Divided powers in the Witt ring of symmetric bilinear forms

TL;DR

This paper derives explicit formulas for divided power operations in the Witt ring of symmetric bilinear forms, expressing them as linear combinations of exterior powers with coefficients involving tangent numbers, linking to Milnor K-theory modulo 2.

Contribution

It provides the first explicit formula for divided powers in the Witt ring, connecting them to exterior powers and tangent numbers, and relates these to Milnor K-theory modulo 2.

Findings

Explicit formula for divided powers as linear combinations of exterior powers

Coefficients involve tangent numbers related to Bernoulli numbers

Constructs divided powers on Milnor K-theory modulo 2

Abstract

The Witt ring of symmetric bilinear forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are essentially linear combinations of exterior powers. We find the explicit formula for the divided powers as a linear combination of exterior powers. The coefficients involve the ``tangent numbers'', related to Bernoulli numbers. The divided powers on the Witt ring give another construction of the divided powers on Milnor K-theory modulo 2.