MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY
(SPRING 2014)
LECTURE 3 (FEBRUARY 28, 2014)
FEIQI JIANG
HIDA THEORY FOR GL(2)
NOTES TAKEN BY PAK-HIN LEE
1. p-adic Interpolation of Modular Forms
Before going into Hida theory, let me give a brief description of the p-adic interpolation
of a family of classical modular forms. Let p be a prime > 3, and W = limm W/pm W be
←−
a p-adically complete algebra. Let {φk }k∈I⊂Z be a family of classical modular forms, where
each φk is a modular form of weight k. Consider
W [[Z×
lim W [(Z/pm Z)× ],
p ]] = ←
−
m
which satisfies the universal property
×
×
Homcont (Z×
p , B ) = Homalg (W [[Zp ]], B)
k
×
for any W -algebra B. The character [k] : Z×
p → Zp given by z 7→ z induces
b
k : W [[Z×
p ]] → Zp .
We say that Φ ∈ W [[Z×
p ]][[q]] interpolates {φk }k∈I⊂Z if
X
b
k(an )q n
φk =
k∈I
for all k ∈ I. We are interested in the following question: if f is a classical modular form of
Pb
weight k, when does there exist Φ ∈ W [[Z×
k(an )q n ? This is partially
p ]][[q]] such that f =
answered by Hida in his “Vertical Control Theorems”.
Let’s talk a bit about modular forms. Let A be a Z[ 61 ]-algebra. We denote by (E, ω)/A
an elliptic curve E/A together with a nowhere vanishing differential form ω. Consider the
functor P1 : Sch/Z[ 1 ] → Sets taking S to [(E, ω)/S ]. We claim this can be represented by
6
some scheme.
Theorem 1. P1 can be represented by the affine scheme M1 /Z[ 1 ] = Spec R, where
6
1
1
R = Z , g2 , g3 , 3
.
6
g2 − 27g32
Last updated: March 1, 2014. Please send corrections and comments to [email protected].
1
Let (E, !)/M1 be the universal elliptic curve, i.e.
E = Proj(R[X, Y, Z]/(ZY 2 − 4X 3 + g2 XZ 2 + g3 Z 3 ))
and
dX
Y
is the universal differential. Then for S = Spec A, M1 (Spec A) = HomZ[ 1 ]-alg (R, A) corre6
sponds to P1 (S) via the correspondence
!=
(φ : S → M1 ) ←→ φ∗ (E, !).
Now we introduce the level-N structures. We will work over Z[ N1 ]. Let E[N ] ⊂ E/S be
the N -torsion. This is locally free of rank N 2 over S. A level N structure is defined to be a
homomorphism of group schemes
φ : (Z/N Z)2 → E[N ].
Let’s consider another functor PN : Sch → Sets given by PN (S) = (E, φN , ω) where (E, ω) ∈
P1 (S) and φN is a level N structure.
Theorem 2. PN is represented by an affine scheme MN = Spec(RN ) which is finite and
étale over M1 .
Recall the universal elliptic curve (E, !) over M1 . If (E, ω, φN )/S ∈ PN (S), there exists a
unique u : S → M1 such that (E, ω) = u∗ (E, !).
For a fixed elliptic curve E, the functor defined by PE/S (T ) = {level N structures over T }
is representable by an affine scheme over S. Thus PE/M1 is representable by an affine scheme
over M, so PN is representable by Spec(RN ) where RN is finite and étale over R.
Define an action Gm × PN → PN by (here Gm (A) = A× )
(λ, (E, φN , ω)) 7→ (E, φN , λω)/A .
This gives an action on MΓ(N ) and RΓ(N ) . Thus we have a schematic representation of Gm
MΓ(N ) : B/A → RΓ(N ) ⊗A B.
For k ∈ Z, set
MΓ(N ),k (B)/A := {χ ∈ MΓ(N ) (B) : λ · χ = λk χ for λ ∈ Gm (A)}.
Then
MΓ(N ) =
M
MΓ(N ),k
k
and
RΓ(N ) =
M
Rk (Γ(N ); A).
k
We can view each f ∈ Rk (Γ(N ); A) as a functorial map
f
MΓ(N ) (B) = PN (B) → A1 (B) = B.
This can be viewed as a modular form, because it satisfies
(G0) f (E, φN , λω)/B = λk f (E, φN , ω)/B .
(G1) If (E, φn , ω)/B ∼
= (E 0 , φ0N , ω 0 )/B , then f (E, φN , ω) = f (E 0 , φ0N , ω 0 ).
0
(G2) If ρ : B → B is a morphism of A-algebras, then f ((E, φN , ω)/B ×B B 0 ) = ρ(f (E, φN , ω)/B ).
2
We have ∆−1 ∈ R ⊂ RΓ(N ) , which has a grading
M
RΓ(N ) =
Rk (Γ(N ); A)
k∈Z
so RΓ(N ) contains a unit of degree 12.
Consider
EΓ1 (N ) (A) := [(E, φN : µN → E[N ])/A ]
which is just Gm \PΓ1 (N ) (A) and can be represented by Proj(RΓ1 (N ) ).
Theorem 3 (Shimura and Igusa). EΓ1 (N ) is represented by
Y1 (N ) := Gm \MΓ (N ) = Proj(RΓ (N ) )/A ∼
= Spec(R0 (Γ1 (N ); A)),
1
1
which is locally free of finite rank over M1 . (The above holds for any Z[ N1 ]-algebra A.)
Over C, we have the upper half plane H = {z ∈ C : Im(z) > 0} and
Y1 (N )(C) = Γ1 (N )\H.
H 0 (Y1 (N ), ω k ) corresponds to modular forms of weight k. We can compactify Y1 (N ). Define
1
]) to be the integral closure of Z[ 16 , g2 , g3 ] in the graded algebra RΓ1 (N ) /Z[ 1 ] . We
GΓ1 (N ) (Z[ 6N
6N
thus get
X1 (N )/A = Proj(GΓ1 (N ) (A)).
Now we have a geometric description of modular forms. To define holomophicity at infinity,
we need to consider a special test object called the Tate curve
O := (Tate(q)/Z((q)) , ω can , φN )
where Z((q)) = Z[[q]][ 1q ],
1
2
3
2
3
Tate(q) = Proj Z((q))
[X, Y, Z]/(ZY − 4X + g2 (q)XZ + g3 (q)Z )
6
and
dX
.
ω can =
Y
1
Thus f (O) ∈ A((q)) for Z[ 6N
]-algebra A. Holomorphicity at infinity is defined as
(G3) f (O) ∈ Z[[q]].
Now let A be a ring of characteristic p. Recall that the Hasse invariant H detects whether
an elliptic curve is supersingular: if H(E, ω)/A = 0, then E is supersingular; otherwise E is
ordinary.
2. Igusa Tower
Let W = Zp and Wm = W/pm W . Fix N such that p - N . Let H be the Hasse invariant.
Define M = X1 (N )/W and Mm = X1 (N )×W Wm . The ordinary locus is defined as Sm ⊂ Mm ,
which is the open subscheme where H(E, ω) is invertible. Note that H(E, ω) is a modular
form of weight 1 − p:
H(E, λω) = λ1−p H(E, ω).
The important point here is that Sm = Spec(Vm,0 ) is affine. Consider the functor
Eα0
ord
(A) = [(E, P, φN )]/A
3
where α ∈ Z, P is a point of order pα and φN is a level N structure. A similar functor is
Eαord (A) = [(E, φpα : µpα → E[pα ], φN )]/A .
We have
Eα0
ord
(A) ∼
= Eαord (A) = (E[pα ]ét − E[pα−1 ]ét )/Sm (A).
We write
Tm,α = E[pα ]ét − E[pα−1 ]ét = Isom(E[pα ]ét → (Z/pα Z))
which is an étale cover of Sm of degree pα−1 (p − 1). A result by Igusa says that Tm,α are
irreducible. Since the Sm are affine, Tm,α are affine as well. We let Vm,α be such that
Tm,α = Spec(Vm,α ). We get a tower of Wm -algebras
Vm,0 ⊂ Vm,1 ⊂ · · ·
with Gal(Vm,α /Vm,0 ) ∼
= (Z/pα Z)× . We can define
Vm,∞ = lim Vm,α ,
−→
α
Tm,∞ = Spec(Vm,∞ ),
V = lim Vm,∞ ,
−→
m
and
VΓ1 (N ) = lim Vm,∞
←−
m
If f ∈ Vm,α = OTm,α , then f can be viewed as a map on Tm,α as follows. Given f ∈
H 0 (Y1 (N ), ω k ), each point in Y1 (N ) is of the form (E, φN ). There exists ι : Spec(A) → Y1 (N )
such that
ι∗ (E, ♦N , !) = (E, φN , ω).
The element ι∗ f ∈ A can be thought of as the value of f at (E, φN , ω). This function f
satisfies the modular form properties G0, G1, G2. Thus
H 0 (X1 (N )/A , ω k ) = Gk (Γ1 (N ), A),
the space of modular forms of weight k over A. Similarly, we can show
Vm,∞ [k] = {f ∈ Vm,∞ : f (E, z −1 φp , φN ) = z k f (E, φp , φN ) for all z ∈ Z×
p = Gal(Vm,∞ /Vm,0 )}
is precisely H 0 (Sm , ω k ). Since V = limm Vm,∞ , we have
−→
!
V[k] =
!
lim Vm,∞ [k] = H 0 S/W , ω k ⊗ lim Z/pm Z
−→
−→
m
= H 0 (S/W , ω k ⊗ Qp /Zp )
m
which is equal to H 0 (S/W , ω k ) ⊗ Qp /Zp since S is free.
There is an ordinary projection e : V → V. It turns out that
rankW (Hom(eV[k], Tp = Qp /Zp ))
is finite. Let V ord be the Pontryagin dual of eV. We can decompose
Z×
p = ΓT × ∆
where ΓT = 1 + pZp ∼
= Zp and ∆ are the roots of unity. We can write
W [[Z×
p ]] = W [[ΓT ]][∆].
4
Since ΓT is generated by γ = 1 + p, W [[ΓT ]] ∼
= W [[X]] via 1 + p 7→ 1 + X.
For k ∈ Z, we consider V ord [χ] where χ is a character on ∆.
×
V ord [χ]/(X + 1 − γ k )V ord [χ] ∼
W
= V ord ⊗
W [[Zp ]],k
∼
= HomW (eV, W ) ⊗W [[Z×p ]],k W
∼
= HomW (H 0 (X1 (N ), ω k ), W )
ord
ord
HomW (Gk (Γ1 (N ), W ), W )
=
assuming k ≥ 3. We can decompose
V ord =
M
V ord [χ].
˜
χ∈∆
If z 7→ z k induces χ, then
V ord [χ] ⊗Λ,k W = V ord ⊗W [[Z×p ]],k W
where Λ = W [[ΓT ]] = W [[X]].
0
Theorem 4 (Vertical Control Theorem). Write Hord
for eH 0 and Gord
for eGk . Then
k
V ord [χ] is Λ-free of finite rank, and
V ord ⊗W [[Z×p ]],k W ∼
= HomW (Gord
k (Γ1 (N ), W ), W ).
P
Let me now say how this relates to p-adic families. Let Φ(q) = n an (T )q n ∈ Λ[[q]] where
an ∈ Λ = W [[T ]]. Write G(χ, Λ) = HomΛ (V ord [χ], Λ). For φ ∈ G(χ, Λ), we have
X
Φφ (T, q) =
φ(a(n))(T )q n
where a(n) : V ord → Λ. A consequence of the VCT is
Theorem 5.
(1) G(χ, Λ) is free of finite rank.
−k
(2) G(χ, Λ)⊗Λ,k W ∼
, W ), where ω is the Teichmüller character.
= Gk (Γ1 (N )∩ΓP
0 (p), χω
k
(3) Under this isomorphism, φ 7→ Φφ (γ − 1, q).
3. Supplementary Remarks by Prof. Urban
Given a character χ, we can define hord
χ , the Hecke algebra acting on G(χ, Λ), which is
free over Λ. There is a duality between modular forms and Hecke algebra, given by
(f, T ) 7→ a(1, f |T ).
Thus
h∼
= HomΛ (G(χ, Λ), Λ)
which is free of finite rank over Λ. There is an action of Λ on h.
Let Pk be the kernel of the map Λ → Zp given by T 7→ (1 + p)k − 1. The control theorem
for modular forms, translated into Hecke algebras, says that
hord /Pk ∼
= hord (χω −k )
χ
k
which is the Hecke algebra acting on Gk (Γ1 (N p), χω −k ). An eigenform of weight k and level
N p and nebentypus χω −k is just a character
−k
hord
) → Qp .
k (χω
5
The diagram
hord
χ
/
/
−k
hord
)
k (χω
O
/
Λ
Qp
O
Λ/Pk = Zp
commutes by definition. The composition of the top row can be thought of as xf ∈
ord
Spec(hord
χ )(Qp ), which has dimension 2. Let Spec(I) be an irreducible component of Spec(hχ )
containing the point xf . We know I is going to be torsion-free over Λ, and therefore is of
dimension 2. We have a map φI : hord
I. The kernel ker(xf ) contains Pk which is the
χ
kernel of the bottom map.
xf ∈ Spec(I)(Qp ) ⊂ Spec(hord
χ )(Qp )
∈ Spec(Λ)(Qp )
[k]
I corresponds to some minimal prime ideal of hord
χ contained in ker(xf ). Here we use the
Cohen–Seidenberg theorem.
We get an I-adic form
∞
X
FI (q) =
φI (Tm )q m .
The map xf : h
m=1
ord
χ
→ Qp factors through φI : hord
χ I as follows:
hord
χ
φI
xf
//
I
xf
Qp
So xf (FI (q)) is the q-expansion of f .
We have a canonical diagram
hord
χ /Pk
6
hord
χ
0
0
−k
= hord
)
k0 (χω
(/
x0 ◦φI
O
/
Λ
and x0 (FI ) =
P∞
m=1 (x
0
Qp
O
Λ/Pk0
◦ φI )(Tm ) is the q-expansion of an eigenform of weight k 0 .
6