SERRE WEIGHTS: A COHOMOLOGICAL DEFINITION
MISJA F.A. STEINMETZ
Abstract. This is an expository note explaining the ‘new’ definition of Serre weights as irreducible
Fp -representations as can be found in papers on generalisations of Serre’s (modularity) conjecture, such
as [1].
Contents
1. Eichler-Shimura Isomorphism
2. Serre Weights
References
1
2
3
1. Eichler-Shimura Isomorphism
I will refer the reader of this section to different sources for much of the details of this section. For
example, a good overview of the important theory can be found in [2, §12].
Let Γ be a congruence subgroup and let Mk (Γ) (resp. Sk (Γ)) denote the spaces of modular forms
(resp. cusp forms) for Γ of weight k over C. Let S k (Γ) denote the space of complex conjugates of cusp
forms.
Theorem 1.1. [Eichler-Shimura Isomorphism] We have an isomorphism
Mk (Γ) ⊕ S k (Γ) → H 1 (Γ, Symk−2 C2 ),
where we are using group cohomology on the right hand side and Symk−2 C2 denotes the (k − 2)th
symmetric power of the standard representation of Γ ⊂ SL2 (Z) on C2 .
The isomorphism can be explicitly described as follows. For any f ∈ Mk (Γ) we define a class in
H 1 (Γ, Symk−2 C2 ) by the cocycle
k−2
Z γ·z0
z
γ 7→
f (z)
dz,
1
z0
where z0 is a basepoint and v k−2 denotes the image of v ⊗ · · · ⊗ v in Symk−2 C2 for any vector v ∈ C2 .
We have a similar construction on the antiholomorphic cusp forms.
Moreover, we can define an action of the abstract Hecke ring R(Γ, ∆), i.e. the ring of double cosets
ΓδΓ for δ ∈ ∆ where ∆ is an appropriate semigroup. For example, if Γ = Γ1 (N ), we take
a b
∆ = ∆1 (N ) :=
∈ M2 (Z) | det > 0, c ≡ a − 1 ≡ 0 mod N .
c d
We now denote Γδ := Γ∩δΓδ −1 and Γδ := δ −1 Γδ∩Γ. We then define the action of ΓδΓ on H 1 (Γ, Symk−2 C2 )
as the composite
H 1 (Γ, Symk−2 C2 )
res
H 1 (Γδ , Symk−2 C2 )
conjδ
H 1 (Γδ , Symk−2 C2 )
cores
H 1 (Γ, Symk−2 C2 ),
where the middle map is gotten from conjugation by δ. One can check that this composite only depends
on the double coset ΓδΓ, therefore by extending linearly we get an action of the abstract Hecke ring
R(Γ, ∆). We have the following very important theorem.
Date: January 2017.
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Theorem 1.2. The Eichler-Shimura isomorphism of Theorem 1.1 commutes with the action of the
abstract Hecke ring R(Γ, ∆).
A trivial consequence is that Hecke eigenforms correspond to an ’eigenclass’ of the Hecke operators
in H 1 (Γ, Symk−2 C2 ). We will use this fact in the next section to give a new definition of the weight in
Serre’s conjecture.
We should note that, although in the above we describe the case for classical modular forms, all these
constructions work well for mod p modular forms with the logical substitutions such as Fp instead of C.
We will use this fact in the next section without further explanation.
2. Serre Weights
Now, let us introduce some notation. Let ρ : GQ → GL2 (Fp ) be a mod p Galois representation.
Write N = N (ρ) for the Artin conductor of ρ. We want to redefine what it means for ρ to be associated
to an eigenform f ∈ Sk (Γ1 (N )) for some weight k ≥ 2 using the group cohomology from the previous
section. Denote the abstract Hecke polynomial ring Fp [. . . , T` , . . . ] for ` - pN by T̃. We saw that for
2
any k ≥ 2, this polynomial ring acts on the space H 1 (Γ1 (N ), Symk−2 Fp ). We denote the image of T̃ in
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End(H 1 (Γ1 (N ), Symk−2 Fp )) by T. Furthermore, let m̃ denote the maximal ideal
h{T` − trρ(Frob` ) : ` - pN }i ⊂ T̃.
Note that this ideal is the kernel of the evaluation map T̃ → Fp : T` 7→ trρ(Frob` ), hence it is certainly
maximal. Let m denote the image of m̃ in T.
Now, by the Eichler-Shimura isomorphism, a mod p eigenform of weight k corresponds to an eigenclass
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in H 1 (Γ1 (N ), Symk−2 Fp ) (i.e. a class which is a simultaneous eigenvalue of all the Hecke operators in T)
with the same eigenvalues at the Hecke operators. So proving that ρ is modular of weight k is equivalent
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to showing there exists an eigenclass in H 1 (Γ1 (N ), Symk−2 Fp ) with eigenvalues trρ(Frob` ) for all l - pN.
Such a class clearly exists if and only if
2
H 1 (Γ1 (N ), Symk−2 Fp )[m] := {[x] ∈ H 1 : T` · [x] − trρ(Frob` )[x] = 0 for all l - pN } =
6 0,
2
i.e. if the kernel in H 1 (Γ1 (N ), Symk−2 Fp ) of multiplication by m is non-zero.
2
But since T is an Artinian ring, H 1 (Γ1 (N ), Symk−2 Fp ) is an Artinian module and it decomposes as
M
2
2
H 1 (Γ1 (N ), Symk−2 Fp ) =
H 1 (Γ1 (N ), Symk−2 Fp )m
m
where the direct sum runs over the finite number of maximal ideals of T. Since multiplication by m on
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the localised component at n 6= m does not have a kernel, the condition H 1 (Γ1 (N ), Symk−2 Fp )[m] 6=
2
0 becomes equivalent to H 1 (Γ1 (N ), Symk−2 Fp )m [m] 6= 0, in particular the former condition implies
2
H 1 (Γ1 (N ), Symk−2 Fp )m 6= 0. Furthermore, we know by Artinianness that there exists k ≥ 1 such that
2
mk H 1 (Γ1 (N ), Symk−2 Fp )m = 0.
2
2
Hence, if H 1 (Γ1 (N ), Symk−2 Fp )m 6= 0, then also H 1 (Γ1 (N ), Symk−2 Fp )m [m] 6= 0. Therefore, we have
shown that
2
2
H 1 (Γ1 (N ), Symk−2 Fp )[m] 6= 0 ⇐⇒ H 1 (Γ1 (N ), Symk−2 Fp )m 6= 0.
Now we are in good shape, since localisation is exact.
Claim. Let Γ := Γ1 (N ). If A is a Γ module, then the Hecke action on H 0 (Γ, A) is Eisenstein. Hence, if ρ
is irreducible and m ⊂ T the maximal ideal it generates as above, then H 0 (Γ, A)m = 0 for any Γ-module
A.
Proof. For any δ ∈ ∆1 (N ), where as
a
∆1 (N ) :=
c
usual
b
∈ M2 (Z) | det > 0, c ≡ a − 1 ≡ 0 mod N ,
d
2
we recall that the Hecke action of ΓδΓ on H 0 (Γ, A) is defined as the composition
H 0 (Γ, A)
res
H 0 (Γδ , A)
conjδ
cores
H 0 (Γδ , A)
H 0 (Γ, A),
where, as usual, we denote Γδ := Γ ∩ δΓδ −1 and Γδ := δ −1 Γδ ∩ Γ (also see [2, §12.4]). The map in
the middle is induced by the conjugation by δ map g 7→ δ −1 gδ which maps Γδ → Γδ . The last map is
x7→[Γ:Γδ ]x
also sometimes called the trace or transfer map; it follows from the map Γ −−−−−−−−−→ Γδ . Using the
identification H 0 (G, A) = AG , i.e. the G-invariant elements of A, we find that the maps above are given
by
AΓ
a7→a
δ
AΓ
a7→δ·a
AΓδ
a7→[Γ:Γδ ]·a
AΓ .
1 0
If δ =
then [Γ1 (N ) : Γ1 (N )δ ] = p + 1 (since p - N ; see [3, p. 170]). Hence, it follows that the
0 p
Hecke operator Tp acts on H 0 (Γ, A) as multiplication by p + 1 and it is Eisenstein.
Since any irreducible representation corresponds to a cuspidal eigenform, it now follows easily that
0
H (Γ, A)m = 0 for any such representation.
Claim. If Γ is ‘sufficiently small’, then H 2 (Γ, A)m = 0 for any Γ-module A.
Proof. (Sketch) If Γ is small, then the quotient map H → H/Γ is a universal cover of the non-compact
modular curve H/Γ (since Γ is the fundamental group of the modular curve in those cases). Therefore,
H 2 (Γ, A) = H 2 (H/Γ, A). But now H 2 (H/Γ, A) = Hc0 (H/Γ, A) by Poincaré Duality and the latter space
vanishes by non-compactness of the modular curve.
From the two claims and exactness of localisation, it now follows that if we have an exact sequence of
Γ1 (N )-modules 0 → A → B → C → 0 then we get a short exact sequence of cohomology groups
0
H 1 (Γ1 (N ), A)m
H 1 (Γ1 (N ), B)m
H 1 (Γ1 (N ), C)m
0.
Hence, if we assume 0 = V0 ⊂ V1 ⊂ · · · ⊂ Vn = Symk−2 Fp is a Jordan-Hölder composition series with
each Wi+1 := Vi+1 /Vi irreducible, then from the short exact sequence on the cohomology groups we see
2
that H 1 (Γ1 (N ), Symk−2 Fp )m 6= 0 if and only if H 1 (Γ1 (N ), Wj )m 6= 0 for some 1 ≤ j ≤ n.
2
This leads us to the long awaited definition of Serre weights. Note that Symk−2 Fp is the (k − 2)th
2
symmetric power of the standard representation of GL2 (Fp ) on Fp .
Definition 2.1. We say that an irreducible Fp -representation V of GL2 (Fp ) is a Serre weight (or
sometimes simply a weight) associated to our representation ρ if ρ is associated to an eigenclass in
H 1 (Γ1 (N ), V ). We denote the set of Serre weights by W (ρ). In other words, V ∈ W (ρ) if
H 1 (Γ1 (N ), V )m 6= 0,
where m ⊂ T is the maximal ideal obtained as the kernel of the evaluation map T` 7→ trρ(Frob` ) as
defined above.
Remark 2.2. If we are not assuming Γ is sufficiently small, then it is possible to find examples of representations for which we cannot pass to the Jordan-Hölder factors as above. See for example DiamondReduzzi [4, p. 17].
References
[1] Kevin Buzzard, Fred Diamond and Frazer Jarvis, On Serre’s conjecture for mod ` Galois representations over totally
real fields, Duke Math. J., 155 (1), pp. 105 – 161, 2010.
[2] Fred Diamond and John Im, Modular forms and modular curves, In: Canadian Mathematical Society Conference
Proceedings, 17, pp. 39 – 133, 1995.
[3] Fred Diamond and Jerry Shurman, A First Course in Modular Forms, Springer New York, 2006.
[4] Fred Diamond and Davide Reduzzi, Crystalline lifts of two-dimensional mod p automorphic Galois representations,
https://arxiv.org/abs/1509.04979, 2015
London School of Geometry and Number Theory and Department of Mathematics, King’s College London,
Strand, London, WC2R 2LS
E-mail address: [email protected]
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