7 Beilinson-Flach Elements

7.1 Some Preliminaries

We write Zp[[Γ⁻]] = Zp[[T]], γ⁻ 7→ 1 +T. Recall g be the Hida family of normalized CM forms attached to characters ofΓKwith the coecient ring Λg :=Zp[[T]](the trivial character of ΓK is a

specialization of this family). We writeL_g for the fraction ring ofΛ_g. As in [30] letM(f)^∗ (M(g)^∗) be the part of the cohomology of the modular curves which is the Galois representation associated to f (g). The corresponding coecients for M(f)^∗ andM(g)^∗ is Qp and L_g. (Note that the Hida familyg is not quite a Hida family considered in loc.cit. It plays the role of a branchathere). Note also thatgis cuspidal (which is called generically non-Eisenstein in an earlier version) in the sense of [32]. We have M(g)^∗ is a rank two L_g vector space and, there is a short exact sequence of L_g vector spaces with G_Qp action:

0→Fg⁺ →M(g)^∗ →Fg⁻→0

withF_g^±being rank oneL_g vector spaces such that the Galois action onF_g⁻ is unramied. Sinceg is a CM family withp splits inK, the above exact sequence in fact splits asG_Qp. For an arithmetic specialization g_φ of g the Galois representation M(f)^∗ ⊗M(g_φ)^∗ is the induced representation from GK to G_Q of M(f)^∗ ⊗ξgφ where ξgφ is the Hecke character corresponding to gφ. This identication will be used implicitly later. We also write DdR(f) = (M(f)^∗⊗BdR)^GQp. We will write H_Iw¹ (K_∞,−) := lim←−K⊆K⁰⊆K∞H¹(K⁰,−).The transition map is given by co-restriction. For f let D_dR(f) be the Dieudonne module for M(f)^∗ and letη_f^∨ be any basis of Fil⁰D_dR(f). Letω_f^∨ be a basis of _Fil^D0^dRD_dR^(f)(f) such thathω^∨_f, ω_fi= 1.

7.2 Yager modules

We mainly follow [38] to present the theory of Yager modules. Let K/Qp be a nite unramied extension. For x ∈ O_K we dene y_K/Qp(x) = P

σ∈Gal(K/Qp)x^σ[σ] ∈ O_K[Gal(K/Qp)] (note our convention is slightly dierent from [38]). Let Q^urp /Qp be an unramied Zp-extension with Galois groupU. Then the above map induces an isomorphism of ΛO_F(U)-modules

y_Q^ur

p /Qp : lim←−

Qp⊆K⊆Q^urp

O_F ' S_Qur

p /Qp ={f ∈Zˆ^urp [[U]] :f^u = [u]f}

for any u∈U a topological generator. Here the superscript meansu acting on the coecient ring while[u]means multiplying by the group-like elementu⁻¹. The module S_Q^ur_{p /Qp} is called the Yager module. It is explained in loc.cit that the S_Q^ur

p /Qp is a free rank one module over Zp. Let F be a representation of U then they dened a map ρ : ˆZ^urp [[U]]→ Aut(F ⊗Zˆ^urp ) by mapping u to its action on F and extend linearly. As is noted in loc.cit the image of elements in the Yager module is in(F ⊗Zˆ^urp )^GQp. Recall also that

d:= lim←−

m

d_m ∈lim←−

m

O^×_km

dened in the proof of Lemma 6.2 is a generator of the Yager module for Qp. Then we can dene ρ(d) and let ρ(d)^∨ be the element in Zˆ^urp [[U]] which is the inverse oflim←−m

P

σ∈U/p^mUd^σ_m·σ⁻¹. We have the following

Lemma 7.1. (1)

1 lim←−m

P

σ∈U/p^mUd^σ_m·σ⁻¹ ∈ S_∞.

(2) Proof. Straightforward computation on the Galois action.

7.3 Beilinson-Flach elements

Unlike Kato's zeta element, the Beilinson-Flach elements constructed in [30] are not in the Iwasawa cohomology (in fact they are unbounded classes). So we need to construct from them a bounded family of classes. Our construction can be viewed as a Galois cohomology analogue of Pollack's construction of the ±p-adic L-function.

We rst dene Our H_r is the Mellin Transform

Z

t∈Zp

(1 +X)^tdµ

of r-admissible distributions dµ dened in loc.cit. Let H_r,0 := Zp[[Y]]⊗ H_r(Xv0). We also dene H_0,r to be the completed tensor productZp[[Y]] ˆ⊗H_r(X_v¯0)with respect to the obvious norm onZp[[X]]

and the norm ofH_rmentioned above (note that the denitions forH_r,0 andH_0,r are not symmetric).

We see that log⁻_v0 ∈ H_r,0 and log⁻_¯v0 ∈ H_0,r. In [36] the authors dened Beilinson-Flach elements BFα and BF−α for fα and f−α, as elements inH_r,0⊗H_Iw¹ (Q∞, M(f)^∗⊗M(g)^∗). It is easily seen that the module H_Iw¹ (Q∞, M(f)^∗⊗M(g)^∗) can be identied withH_Iw¹ (K_∞, M(f)^∗).

Now we recall some notations in [43]. Let ES_p(DK) := lim←−rH¹(X₁(DKp^r) ⊗ Q¯,Zp) and GESp(DK) := lim←−rH¹(Y1(DKp^r)⊗Q¯,Zp) which are modules equipped with Galois action of G_Q. HereX₁(DKp^r)andY₁(D_Kp^r)are corresponding compact and non-compact modular curves. Recall in loc.cit there is an ordinary idempotent e^∗ associated to the covariant Hecke operator Up. Let A^∗_∞=e^∗ESp(DK)^Ip =e^∗GESp(DK)^Ip (see the Theorem in loc.cit). LetB^∗_∞(B˜^∗_∞) be the quotient of e^∗ES_p(DK) (e^∗GES_p(DK)) overA^∗_∞.

In an earlier version of [32] the authors dened elements ω_g^∨ ∈(Fg⁺(χ⁻¹_g )⊗Zˆ^urp )^GQp and η^∨_g ∈ (F_g⁻⊗Zˆ^ur_p )^GQp. Here the χg is the central character for g. We briey recall the denitions since

they are more convenient for our use (these notions are replaced by their dual in the current version of [32]). In the natural isomorphism

A^∗_∞⊗_Zp[[T]]Zˆ^urp [[T]]'Hom_ˆ

Z^urp (S^ord(DK, χK,Zˆ^urp [[T]]),Zˆ^urp [[T]])

(see the proof of [43, Corollary 2.3.6], the ω^∨_g is corresponds to the functional which maps each normalized eigenform to1. On the other handη_g^∨ is dened to be the element inB^∗_∞which, under the pairing in [43, Theorem 2.3.5], pairs with ω_g^∨ to the product of local root numbers at prime to p places ofg. This product movesp-adic analytically and is a unit.

Let v₁, v₂ be a Λ basis ofH¹(K_v¯0, M(f)^∗⊗Λ(−Ψ)). Then there aref₁, f₂ ∈ H_r(X_v0)⊗_Zp[[Xv

0]]

Zp[[X_v0, T]]and some f₀ ∈Frac(Zp[[T]])\{0}such thatBF_α−BF−α =α·f₀(f₁v₁+f₂v₂). Let L=L^G_V

f :H¹(G_Qp, M(f)^∗⊗Λ(−Ψ))→ ⊕^p_i=1^t (H_0,r⊗D_cris(V_f))·γ_i

be the regulator map dened in [38, Theorem 4.7]. (We know L(v_i) ∈ H_0,r by [38, Proposition 4.8]). We write Pr^α and Pr^−α for the projection map from D_cris(V_f) to the α or −α eigenspace for Frobenius action ϕ (as numbers, with respect to the basis given by the image of the Neron dierentialω_E in the±α-eigenspaces ofDcris(V_f)). Let

L⁺= Pr^α−Pr^−α

2α ◦ L, L⁺ = Pr^α+ Pr^−α

2 ◦ L.

Then by Proposition 7.5 in the following, we have

f₀f₁L⁺(v₁) +f₀f₂L⁺(v₂) = log⁻_v0log⁻_¯v0L⁺⁺_f,p, f0f1L⁻(v1) +f0f2L⁻(v2) = log⁻_v0log⁺_¯v0L⁺⁻_f,p. We need the following

Lemma 7.3. The L⁺⁻_f,p and L⁻⁺_f,p are not identically zero.

Proof. We just need to know that the LK(E, χ,1) is non zero for some character χ of ΓK whose conductor atv0 is a even power ofp and whose conductor at ¯v0 is an odd power of p. This is just [51, Theorem 2].

We have the following

Lemma 7.4. We havef1, f2∈log⁻_v0·Frac(Z_p[[Xv0, T]]).

Proof. We rst claim thatdet

L⁺(v₁) L⁺(v₂) L⁻(v1) L⁻(v2)

is not identically zero. Suppose it is not the case.

Then we have

log⁻_¯v0L⁺⁺_f,p · L⁻(v1)−log⁺_¯v0L⁺⁻_f,p · L⁺(v1) = 0, log⁻_¯v0L⁺⁺_f,p · L⁻(v2)−log⁺_¯v0L⁺⁻_f,p · L⁺(v2) = 0.

Letζ₁,· · ·, ζ_s be the zeros oflog⁻_p such thatX_v¯0 −ζ_i is a divisor of L⁺⁻_f,p (easily seen to be a nite set sinceL⁺⁻_f,p is not identically zero). Then for any other rootζ oflog⁻_p,L⁺(v1)restricts to the zero function at the lineX¯v0 =ζ. If we expandL⁺(v1)as a power series inX¯v0 andU, then by Weierstrass

preparation theorem (see [70, Theorem 7.3]), the coecient for U^m (any m) is log⁻_¯v0/Q

(X_¯v0 −ζ_i) times some element in Zp[[X_¯v0]] ⊗_Zp Qp whose coecients are uniformly bounded (bound also independent of m). (See [48, Proof of Theorem 5.1]). Thus L⁺(v1)∈ log⁻_¯v0·Frac(Z_p[[X¯v0, U]]) We can apply the same argument to all L^±(v_i) and get

L⁺(vi)∈log⁻_v¯0·Frac(Zp[[Xv¯0, U]]), L⁻(vi)∈log⁺_v¯0·Frac(Z_p[[Xv¯0, U]]).

But det

L⁺(v1) L⁺(v2) L⁻(v₁) L⁻(v₂)

= 0. This contradicts the fact that L is injective, which is proved in [38, Proposition 4.11].

Now let us return to the proof of the lemma. Fix k= 1or 2. If fk is identically 0 then nothing is needed. If not, recall we xed representatives γ1,· · · , γ_p^t of ΓK/Γp. Then from the claim there is an a∈Cp,|a|_p <1such that

06=fk|_X¯v

0=a∈ ⊕^p_i=1^t log⁻_v0·Frac(Z_p[[Xv0]])·γi. (6) It is possible to writef_k=P

jf_kj(X_v0)·g_kj(X_v0, T)(nite sum) wheref_kj(X_v0)∈ H_r(X_v0)⊗

Z^p[[Xv0]]

Frac(Zp[[X_v0]]) and g_kj(X_v0), T) ∈ Zp[[X_v0, T]] such that either f_k1(X_v0) ∈ log⁻_v0·Frac(Zp[[X_v0]]) or {log⁻_v0} ∪ {f_kj} forms a linearly independent set overFrac(Zp[[Xv0]]). Then (6) implies we must have j= 1 andf_k1(Xv0)∈log⁻_v0·Frac(Zp[[Xv0]]). Thus the lemma is true.

So there is an element 06=h ∈Zp[[Xv0, Y]] such that h·fi ∈log⁻_v0·Λ. We dene the bounded cohomology class

BF⁺:= f₁h

2 log⁻_v0v₁+ f₂h

2 log⁻_v0v₂∈H_Iw¹ (K_∞, M(f)^∗) (7) It follows from that the Galois cohomology image of Beilinson-Flach element is geometric that the BF⁺maps toH₊¹(Gv0, M(f)^∗⊗Λ(−Ψ))⊆H¹(Gv0, M(f)^∗⊗Λ(−Ψ)), since for any arithmetic point φsuch thatlog⁻_v0|_φ6= 0, the class is in the nite partH_f¹(Gv0,−).

We take basis v^± of Fg^± with respect to which ω_g^∨ and η_g^∨ are ρ(d)^∨v⁺ and ρ(d)v⁻ (see the discussion for Yager Modules). We use this basis to giveΛ_g-integral structure forM(g)^∗. With this integral structure we can talk about specializingBF⁺ to arithmetic points φ, provided we remove the set ofφ's in a lower dimensional subspace (the zeroes of the denominator forBF⁺ with respect to the basis). The following propositions are proved in [30]. Let Ψ_g be the Λ_g-valued Galois char-acter of GK corresponding to the Galois representation associated to g (i.e. Ind^G_G^K

QΨ_g = M(g)^∗).

Since psplits as v₀v¯₀ inK, there is a canonical identication(Ind^G_G^K

QΨ_g)|_G

Qp 'Ψ_g|_GK

v0 ⊕Ψ_g|_GK

¯ v0

and can take aΛ_g-basis of the right side as{v, c·v}wherecis the complex conjugation. (Note that there are two choices for theΨg and we choose the one so thatΨg|_GK

v0 corresponds toFg⁻).

Convention: we use the basis{v⁺, c·v⁺}to identify the Galois representation ofgwith the induced representationInd^K_G

QΨg.

In the following we dene φin a generic set of arithmetic points corresponding to a nite order character of ΓK to mean all suchφoutside a proper closed sub-scheme of SpecΛ.

Proposition 7.5. For some H₀ ∈ Q¯^×p and φ in a generic set of arithmetic points corresponding to a primitive character of Γ_n×U_m with n an even number (as local Galois group at v¯₀), for any α, β∈ {±√

−p},

H0·Pr^Fg+Pr^β(exp^∗(φ(BFα))) = φ(L_p,α,β)ε(χ⁻¹_φ )

(−p)ⁿ² η_f,β^∨ ⊗φ(ω_g^∨).

Here Pr^β and η_f,β^∨ denote projecting to the β-eigenspace of D_dR(f), exp^∗ is the Bloch-Kato dual exponential map. Theχφmeans composing Ψwith φ. By class eld theory the χφ can be considered as a character ofQ^×_p. For thePr^Fg+, we recall thatM(g)^∗ is split as the direct sum ofF_g⁺andF_g⁻ as Galois modules which are rank one vector spaces overL_g. So if we exclude the set ofφ's in a lower dimensional space it makes sense to talk about projection to (Fg⁻⊗Zˆ^urp )^GQp or(Fg⁺(χ⁻¹_g )⊗Zˆ^urp )^GQp components at φ.

Proof. It follows from the explicit reciprocity law in [30, Theorem 7.1.4, Theorem 7.1.5] together with the interpolation property of the big regulator map [38, Theorem 4.15]. Note thatαⁿ=βⁿ= (−p)ⁿ² and that the βg in loc.cit corresponds to the χ_φ(u)ⁿ part (u being the arithmetic Frobenius) of ε(χ⁻¹_φ ).

The proposition has the following corollary using Proposition 6.11.

Corollary 7.6. We use the convention before Proposition 7.5. Then for some H₁∈Q^×p we have Col⁺_v¯0(BF⁺) =h·H1·L⁺⁺_p .

(the h is dened in (7)).

Proof. First recall that ω^∨_g is the ρ(d)^∨v⁺ for the basis v^± we have chosen. If we take H1 to be

Ω⁺_EΩ⁻_E

hf,fi times some element inQ¯^×p (recallη^∨_f is dened up to a scalar). Then the corollary follows.

Proposition 7.7. There is a non-zero element 06=H₂∈Zˆ^urp [[T]], such that for φ in a generic set of arithmetic points corresponding to a primitive character ofΓn×Um (as Galois group atv0) with n an even number and for α∈ {±√

−p},

φ(H₂)Pr^Fg−log_v0φ(BF_α) = 1

α·(−p)ⁿ² φ(L²_f,K)·ε(χ⁻¹_φ )ω_f^∨⊗φ(η_g^∨).

Here log_v0 is the Bloch-Kato logarithm map atv₀.

Proof. This again follows from [30, Theorem 7.1.4, Theorem 7.1.5]. Note that the arithmetic points at which the interpolation formulas are proved there are not quite theφ's considered here. In fact those points in loc.cit correspond to the product of a nite order character ofΓand some character of U which is not of nite order. We may use the lemma below to get the result we need. We also need to compare the p-adic L-function in loc.cit with the one in [67]. In [67] we used the Σ-primitivep-adicL-function which is inΛfor Σa nite set of primes. The originalp-adicL-function is obtained by putting back the Euler factors at Σ. We only know a priory it is in the fraction eld of Λ. There is another construction of this p-adic L-function L^{U rban}_f,K ∈ Λ⊗_Λg L_g by E.Urban [66]

using Rankin-Selberg method. This is thep-adic L-function used in [30]. However the period there is the Petersson inner product of the normalized eigenforms in g instead of the CM period. The ratio of these periods is given byhK· L^Katz_K ∈Zˆ^ur_p [[T]](see [22]). So we may choosehK· L^Katz_K times some constant inQ¯^×p as theH2. The proposition follows.

RecallF_n is the unramied extension ofQp of degree pⁿ.

Lemma 7.8. Let χbe a primitive character of Γ/pⁿΓn and let Kχ be the eld obtained fromQp by joining in the coecient ring of χ. We let

X

τ∈Γ/pⁿΓ

ζ^τω_f ⊗χ(τ) :=ω_f,χ ∈ D_dR^Kχ(T) Fil⁰D^K_dR^χ(T)

⊗_QpK_χ.

Thelim←−nH_f¹(Fn, T(χ))is free of rank one overO_Kχ[[U]]. Moreover iflim←−nH_f¹(Fn, T(χ))→(lim←−nO_Fn· ω_f,χ) ⊗_Zp Qp is a homomorphism of O_Kχ[[U]]-modules and for a Zariski dense set points φ ∈ SpecO_Kχ[[U]] with the associated unramied Galois character denoted as ρφ, the specialization to φ:O_Kχ[[U]]→Cp of this map is the Bloch-Kato logarithm map

H_f¹(Qp, T(χρ_φ))→ D_dR(T(χρ_φ)) Fil⁰DdR(T(χρφ)).

Proof. Let Kχ,n be the composed eld of Fn and Kχ and let Kχ,∞ be the union of all Kχ,n and Oˆ_Kχ,∞ be thep-adic completion of the integer ring ofKχ,∞. We rst observe that there is an integer m⁰⁰such that the image ofOˆ_Kχ,n·ω_f under the Bloch-Kato exponential map lies inp^−m00H⁽Kχ,∞, T). This follows from the explicit formula for the logarithm map of the formal groupE ofEˆ. From this we know there is an integer m⁰ with

exp : ˆO_F∞·ω_f,χ→p^−m0H¹(F∞, T(χ)). (8) and

lim←−

n

O_Fn·ω_f,χ→p^−m0lim←−

n

H¹(Fn, T(χ)).

Suppose ρ is an unramied character of G_Qp with ρ(u) = 1 +m so that |m|_p < 1 is in a nite extensionL/Qp. Recall thatd∈lim←−nO_Fn andρ(d) := lim←−n

P

σ∈U/pⁿUρ(σ)d^σ_n. Dene ω_f,χρ:=ρ(d)·ω_f,χ = lim←−

n

X

σ

d^σ_n·ω_f,χ⊗ρ(σ)∈(lim←−

n

O_Fn)ω_f ⊗_ZpO_Lχ for Lχ=Kχ(m). Then the boundedness (8) implies

ρ(d)·ωf,χ →lim←−

n

X

σ∈U/pⁿU

ρ(σ) exp(d^σ_n·ωf,χ)

gives the Bloch-Kato exponential map forV(χρ). Now it follows from the fact that the exponential mpa for V(χρ) is an isomorphism from Qpω_f,χρ⊗_Qp L_χ to H_f¹(Qp, V(χρ)) and some elementary theory of O_Kχ[[U]]-module structures that there is an integerm such that

exp(lim←−

n

O_Fn·ω_f,χ)

isp^mtimes a rank oneO_Kχ[[U]]-direct summand of the free rank twoO_Kχ[[U]]modulelim←−nH¹(F_n, T(χ)). These altogether give the lemma.

Corollary 7.9. We use the convention before Proposition 7.5. Then for some06=H₃ ∈FracΛ_g we have

LOG⁺_v0(BF⁺) =h·H3·(L²_f,K).

Proof. TakeH₃ as−_H¹

2 ·_c·v^v−₊ and use Proposition 6.11.

Remark 7.10. Both H2 and _c·v^v−+ are elements in the fraction eld of Λg. In the next section we are going to carefully study them to get a rened main theorem. In particular we will use Rubin's work on CM main conjecture to prove that 1/H₃ is almost integral.

Dans le document IWASAWA Main Conjecture for Supersingular Elliptic curves (Page 36-43)