From higher dimensional class field theory to anew theory of motives
Shuji Saito
University of Tokyo

Departmental colloquia

S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of motives
13 March 2023

1 / 19Classical CFT
Consider√
Gal(Q( 2)/Q) ∼= {±1}√√−1)andQ(2).
These do not tell us arithmetic differenceofQ(√On the other hand, from the fact Q( −1) ⊂ Q(ζ4 ), we get√
Gal(Q( −1)/Q) ∼= {±1},√Gal(Q( −1)/Q) ∼= (Z/4Z)×√
Similarly, from the fact Q( 2) ⊂ Q(ζ8 ), we get√
Gal(Q( 2)/Q) ∼= (Z/8Z)× /{±1}S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of motives
13 March 2023

(1)

(2)

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The latter isomorphisms imply the following
Theorem 0.1 (Fermat)
For an odd prime p, we have equivalences:∃x, y ∈ Z, p = x2 + y 2 ⇔ p ≡ 1 mod 4,∃x, y ∈ Z, p = x2 − 2y 2 ⇔ p ≡ ±1 mod 8.
(1) and (2) follow from the isomorphism (which follows from theirreducibility of cyclotomic polynomials)'cn : (Z/nZ)× −→ Gal(Q(ζn )/Q) ; a → σa(3)where σa ∈ Gal(Q(ζn )/Q) is determined by σa (ζn ) = ζna .

S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of motives
13 March 2023

3 / 19In order to describe π1ab (U ) for non-proper U , Kerz-Saito introduced
a new theory of Chow groups called Chow groups with modulus. Fora smooth variety U over a field k, we consider pairs (X, D) where X
is normal and proper over k and D is an effective Cartier divisor onX such that U = X \ |D|. Such a (X, D) is called a modulus pairwith interior U . The Chow group of zero-cycles of a modulus pair
(X, D) is defined asMM δ
CH0 (X|D) = Coker
Z ,(9)
k(C)D −→ Z0 (U ) =x∈U(0)C⊂U

where the direct sum ranges over integral curves on U and k(C)D isa subgroup of k(C)× of rational functions f on C such thatf ≡1

mod D ×X C̃,where C̃ is the normalization of the closure of C in X.
S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13
motivesMarch 2023

13 / 19

Theorem 0.3 (Kerz-S (2016), Binda-Krishna-S (2022))
Assume k is finite. Then, the Frobenius map ρ̃U from (7) induces anisomorphism of finite groups'CH0 (X|D)0 −→ π1ab (X, D)0 ,where π1ab (X, D) is the quotient of π1ab (U ) classifying the coverings
V → U whose Artin conductor is bounded by D. In particular wehave an isomorphism of pro-finite topological groups'lim CH0 (X|D)0 −→ π1ab (U )0 ,←−(X,D)

where the limit is over all modulus pairs with interior U , or over all
effective Cartier divisors such that X \ U = |D| for a fixed X.
S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13
motivesMarch 2023

14 / 19

Cohomology theory of modulus pairs
In algebraic geometry, commonly considered sheaf theories on
categories Sch ⊃ Sm (over some fixed base). Motivated by
Kerz-Saito’s theorem, Kahn-Miyazaki-Yamazaki-S started in 2018 aproject to construct general cohomology theories of modulus pairs(X, D): X are proper scheme over a field k and D are effective
Cartier divisors such that X \ |D| ∈ Sm (the interiors are not fixed
as before). We defined what morphisms between modulus pairs are
so that they form a category MSm with good properties (e,g, admits
fiber products).
Miyazaki defined a Grothendieck topology MNis on MSm so thatthe category ShvMNis (MSm) of sheaves of abelian groups on the site
MSm is Grothendieck abelian so that we can apply homologicalalgebra to define cohomology of (X, D) with coefficientF ∈ ShvMNis (MSm).
S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13
motivesMarch 2023

15 / 19

Application to ramification theory
A remarkable feature of the new theory is its relation to ramification
theory. To explain this, we need to introduce reciprocity sheaves.
We have an easy functor ω ∗ : ShvNis (Sm) → ShvMNis (MSm):ω ∗ F (X, D) = F (X \ D) for F ∈ ShvNis (Sm) and (X, D) ∈ MSm,KMSY defined a mysterious functor
ψ : ShvNis (Sm) → ShvMNis (MSm) asψF := the maximal -invariant object of ω ∗ F,
 = (P1 , ∞), G is -invariant if G(X, D) = G((X, D) ⊗ ).
Remark 0.4
It is impossible to compute ψF just from the definition.
S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13
motivesMarch 2023

16 / 19

For U ∈ Sm, {ψF (X, D)}(X,D) with (X, D) ranging over modulus
pairs with interior U , gives a filtration on F (U ).
Definition 0.1
F ∈ ShvNis (Sm) is called reciprocity sheaves if the filtration isexhaustive and admits transfers (push-forward for finite maps).
Let RSCNis be the category of reciprocity sheaves.
Example 0.5
The following are reciprocity sheaves.
1
A1 -invariant sheaves with transfers.
2Ωj , Wn Ωj .
3F (X) = {rank 1-connections over X}/ '.
4
G finite abelian group, F (X) = {étale G-torsors over X}/ '.
5
G finite flat group scheme, F (X) = {flat G-torsors over X}/ '.
S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13motives
March 2023

17 / 19

Theorem 0.6 (Rülling-S (2022))
For F ∈ RSCNis ⊂ ShvNis (Sm) and (X, D) ∈ MSm such that X
smooth projective and |D| is a SNCD,(D)ψF (X, D) = F AS (X, D) := {a ∈ F (U )| p∗1 (a) − p∗2 (a) ∈ F (PX )},(D)where U = X \ D and p1 , p2 : U × U → X are projections, and PX
is the dilatation of X along D: The blow-up of X × X in D
embedded diagonally into X × X and with the strict transforms ofX × D and D × X removed. Note the open immersion(D)
U × U → X × X lifts to U × U → PX .

AS stands for Abbes and T.Saito who studied ramification theory forG-torsor (even for non-abelian case) using dilatations.

S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13
motivesMarch 2023

18 / 19

Using the theorem, one can compute ψF for some examples, giving aunified interpretation to different phenomena such as poles ofdifferential forms, Artin conductors, and irregularities of connections.
Example 0.7(1) ψΩj (X, D) = ΩjX (log |D|)(D − |D|) if ch(k) = 0.
A formula exists also in case ch(k) = p > 0 but omitted.
(2) For F (U ) = Homcont (π1ab (U ), Q/Z),ψF (X, D) = Homcont (π1ab (X, D), Q/Z) ⊂ F (U ).
Recall π1ab (X, D) is defined by Artin conductors.
(3) For F (U ) = {line bundles on U with connections}/ ',ψF (X, D) is computed by using the irregularities of connections.

S. Saito (University of Tokyo)

From higher dimensional class field theory to a new theory of 13motives
March 2023

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