Quadratic forms, quadrics, sums of squares and Kato-cohomology in positive characteristic

Principal Investigator

Prof. Dr. D. Hoffmann (TU Dortmund)

Project description

Conics are curves obtained as the intersection of the surface of a cone with a plane giving rise to ellipses, parabolas and hyperbolas. In projective geometry, they can be described as the zeros of a homogeneous polynomial of degree two in three variables. Such homogeneous quadratic polynomials in several variables are called quadratic forms, and just as for conics, they give rise to geometric objects called quadrics and they can be defined over any field. There are well developed theories for quadratic forms and for quadrics using algebraic and geometric methods, respectively, that interact in subtle ways giving rise to a rich theory combining both algebraic and geometric aspects. One part of the project deals with the classification of quadrics according to various ways of comparing them geometrically, such as isomorphism, (stable) birational equivalence, or motivic equivalence. Motivic equivalence of quadrics is a fairly recent concept that grew out of Voevodsky's Fields Medal winning work on motivic homotopy theory. Depending on the equivalence relation, the classification can be finer or coarser. We plan to compare these different classification methods, focusing in particular on the case where the base field has characteristic 2, i.e. where 2=0. In this case, the known results are far less complete than in characteristic not 2. An important algebraic tool for classifying quadratic forms in characteristic 2 is the theory of certain algebraic objects called Kato's cohomology groups that can be defined for any base field of positive characteristic. The second part of the project deals with properties of these cohomology groups, in particular, how they behave under extension of the base field and when certain elements in such a cohomology group annihilate other elements. A different aspect of quadratic forms concerns the study of representations of elements by quadratic forms over any commutative base ring. A classic example is Lagrange's theorem that states that four is the least positive integer n such that each positive integer can be written as a sum of n squares of integers. Here, the quadratic form is given by the sum of four squares over the ring of integers. One defines the Pythagoras number p of a ring as the least positive integer n such that each sum of squares in that ring can be written as a sum of n squares (or infinity if no such n exists). Lagrange's theorem then states that the Pythagoras number of the ring of integers is 4. The level s of a ring is the least positive integer n such that -1 can be written as a sum of n squares (or infinity if no such n exists). If s is finite, s and p are related in a subtle way: p is at least s and at most s+2. We study which values for s, p and other invariants related to the notion of level can be realized by rings with finite s, and to determine these values explicitly for certain types of rings.

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