The $\ell$-adic trace formula for dg-categories and Bloch's conductor conjecture

TL;DR

This paper develops an $$-adic trace formula for smooth, proper dg-categories over certain algebraic bases, and proposes a new approach to prove Bloch's conductor conjecture, advancing the understanding of algebraic and categorical invariants.

Contribution

It introduces an $$-adic trace formula for dg-categories over $$-algebras and offers a novel strategy to prove Bloch's conductor conjecture.

Findings

Established an $$-adic trace formula for dg-categories

Proposed a new approach to Bloch's conductor conjecture

Extended the trace formula to $$-algebras with $$- and $$-algebra structures

Abstract

Building on recent results by A. Blanc, M. Robalo and the authors, we present an $\ell$-adic trace formula for smooth and proper dg-categories over a base $\mathbb{E}_{\infty}$-algebra $B$. We also give a variant when $B$ is only an $\mathbb{E}_2$-algebra. As an application of this trace formula, we propose a strategy of proof of Bloch's conductor conjecture. This is a research announcement and detailed proofs will appear elsewhere.