Arithmetic of local systems

[Workshop] Arithmetic of local systems

November 8-12, 2027 RIMS Kyoto, JP Organizers: B. Collas (RIMS Kyoto, JP), Y. Yatagawa (IS Tokyo, JP)

This workshop investigates the interface between the arithmetic of the étale fundamental group and that of local systems -- two theories developed with independent vocabularies but similar mechanisms.
The following questions can be read as guide:

1. Wild ramification appears as inertia subgroups of the fundamental group, and as a cycle on the cotangent bundle: how much of the one survives in the other? (Abbes–Saito filtration, refined Swan conductor, singular support and characteristic cycle, Milnor and conductor formulas, epsilon factors);
2. Euler characteristics are read off a single sheaf by Grothendieck–Ogg–Shafarevich, and off a whole tower of coverings by Deuring–Shafarevich: what does each of the two determine about the variety? (Swan conductors, p-rank, generalized Hasse–Witt invariants of coverings);
3. Rank-one local systems are exactly the characters of the abelian fundamental group: does bounding their ramification by a modulus mean the same thing on both sides? (class field theory with modulus, reciprocity sheaves, algebraic loops on schemes, mixed characteristic);
4. Monodromy is a representation of the fundamental group on one side and a Tannaka group of perverse sheaves on the other: how do the arithmetic and geometric monodromy groups compare, and which formalism remembers what? (systems of étale covers and their inertia subgroups, étale homotopy type and pro-étale refinements, higher-categorical exodromy, convolution).

Both sides then face one problem of reconstruction: the Jacobian linearises the abelianised fundamental group, while the Prym varieties of its coverings carry a whole family of ℓ-adic representations indexed by the tower. Which of these linear shadows retains enough -- anabelian on one side, Tannakian on the other?

Keywords: étale fundamental group; local systems; wild ramification; singular support and characteristic cycle; conductor formulas; fundamental group with modulus; reciprocity sheaves; higher-categorical monodromy; perverse sheaves and convolution; Tannaka formalism; Prym varieties; anabelian reconstruction.