$K$-theoretic torsion and the zeta function

TL;DR

This paper extends Milnor's classical identity linking Reidemeister torsion and the Lefschetz zeta function to higher algebraic K-theory, introducing higher torsion invariants and zeta functions for families of endomorphisms.

Contribution

It generalizes a fundamental identity to higher algebraic K-theory, involving new higher torsion invariants and zeta functions for parametrized families of endomorphisms.

Findings

Derived a higher K-theoretic identity relating torsion and zeta functions

Introduced the concept of endomorphism torsion in higher algebraic K-theory

Provided examples of non-trivial endomorphism torsion

Abstract

We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.