of degree 4,6,8,12,14 or 16 is the dynamical degree of an automorphism of a non-projective K3 surface. We define a notion of signature of an automorphism, and use it to give a necessary and sufficient condition for Salem numbers of degree 10 and 18 to be realized as the dynamical degree of such an automorphism. The first part of the paper contains results on isometries of lattices.Comment: This paper contains several (but not all) of the results of arXiv:2107.07583, as well as new results. arXiv admin note: text overlap with arXiv:2107.07583. Changes in version 2 : added reference to Takada's result on degree 20 Salem numbers, changes (mainly) in Sections 7 and 15. Changes in version 3 : added two sections on automorphisms of projective K3 surfaces (sections 26 and 27

Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups

Let $R$ be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an $R$-algebra with involution, which are rationally ismorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck--Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat--Tits theory.Comment: 27 pages. Changes from previous version: Section 5 was split into two sections, several proofs have been simplified, other mild modification