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Building on recent results by A. Blanc, M. Robalo and the authors, we presentan $\ell$-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 astrategy of proof of Bloch's conductor conjecture. This is a researchannouncement and detailed proofs will appear elsewhere.

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