# Reciprocity sheaves and their ramification filtrations

## TL;DR

This paper introduces a new motivic conductor for reciprocity sheaves, unifying and extending classical conductors like Artin and irregularity, and applies it to various geometric and arithmetic contexts.

## Contribution

It defines a motivic conductor for presheaves with transfers, extending classical conductors and establishing a minimal, universal framework for ramification filtrations.

## Key findings

- The motivic conductor extends classical conductors such as Rosenlicht-Serre and Artin.
- It provides a unified approach to ramification filtrations for various sheaves.
- The machinery introduces a new conductor for torsors under finite flat group schemes.

## Abstract

We define a motivic conductor for any presheaf with transfers $F$ using the categorical framework developed for the theory of motives with modulus by Kahn-Miyazaki-Saito-Yamazaki. If $F$ is a reciprocity sheaf this conductor yields an increasing and exhaustive filtration on $F(L)$, where $L$ is any henselian discrete valuation field of geometric type over the perfect ground field. We show if $F$ is a smooth group scheme, then the motivic conductor extends the Rosenlicht-Serre conductor; if $F$ assigns to $X$ the group of finite characters on the abelianized \'etale fundamental group of $X$, then the motivic conductor agrees with the Artin conductor defined by Kato-Matsuda; if $F$ assigns to $X$ the group of integrable rank one connections (in characteristic zero), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with {\em perfect} residue field can be uniquely extended to all such fields without any restriction on the residue field. For example the Kato-Matsuda Artin conductor is characterized as the canonical extension of the classical Artin conductor defined in the perfect residue field case.