Zeta morphisms for rank two universal deformations
Kentaro Nakamura
Saga University

January 25, 2021

Kentaro Nakamura (Saga)

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Notationp : a prime number.
ι∞ : Q ,→ C,

ιp : Q ,→ Qp : fixed embeddings.

L : a (suﬃciently large) finite extension of Qp ,O = OL ,ϖ ∈ O : a uniformizer,

F = O/(ϖ).

Γ = Gal(Q(ζp∞ )/Q),Λ = O[[Γ]] : the Iwasawa algebra of Γ.

For a field F , we set GF = Gal(F sep /F ).

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Kato’s zeta morphisms
∑nnew
f= ∞
n=1 an q ∈ Sk (Γ1 (Nf )) : a normalized Hecke eigen cusp
newform of level Nf ≧ 1, weight k ∈ Z≧2 .
ρf : GQ → GL2 (O) : a Galois representation associated to f , i.e. odd
and unramified outside Σf = prime(Nf ) ∪ {p} satisfying
tr(ρf (Frobℓ )) = alfor all ℓ ̸∈ Σf .
Hi (ρ∗f (1)) := limk≧0 H i (Z[1/p, ζpk ], (jk )∗ (ρ∗f )(1)) : Iwasawa←−cohomology of ρ∗f (1). This is a Λ-module.
(jk : Spec(Z[1/Σf , ζpk ]) ,→ Spec(Z[1/p, ζpk ]) : the canonicalinclusion)

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Kato defined a non zero Euler system, i.e. a collection of elements
znpk ∈ H 1 (Q(ζnpk ), ρ∗f (1))
for k ≧ 0, n ≧ 1 such that (n, Σf ) = 1 satisfying the Euler system norm
relation, and proved the following :Theorem (12.4 of Kato (04))
H2 (ρ∗f (1)) is a torsion Λ-module.
H1 (ρ∗f (1))[1/p] is a free Λ[1/p]-module of rank one.
We can define an element{zpk }k≧1 ∈ H1 (ρ∗f (1)),
but it is not canonical since {znpk }n,k depends on many choices c, d ≧ 2such that (cd, 6pNf ) = 1, 1 ≦ j ≦ k − 1 and α ∈ SL2 (Z), etc. appearingin Kato’s article.
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Dividing its dependent factors (and the L-factors at the bad primes ℓ ̸= p),
he constructed the following map which we call the zeta morphism for ρf .

Theorem (12.5 of Kato (04))
There is a canonical O-linear mapz(f ) : ρ∗f → H1 (ρ∗f (1))[1/p]interpolating, via Bloch-Kato’s dual exponentials, all the critical values ofL{p} (f, χ, s) =∞∑n=1,(n,p)=1an χ(n)ns∼×for all the finite characters χ : Γ(→ Z×p ) → C . If p is odd andρf = ρf (modϖ) is absolutely irreducible, one has the inclusionCharΛ (H1 (ρ∗f (1))/Λ · Im(z(f ))) ⊆ CharΛ (H2 (ρ∗f (1)))
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Conjecture (12.10 of Kato (04), Kato (93))(1) (Kato main conjecture, KMC)
CharΛ (H1 (ρ∗f (1))/Λ · Im(z(f ))) = CharΛ (H2 (ρ∗f (1)))(2) (Roughly speaking,) Such zeta morphisms exist for all the families of
p-adic representations of GQ which are unramified outside a finite setof primes.
When f is ordinary at p, KMC is equivalent to the usual Iwasawa
main conjecture, i.e. the equality(p-adic L-function) = CharΛ ((cyclotomic Selmer group)∨ ),which is (up to now) formulated only for f whose p-component πp (f )is principal series (B. Mazur, R. Greenberg, S. Kobayashi,
Lei-Loeﬄer-Zerbes).
We can consider KMC for arbitrary f , e.g. even for f whose πp (f ) issupercuspidal.
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Zeta morphisms for rank two universal deformations
Σ : a finite set of primes containing p, Σ0 = Σ \ {p}.
ρ : GQ → GL2 (F) : odd, absolutely irreducible, unramified outside Σ.
Comp(O) : the category of commutative local Noetherian complete
O-algebras with finite residue field.
ρunivΣ : GQ → GL2 (RΣ,ρ ) : the universal deformation for thedeformations ρ : GQ → GL2 (A) (A ∈ Comp(O)) of ρ which areunramified outside Σ (no condition at the primes in Σ).
XΣ (ρ) = Spec(RΣ,ρ [1/p])0 : the set of closed points.
XmodΣ (ρ) : the subset of modular points.
For f ∈ Sknew (Γ1 (N )), we set
Pf,ℓ (T ) = det(1 − Frobℓ · T | ρIfℓ ) ∈ O[T ].

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Theorem (Main Theorem, Nakamura (20))
Assume the following :(i) ρ|GQ(ζp ) is absolutely irreducible, (ii) p ≧ 3, (iii) EndF[GQp ] (ρ) = F,
( ±1 )
χp∗(iv) ρ|GQp is not of the form⊗ η (η : GQp → F× ).
0 1
Then, there exists a RΣ,ρ -linear map∗
1univ ∗ZΣ,ρ : (ρunivΣ ) → H ((ρΣ ) (1))

satisfying the following : for any xf ∈ Xmod
Σ (ρ), one hasx∗f (ZΣ,ρ ) =∏Pf,ℓ (Frobℓ ) · z(f )ℓ∈Σ0∏mod (ρ)) whichℓ∈Σ0 Pf,ℓ (Frobℓ ) · z(f ) (xf ∈ X0∑∞
are related with LΣ (f, χ, s) = n=1,(n,Σ)=1 annχ(n).
sNamely, ZΣ,ρ interpolates

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Related works on the construction of zeta morphisms (or Euler systems)
for families.
Ochiai, Fukaya-Kato : Hida families (of ordinary p-adic modularforms).
Hansen, Ochiai, S. Wan : Coleman-Mazur eigencurves (families ofoverconvergent of p-finite slope modular forms).
Fouquet, S. Wan : universal deformations.
RemarkFukaya-Kato generalized the construction of zeta morphisms to Hidafamilies. We construct our zeta morphisms ZΣ,ρ combining (ageneralization of) Fukaya-Kato’s method and many deep results in thetheory of p-adic Langlands correspondence for GL2/F for F = Q, Qp , Qℓ(Colmez (10), Emerton (11), Paškūnas (13), Emerton-Helm (14)).

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Application to KMC∑nnewFor fi = ∞n=1 an (fi )q ∈ Ski (Ni ) (i = 1, 2), we say that f1 and f2 arecongruent ifaℓ (f1 ) ≡ aℓ (f2 ) (modϖ)
for all but finitely many primes ℓ.

Theorem (Na)Assume f1 and f2 are congruent, ρf1 satisfies all the assumptions in ourmain theorem, andz(f1 ) (modϖ) ̸= 0.
Then one also hasz(f2 ) (modϖ) ̸= 0.
Moreover, one has the following equivalence
KMC for f1 holds⇐⇒

KMC for f2 holds.

It is expected that the assumption z(f1 ) (modϖ) ̸= 0 always holds.
Kentaro Nakamura (Saga)

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10 / 24Known results (Assume that ρf1 is absolutely irreducible and µ(f1 ) = 0)
Greenberg-Vatsal (00) : congruent elliptic curves E1 and E2 withgood ordinary reduction at p (i.e. of weight two).
Emerton-Pollack-Weston (06) : congruent eigenforms which areordinary at p (of arbitrary weights).
many related results in many related settings · · ·Kim-Lee-Ponsinet (19) : congruent eigenforms which are of finite
slope (not ordinary in general) but with a fixed weight 2 ≦ k ≦ p − 1.
(Na) : all the congruent eigenforms with arbitrary levels and weights.
Therefore, we can compare (under the assumption that z(f1 ) mod ϖ ̸= 0)
known IMC (=KMC) for ordinary case (Kato, Skinner-Urban),or of finite slope case (Kato, X.Wan)
with
unknown KMC, e.g. for supercuspidal case.
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Outline of 「Main theorem ⇒ Theorem on KMC」
a congruence between f1 and f2 and the irreducibility of ρf1 imply∼ρf1 → ρf2 = : ρ, and we can consider
xf1 , xf2 ∈ XmodΣ (ρ) for suﬃciently large Σ.
By our main theorem, one obtains a congruence∏∏
Pf1 ,ℓ (Frobℓ ) · z(f1 ) ≡Pf2 ,ℓ (Frobℓ ) · z(f2 ) (modϖ).
ℓ∈Σ0ℓ∈Σ0

Kim-Lee-Ponsinet proved that a congruence between zeta morphisms
as above implies the equivalence of KMC.
Remark
∪We can also formulate KMC for arbitrary points in X(ρ) = Σ XΣ (ρ), andobtain the equivalence of KMC between (almost) all the points in X(ρ).

Kentaro Nakamura (Saga)

Zeta morphisms

January 25, 2021

12 / 24The proof of the main theorem
We mainly explain how to construct our zeta morphism∗1univ ∗ZΣ,ρ : (ρunivΣ ) → H ((ρΣ ) (1)).

Notationb we set
For each open compact subgroup K of GL2 (Z),Y (K)(C) = GL2 (Q)\H± × (GL2 (Af )/K) andH 1 (K) = H 1 (Y (K)(C), O).
For N ≧ 1, we set
{()}
1 0b
K(N ) = g ∈ GL2 (Z) g ≡(modN ) and
0 1{()}b g ≡ ∗ ∗ (modN ) .
K1 (N ) = g ∈ GL2 (Z)0 1∏∏Gℓ = GL2 (Qℓ ), GΣ = ℓ∈Σ Gℓ , GΣ0 = ℓ∈Σ0 Gℓ .
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Fukaya-Kato’s construction for Hida families
Theorem (Fukaya-Kato (12))
For each N ≧ 4, there exists a canonical Hecke equivariant O-linear map
z1,N : H 1 (K1 (N )) → H1 (H 1 (K1 (N ))(1)) ⊗Λ Frac(Λ)
interpolating the operator valued L-functions
on H 1 (Y1 (N )(C), C).

∑n≧1,(n,p)=1Tn ·χ(n)actingns

The set of maps {z1,N pk }k≧0 is compatible with the corestrictions
H 1 (K1 (N pk+1 )) → H 1 (K1 (N pk )) (k ≧ 0), and we can control thedenominators of the images, so we can take the limit
z1,N p∞ : lim H 1 (K1 (N pk )) → H1 (lim H 1 (K1 (N pk ))(1)) ⊗Λ Λ[1/λ]
←−
←−
k≧0

k≧0

for some λ ∈ Λ = Λ[[limk≧1 (Z/N pk )× ]]. Applying Hida’s ordinary←−
projection defined using a Hecke operator at p (Up -operator), we canobtain the zeta morphisms for Hida families.
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Strategy for the construction of ZΣ,ρ∗(ρunivH 1 (K(N pk )) by (Pontryagin dual of)
Σ ) appears in lim←−k≧0Emerton’s theory of the completed cohomology.

We generalize Fukaya-Kato’s map z1,N to H 1 (K(N )).
(a subtle part) To control denominators, we need to restrict it to theρ-part (non-Eisenstein part).
Taking its limit, we obtain a equivariant zeta map for the ρ-part oflimk≧0 H 1 (K(N pk )).
←−Using deep results in family versions of p-adic local Langlands
correspondence for Gℓ (Paškūnas for ℓ = p, Emerton-Helm for∗ℓ ∈ Σ0 ), we can factor out the (ρunivΣ ) -part.
To compare our zeta morphisms ZΣ,ρ with Kato’s ones z(f ), we needPaškūnas’ another result (Paškūnas (15)). This part is the mosttechnical part of the article, but we omit to explain it in this talk.
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A refined local-global compatibility (Emerton)
For each N0 ≧ 1 such that prime(N0 ) = Σ0 , we set
e 1BM (KΣ (N0 )) := lim H 1 (K(N0 pk ))(1)H0←−
k≧0w.r.t. the corestrictions H 1 (K(N0 pk+1 ))(1) → H 1 (K(N0 pk ))(1) (k ≧ 1),andBMe BM (KΣ (N0 ))e 1,Σ
:= lim H
H0−→ 1N0e BM (KΣ (N0 )) → He BM (KΣ (N ′ )) for N0 |N ′ .
w.r.t. the restrictions H
001
100e BM is equipped with actions of GQ , Gp and GΣ = ∏HG,andℓ0ℓ∈Σ01,Σ
(homological) Hecke actions at the primes ℓ ̸∈ Σ. Using its Hecke actions,
one can define its ρ-parte BM
H1,Σ,ρ
which is a topological RΣ,ρ [GQ × GΣ ]-module.
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The following is the dual version of Emerton’s theorem.

Theorem (A refined local-global compatibility, Emerton (11))
There exists a topological RΣ,ρ [GQ × GΣ ]-linear isomorphism∼∗e BM →H(Πp )∗ ⊗RΣ,ρ (ρuniveΣ0 ,1,Σ,ρΣ ) ⊗RΣ,ρ πwhereΠp is the representation of Gp corresponding to ρuniv
Σ |GQp ,πΣ0 is the representation of GΣ0 corresponding to {ρuniv
Σ |GQℓ }ℓ∈Σ0by the family version of p-adic local Langlands correspondence defined byColmez (10) (+many people) for Πp and Emerton-Helm (14) for πΣ0 .

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GΣ -equivariant zeta morphisms
Theorem (Na)For each N0 ≧ 1 and k ≧ 1 as before, there exists a canonical Hecke
equivariant O-linear map
zN0 pk ,ρ : H 1 (K(N0 pk ))ρ (1) → H1 (H 1 (K(N0 pk ))ρ (2))
characterized by a similar interpolation property using the L-functions
removing its Euler factors at ℓ ∈ Σ, which is compatible with corestrictionsfor k ≧ 1 and restrictions for N0 .
(A subtle point) We can define the map zN0 pk ,ρ over Λ (not overFrac(Λ)) after taking the ρ-part.
By this integrality and the compatibilities, we can define the following
maps.
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We sete 1BM (KΣ (N0 ))ρ → H1 (He 1BM (KΣ (N0 ))ρ (1))
zN0 p∞ ,ρ := lim zN0 pk ,ρ : H
00←−
k≧1andBMBMe 1,Σ,ρ
e 1,Σ,ρ
zΣ,ρ := lim zN0 p∞ ,ρ : H→ H1 (H(1)).
−→N0Theorem (Na)The map zΣ,ρ is continuous and RΣ,ρ [GΣ ]-linear.
All the equivariances for Fukaya-Kato’s and our maps follow from theinterpolation property, which follows from Kato’s very deep result, i.e.
the explicit reciprocity law.
Sharifi-Venkatesh recently studies a diﬀerent, a motivic approach todefine zeta morphisms (for K1 (N )). I hope their idea enables us toprove the equivariances more directly.
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19 / 24∗e BM
Factoring out the (ρunivΣ ) -part from H1,Σ,ρ

Since one has an isomorphism∼BM∗e 1,Σ,ρ
ψ1 : H→ (Πp )∗ ⊗RΣ,ρ (ρuniveΣ0 ,Σ ) ⊗RΣ,ρ π

it suﬃces to remove (Πp )∗ -and πeΣ0 -parts.
Removing πeΣ0 -part : For π a smooth admissible representation of Gℓ(ℓ ̸= p) defined over Qp , we set Ψℓ (π) the largest quotient on which()
1 QℓUℓ =acts by a fixed non-trivial additive character Uℓ → Qp .
0 1
Emerton-Helm extended this functor for smooth admissible representationsof Gℓ defined over more general Zp -algebras, e.g. for πeΣ0 . We setΨΣ0 (eπΣ0 ) := Ψℓ1 ◦ · · · ◦ Ψℓd (eπΣ0 )for Σ0 = {ℓ1 , · · · , ℓd }. By the characterization property of theircorrespondence, one has a RΣ,ρ -linear map∼ψ2 : ΨΣ0 (eπΣ0 ) → RΣ,ρ(genericity of πeΣ0 ).
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Removing (Πp )∗ -part :C(O) : the category which is the Pontryagin dual of the category of
locally admissible Gp -representations on torsion O-modules(Emerton).
ρp : GQp → GL2 (Rp ) : the universal deformation of ρ|GQp .
Πuniv: the representation of Gp over Rp corresponding to ρp .
pTheorem (Paškūnas (13), a very rough form)
P := (Πuniv)∗ is a projective object in C(O).
pRp = EndC(O) (P ).
By the universality for ρp , one has Rp → RΣ,ρ and∼b Rp RΣ,ρ .
(Πp )∗ → P ⊗Hence, one also has∗
BM ∼e 1,Σ,ρ
b Rp (ρuniv→ P⊗
ψ1 : HeΣ0 .
Σ ) ⊗RΣ,ρ π
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Definition of ZΣ,ρThe isomorphisms ψ1 and ψ2 induce the following isomorphisms.

Corollarye BM ) ∈ C(O), andOne has ΨΣ0 (H1,Σ,ρ∼∗e BM )) →(ρuniv
HomC(O) (P, ΨΣ0 (HΣ ) .
1,Σ,ρ

e BM (1))) ∈ C(O), andOne has ΨΣ0 (H1 (H1,Σ,ρ∼BM∗
e 1,Σ,ρ
HomC(O) (P, ΨΣ0 (H1 (H(1)))) → H1 ((ρunivΣ ) (1)).

Applying HomC(O) (P, ΨΣ0 (−)) to the continuous RΣ,ρ [GΣ ]-linear map
BMBMe 1,Σ,ρ
e 1,Σ,ρzΣ,ρ : H→ H1 (H(1)),

we can finally define∗
1univ ∗ZΣ,ρ := HomC(O) (P, ΨΣ0 (zΣ,ρ )) : (ρunivΣ ) → H ((ρΣ ) (1)).
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References
P. Colmez (10)Représentations de GL2 (Qp ) et (φ, Γ)-modules, Astérisque 330 (2010), 281-509.
M. Emerton (11)Local-global compatibility in the p-adic Langlands programme for GL2/Q , available
at http://www.math.uchicago.edu/ emerton/pdﬃles/lg.pdf.
M. Emerton, D. Helm (14)The local Langlands correspondence for GLn in families, Ann. Sci. Ec. Norm.
Super. (4) 47(4) (2014), 655-722.
T. Fukaya, K.Kato (12)On conjectures of Sharifi, available at https://www.math.ucla.edu/sharifi/sharificonj.pdf.
K. Kato (93)
Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR .
Arithmetic algebraic geometry, Lecture Notes in Mathematics 1553,Springer-Verlag, Berlin (1993), 50-163.
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K. Kato (04)p-adic Hodge theory and values of zeta functions of modular forms, Astérisque(2004), no. 295, ix, 117-290, Cohomologies p-adiques et applications arithmétiques.
III.
K. Nakamura (20)Zeta morphisms for rank two universal deformations, preprint in arXiv 2006.13647.
V. Paškūnas (13)The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci. 118(2013), 1-191.
V. Paškūnas (15)On the Breuil-Mézard conjecture, Duke Math. J. 164 (2015), no. 2, 297-359.

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