L-Invariants of Tate Curves: Hida's Lecture Notes, Study notes of Cryptography and System Security

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HARUZO HIDA

1. Lecture 3

1.1. Extensions of Q p

by its Tate twist. Let K ⊂ C p

be a finite extension of

Q

p

and W be a two dimensional vector space over K with a K-linear action of

D := Gal(Q p

/Q

p

). We start with an extension of local Galois modules

0 → K(1) → W → K → 0

over D. This type of extensions (for K = Q p

) can be obtained by the p-adic Tate

module W = T p

E ⊗

Zp

Q

p

of an elliptic curve E /Qp

with multiplicative reduction.

We prepare some general facts. The following is a description of a result in [GS1]

Section 2 (see also [H07]). We write H

i (?) for H

i (D, ?). By definition, H

1 (M) =

Ext

1

K[D]

(K, M) for a D-module M, and hence, there is a one-to-one correspondence:

{

nontrivial extensions

of K by M

}

↔

{

1-dimensional subspaces

of H

1

(M)

}

.

From the left to the right, the map is given by (M ↪→ X  K) 7 → δX (1) for

the connecting map K = H

0 (K)

δ X

−→ H

1 (M) of the long exact sequence attached

to (M ↪→ X  K). Out of a 1-cocycle c : D → M, one can easily construct

an extension (M ↪→ X  K) taking X = M ⊕ K and letting D acts on X by

g(v, t) = (gv + t · c(g), t), and [c] 7 → (M ↪→ X  K) gives the inverse map.

By Kummer’s theory, we have a canonical isomorphism:

H

1

(K(1))

∼

=

(

lim

←−

n

Qp

×

/(Qp

×

)

p

n

)

⊗Z

p

K.

We write γ q

∈ H

1 (K(1)) for the cohomology class associated to q ⊗ 1 for q ∈ Q p

×

.

The class γ q

is called the Kummer class of q. A canonical cocycle ξ q

in the class γ q

is

given as follows. Then ξq(σ) = lim

←−n

(q

1 /p

n

)

σ− 1

having values in Zp(1) ⊂ K(1).

In summar, we have

Proposition 1.1. If W is isomorphic to the representation σ 7 →

(

N (σ) ξq (σ)

0 1

)

with 0 <

|q|p < 1 , then for the extension class of [W] ∈ H

1

(K(1)), we have K[W] = Kγq. In

particular, Kγ q

is in the image of the connecting homomorphism H

0 (K)

δ 0

−→ H

1 (K(1))

coming from the extension K(1) ↪→ W  K.

Corollary 1.2. Let E /Qp

be an elliptic curve. If E has split multiplicative reduction

over W , the extension class of [TpE ⊗ Q] is in Qpγq E

for the Tate period qE ∈ Qp

×

.

Date: August 6, 2008.

The third lecture at TIFR on August 4, 2008. The author is partially supported by the NSF

grant: DMS 0244401, DMS 0456252 and DMS 0753991.

1

Let

˜

K := K[ε] = K[t]/(t

2 ) with ε ↔ (t mod t

2 ). A

˜

K[D]-module

˜

W is called an

infinitesimal deformation of W if

˜

W is

˜

K-free of rank 2 and

˜

W/ε

˜

W

∼

= W as K[D]-

modules. Since the map ε :

˜

W  W ⊂

˜

W given by v 7 → εv is Galois equivariant, we

have an exact sequence of D-modules

0 → W →

˜

W → W → 0.

Each infinitesimal deformation gives rise to an infinitesimal character ψ : D →

˜

K

×

with ψ mod (ε) = 1. Define

˜

K(ψ) for the space of the character ψ. Obviously,

dψ

dt

: D → K is a homomorphism; so,

dψ

dt

∈ Hom(D, K) = H

1

(K). Since the extension

˜

K(ψ) is split if and only if

dψ

dt

= 0, we get

Proposition 1.3. The correspondence

˜

K(ψ) ↔

dψ

dt

∈ H

1 (K) gives a one-to-one

correspondence:

{

Nontrivial infinitesimal

deformations of K

}

↔

{

1-dimensional

subspaces of H

1

(K)

}

.

Note that

H

1

(D, K)

∼

= Hom(D, K) = Hom(D

ab

, K)

∼

= K

2

,

where, as we have seen in Lecture 2, the last isomorphism is given by

Hom(D

ab

, K) 3 φ 7 → (

φ([γ, Qp])

log p

(γ)

, φ([p, Qp])) ∈ K

2

.

This follows from local class field theory. Since the Tate duality 〈·, ·〉 is perfect, for

any line ` in H

1 (D, K), one can assign its orthogonal complement `

⊥ in H

1 (D, K(1)).

Proposition 1.4. The correspondence of a line in H

1 (D, K) and its orthogonal com-

plement in H

1 (D, K(1)) gives a one-to-one correspondence:

{

Nontrivial extensions

of K by K(1) as K[D]-modules

}

↔

{

nontrivial infinitesimal

deformations of K over D

}

.

Theorem 1.5. Let E /L

be an elliptic curve with split multiplicative reduction defined

over a finite extension L/Q p

, and let ψ : Gal(Q p

/Q

p

) →

˜

Q

p

×

be a nontrivial character

which is congruent to 1 modulo ε. Let W = T p

E ⊗ Q

p

for the p-adic Tate module

T

p

E of E and q E

∈ L

× be the Tate period of E. Then the following statements are

equivalent:

(a)

dψ

dt

(σ N L/Qp

(q E

)

) = 0 for σ q

= [q, Q p

]

− 1 ;

(b) W corresponds to

˜

Q

p

(ψ) under the correspondence of Proposition 1.4;

(c) There is a deformation

˜

W of W and a commutative diagram of

˜

Qp[D]-modules

with exact row:

˜

Qp(1)

↪→

−−−→

˜

W



−−−→

˜

Qp(ψ)

mod ε





y mod ε





y





y mod ε

Q

p

(1) −−−→

↪→

W −−−→



Q

p

.

Normalize the Artin symbol [x, Q p

] so that

• N ([u, Q p

]) = u

− 1 for u ∈ Z

×

p

,

• [p, Q p

] is the arithmetic Frobenius element.

By an explicit form of Tate duality, we have 〈γ q

, φ〉 = φ(σ q

) for γ q

∈ H

1 (D, Q p

(1))

and φ ∈ Hom(D, Q p

) = H

1 (D, Q p

).

Proof. The case L = Qp is treated in [GS1] 2.3.4 and the gneral fact is in [H07].

For simplicity, we assume L = Qp. Since 〈γq , φ〉 = φ(σq) for φ ∈ H

1

(D, Qp) =

Hom(D, Qp) and γq ∈ H

1

(D, Qp(1)), applying these formulas to φ =

dψ

dt

, we get (a)

⇔ (b) by the definition of the correspondence in Proposition 1.4.

Here is the argument proving (b) ⇒ (c). Let ξQ be a 1-cocycle representing γQ for

Q = q E

. Then D × D 3 (σ, τ ) 7 → c(σ)

dψ

dt

(τ ) ∈ Q p

(1) is the 2-cocycle representing the

cup product γ Q

∪ [

˜

Q

p

(ψ)] (another expression of the Tate pairing), which vanishes by

(b) (⇔ (a)). Thus it is a 2-coboundary:

(1.1) ξQ(σ)

dψ

dt

(τ ) = ∂Ξ(σ, τ ) = Ξ(στ ) − N (σ)Ξ(τ ) − Ξ(σ)

(⇔ Ξ(στ ) = ξQ(σ)

dψ

dt

(τ ) + N (σ)Ξ(τ ) + Ξ(σ))

for a 1-chain Ξ : D → Q p

(1). Then defining an action of σ ∈ D on

˜

Q

p

2

via the matrix

multiplication by ˜ρ(σ) :=

(

N (σ) ξ Q

(σ)+Ξ(σ)ε

0 ψ(σ)

)

. One checks that this is well defined by

computation (the relation (1.1) shows up in the ε-term of ˜ρ(στ )

?

= ρ˜(σ)˜ρ(τ ) at the

shoulder). The resulting

˜

Q

p

[D]-module

˜

W fits well in the diagram in (c).

Conversely suppose we have the commutative diagram as in (c), which can be

written as the following commutative diagram with exact rows and columns:

0 0 0

↓ ↓ ↓

0 −→ Q

p

(1) −→ W −→ Q

p

−→ 0

↓ ↓ ↓

0 −→

˜

Qp(1) −→

˜

W −→

˜

Qp(ψ) −→ 0

↓ ↓ ↓

0 −→ Q

p

(1) −→ W −→ Q

p

−→ 0

↓ ↓ ↓

0 0 0

The connecting homomorphism d : H

1 (D, Q p

(1)) → H

2 (D, Q p

(1)) vanishes because

the leftmost vertical sequence splits. On the other hand, letting δ ψ

: H

0 (D, Q p

) →

H

1 (D, Q p

) stand for the connecting homomorphism of degree 0 coming from the

rightmost vertical sequence, and letting δ i

: H

i (D, Q p

) → H

i+ (D, Q p

(1)) be the

connecting homomorphism of degree i associated to the bottom row and the top row.

By the commutativity of the diagram, we get the following commutative square:

H

0

(D, Qp) = Qp

δ 0

−−−→ H

1

(D, Qp(1))

δ ψ





y





yd=

H

1 (D, Q p

) −−−→

δ 1

H

2 (D, Q p

(1)).

Since δψ(1) =

dψ

dt

, we confirm

dψ

dt

∈ Ker(δ 1 ). By Proposition 1.1, γQ is the in the

image of δ 0. Thus (a)/(b) follows if we can show that Ker(δ 1 ) is orthogonal to Im(δ 0 ).

Since W = TpE ⊗ Q, W is self dual under the canonical polarization pairing, which

induces a self duality of W and also the self (Cartier) duality of the exact sequence

0 → Q

p

(1)

ι

−→ W

π

−→ Q

p

→ 1. In particular the inclusion ι and the projection π are

mutually adjoint under the pairing. Thus the connecting maps δ 0

: H

0 (D, Q p

) →

H

1 (D, Q p

(1)) and δ 1

: H

1 (D, Q p

) → H

2 (D, Q p

(1)) are mutually adjoint each other

under the Tate duality pairing. In particular, Im(δ 0

) is orthogonal to Ker(δ 1

). 

1.2. How to relate the L-invariant with the logarithm of Tate period. Take

an elliptic curve E with multiplicative reduction over the finite extension L /Qp

. Let

W = T

p

E ⊗

Zp

Q

p

. Write Q i

= N

Fp i

/Qp

(q pi

).

Theorem 1.6 (L-invariant). If Deformation conjecture holds for ρ E

, then Sel F

(V ) =

0 and we have

L(Ad(ρ E

)) =

(

b ∏

i=

log p

(Q

i

)

ord p

(Q

i

)

)

L(1).

We have L(m) = 1 if b = e, and the value L(1) when b < e is given by

L(1) = det

(

∂δ pi

([p, F pi

])

∂Xj

)

i>b,j>b

∣

∣

∣

X 1 =X 2 =···=Xe=

∏

i>b

log p

(γp i

)

[Fp i

: Qp]αp i

([p, Fp i

])

for the local Artin symbol [p, F pi

].

Proof. In the proof, we continue to assume F pj

= Q

p

for all j = 1,... , e. Fix an

index j. Write D j

= Gal(F pj

/F

pj

)

∼

= D. We consider the universal couple (R,^ ρ) of

ρ E

under the conditions (K1–4). Put m j

:= (X

1

,... , X

j− 1

, X

2

j

, X

j+

,... , X

e

) ⊂ R =

K[[X

j

]]

j=1,...,e

for X j

= X

pj

. Consider

˜

W

j

= W/m j

W for W = ρ.

Suppose j ≤ b. We have a Dj -stable filtration 0 = F

2 ˜ Wj ⊂ F

1 ˜ Wj ⊂

˜

Wj = F

0 ˜ Wj.

Let δ j

be the nearly ordinary character

δ j

:= (δ j

mod (X 1

,... , X

j− 1

, X

2

j

, X

j+

,... , X

e

)).

The character δj satisfies δj ≡ αp j

= 1 mod (Xj ) for the trivial character 1 of D.

Since det(ρ) = N (K3), we have

˜

W

j

F

1 ˜

Wj

= δj

∼

=

˜

Qp and F

1 ˜ Wj =

˜

Qp(δ

− 1

j

N ).

The matrix form of the Dj -representation

˜

Wj is

(

δ

− 1

j

N ∗

0 δj

)

. Twist

˜

Wj by δj ; then,

˜

W

j

⊗ δ j

has the matrix form

(

N ∗

0 ψj

)

for ψ j

= δ

2

j

. Then

˜

W

j

⊗ ψ j

is an infinitesimal

extension making the following diagram commutative:

˜

Q

p

(1)

↪→

−−−→

˜

W

j

⊗ ψ j



−−−→

˜

Q

p

(ψ j

)





y





y





y

Q

p

(1) −−−→

↪→

W −−−→



Q

p

.

This diagram satisfies the condition (c) of Theorem 1.5, for Q j

= q E/Fp j

,

∂ψ j

∂X

j

∣

∣

∣

X j

=

([Q

j

, Q

p

]) = 2

(

δ j

∂ψ j

∂X

j

∣

∣

∣

X j

=

)

([Q

j

, Q

p

]) = 0 ⇒

∂δ j

∂X

j

∣

∣

∣

X j

=

([Q

j

, Q

p

]) = 0.

Write Qj = p

a

u for a = ordp(Qj ) and u ∈ Z

×

p

. Then log p

(u) = log p

(Qj ). We have

δ j

([Q

j

, F

pj

]) = δ j

([p, F pj

])

a δ j

([u, F pj

])

= δj ([p, Fp j

])

a

(1 + Xj )

− log p

(N ([u,Fp j

]))/ log p

(γp j

)

= δ j

([p, F pj

])

a

(1 + X j

)

− log p

(u)/ log p

(γp j

)

(because N ([u, F pj

]) = u

− 1 ). Differentiating this identity with respect to X j

, we get

from δj ([u, Fp j

])|X

j =^

= δj ([p, Fp j

])|X

j=^

= αj ([p, Fp j

]) = 1

a

∂δ j

∂Xj

∣

∣

∣

Xj =

([p, Fp j

]) −

log p

(u)

log p

(γj )

= 0.

From this we conclude

(1.2)

∂δ j

([p, F pj

])

∂X

j

∣

∣

∣

Xj =

log p

(γ j

)α pj

([p, F pj

])

− 1

=

log p

(Q

j

)

ord p

(Q

j

)

,

since αj ([p, Fp j

]) = 1 (by split multiplicative reduction of E at pj with j ≤ b).

As already seen, SelF (Ad(ρE )) = 0, assuming that R

∼

= K[[Xp]] p|p

. We will prove

the following factorization in the fourth lecture:

(1.3) L(Ad(ρ E

)) =

b ∏

i=

∂δ i

([p, F pi

])

∂X

i

∣

∣

∣

Xi=

log p

(γ pi

)α pi

([p, F pi

])

− 1

× det

(

∂δ i

([p, F pi

])

∂X

j

)

i>b,j>b

∣

∣

∣

X=

∏

j>b

log p

(γ pj

)α pj

([p, F pj

])

− 1

.

From this and (1.2), the desired formula follows. 

References

[Gr] R. Greenberg, Trivial zeros of p–adic L–functions, Contemporary Math. 165 (1994),

149–

[GS] R. Greenberg and G. Stevens, p-adic L–functions and p–adic periods of modular forms,

Inventiones Math. 111 (1993), 407–

[GS1] R. Greenberg and G. Stevens, On the conjecture of Mazur, Tate, and Teitelbaum, Con-

temporary Math. 165 (1994), 183–

[H07] H. Hida, L-invariants of Tate curves, 2006 (posted at www.math.ucla.edu/~hida)

Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, U.S.A.