Essential dimension, spinor groups and quadratic forms

TL;DR

This paper establishes that the essential dimension of spinor groups increases exponentially with their rank, providing explicit formulas and implications for quadratic form representations.

Contribution

It provides a precise exponential growth formula for the essential dimension of Spin_n when n is not divisible by 4, advancing understanding of algebraic group complexity.

Findings

Essential dimension of Spin_n grows exponentially with n

Explicit formula for essential dimension when n mod 4 ≠ 0

Number of 3-fold Pfister forms needed grows exponentially with n

Abstract

We prove that the essential dimension of the spinor group Spin_n grows exponentially with n; in particular, we give a precise formula for this essential dimension when n is not divisible by 4. We use this result to show that the number of 3-fold Pfister forms needed to represent the Witt class of a general quadratic form of rank n with trivial discriminant and Hasse-Witt invariant grows exponentially with n. This paper overlaps with our earlier preprint arXiv:math/0701903 . That preprint has splintered into several parts, which have since acquired a life of their own. In particular, see "Essential dimension of moduli of curves and other algebraic stacks", by the same authors, and "Some consequences of the Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).