Nonsingular Quadratic Forms Research Articles

On a Noether Normalization for a mod-2 Cohomology Ring

An almost extraspecial group with a nonsingular quadratic form is treated. There are three cases: real, complex, and quaternion. The mod-2 cohomology ring has three subrings: the subring generated by Stiefel-Whitney classes of a unique faithful irreducible real representation, an invariant subring of the orthogonal group, and that of its commutator subgroup. The object of this paper is to show that the three subrings are very close to one another. Particularly in the quaternion case, they coincide with each other module the nilradical of the cohomology ring. It is well known that the first is a Noether normalization for the cohomology ring, while it was shown by us that the third is a subring of universally stable elements defined by Evens and Priddy (with a few exceptions of small order). Thus it may be concluded that the last is nothing but a Noether normalization for the cohomology ring.

Classifying nonsingular quadratic forms of rank three

The determinant on 2 by 2 matrices of trace zero gives a quadratic form. Linear algebra shows that its orthogonal group is the matrix algebra automorphisms together with scalars of square 1. As this holds over all commutative rings, descent theory can be applied to it to get a quick classification of all nonsingular quadratic forms of rank 3. Further descent arguments show how this classification yields the Witt invariant and discriminant.

The calculation of coefficients in the formulas for the transformation of inhomogeneous theta series under the action of Hecke operators

Simple explicit formulas are derived for the coefficients of linear combinations of nonhomogeneous theta series of integral nonsingular quadratic forms, which are images of these theta series under the action of Hecke operators.

The relation of quadratic k-theory to hermitian k:-Theory

It is known that the Hermitian algebraic K-theory of a ring with antistructure is the fixed point theory of a certain involution on the algebraic K-theory of the ground ring. It is shown here that the (unitary) algebraic K-theory of nonsingular split quadratic forms over a ring with antistructure (R,α,u) is the same as the algebraic K-theory of nonsingular Hermitian forms over a related ring with antistructure (S⟪v). Specifically, S = R = R[d]/(d2) is the ring of dual numbers of R, the conjugation β is the antiautomorphism of S given by β{r + r’d) = α(r) - α(r’)d, and the unit of symmetry v= u(l + d). This makes it possible to regard split quadratic algebraic K-theory of R as a fixed point theory of an involution on algebraic K-theory of R . On the level of Ko (Grothendieck) groups, quadratic and split quadratic algebraic K-theory are the same. This is applied to show that the Kervaire-Arf invariant of skew-quadratic forms over Z (here (α,u) = (id,-l)) becomes identified with a generalized Hasse-Witt...

The arithmetic of certain quartic curves

SynopsisLet F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.

Cancellation of quadratic forms over principal ideal domains

Let R be a commutative ring of dimension one in which 2 is invertible. It was proved in [8, Theorem 7.21 that if 4 is a quadratic space (i.e. a non-singular quadratic form on a finitely generated projective module) over R of Witt-index at least two, cancellation holds for q. The aim of this note is to prove a refinement of this result if R is a principal ideal domain. More precisely, we prove that if R is a principal ideal domain and 4 an isotropic quadratic space over R, then cancellation holds for 4.’ (We note that a quadratic form q over a Dedekind domain is isotropic if and only if its Witt-index is at least one.) We give examples to show that, in general, cancellation fails for quadratic spaces of Witt-index one over arbitrary Dedekind domains and for anisotropic spaces over principal ideal domains, thereby showing that our result is in some sense the best possible. I am thankful to M.A. Knus for the interesting discussions I had with him during the preparation of the paper.

On the group of automorphisms of a Hopf map

Let K be an infinite field of characteristic not 2. Let qx, qY be nonsingular quadratic forms on vector spaces X, Y over K, respectively. Assume that there is a bilinear map B:X × Y → Y such that qY(B(x, y)) = qx(x)qY(y). To each such triple {qx, qY, B} one associates the Hopf map h: Z = X × Y → W = K × Y by h(z) = (qx(x) – qY(y), 2B(x, y)), z = (x, y).