Cyclic homology, Serre's local factors and the $\lambda$-operations

TL;DR

This paper demonstrates that cyclic homology with adele coefficients effectively captures Serre's archimedean local factors of complex L-functions of smooth projective varieties over number fields through lambda-operations and regularized determinants.

Contribution

It introduces a novel approach using cyclic homology with adele coefficients to interpret archimedean local factors via lambda-operations and determinants.

Findings

Cyclic homology with adele coefficients models local factors accurately.

Lambda-operations relate to the structure of L-functions.

Regularized determinants encode archimedean factors.

Abstract

We show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring of infinite adeles of K, provides the right theory to obtain, using the lambda-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.