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ofcBF[m,aF,G]under projection to the module HIw1 (Q(μm)⊗Qp,∞,F−−DV1×V2(F⊗G)∗)is zero.

Proof By taking the Vi sufficiently small, we may assume that F−−DV1×V2(F ⊗G)∗

is actually isomorphic to RA(α−1), whereα = αFαG andA = O(V1×V2), and that

α−1_<p1+h_{andα−1 is not a zero-divisor. It suffices, therefore, to show thatL} RA(α−1)

maps the image ofcBF[m,aF,G]to zero.

However, for each pair of integers (,)∈V1×V2with, 1+2hand such that F andG are not twists of each other, we know that the image ofL_RA(α−1₎(cBF[m,aF,G])

vanishes when restricted to (,)×W⊆Max(A)×W, by Proposition3.5.11. Since such pairs (,) are Zariski-dense in Max(A), the result follows.

Remark 7.1.3 Cf. [19, Lemma 8.1.5], which is an analogous (but rather stronger) statement in the ordinary case.

Hence the projection ofcBF[m,aF,G]toF−◦is in the image of the injection H_Iw1(Qp,∞,F−+DV1×V2(F⊗G)∗)→HIw1(Qp,∞,F−◦DV1×V2(F⊗G)∗).

SinceF+DV2(G)∗is isomorphic to an unramified module twisted by anA×-valued char-

acter of the cyclotomic Galois groupΓ, we may define a Perrin-Riou logarithm map for

F−+_{DV1×V2(F⊗G)}∗_{by reparametrising the corresponding map for its unramified twist,}

exactly as in Theorem 8.2.8 of [19]. That is, if we define D(F−+M(F⊗G)∗)=F−+D(F⊗G)∗(−1−κV2)

_Γ=1

,

which is free of rank 1 overO(V1×V2), then we obtain the following theorem: Theorem 7.1.4 There is an injective morphism ofO(V1×V2×W)-modules

L:H_Iw1(Qp,∞,F−+DV1×V2(F⊗G)∗)→D(F−+M(F⊗G)∗) ˆ⊗O(W),

with the following property: for all classical specialisations f, g ofF,G, and all characters ofΓ of the formτ =j+ηwithηof finite order and j∈Z, we have a commutative diagram

H_Iw1 Qp,∞,F−+DV_1×V2(F⊗G)∗ _-L _D(F−+ M(F⊗G)∗) ˆ⊗O(W) H1(Qp,F−+D(f ⊗g)∗(−j−η)) ? - _F−+_Dcris(M(f?_⊗g)∗_(−εg,p))

in which the bottom horizontal map is given by ⎧ ⎪ ⎨ ⎪ ⎩ 1− _αpj fβg 1− αfβg p1+j ₋1 if r=0 p1+j αfβg r G(ε)−1 if r>0 ⎫ ⎪ ⎬ ⎪ ⎭· ⎧ ⎨ ⎩ (−1)k−j (k−j)! log if jk, (j−k−1)! exp∗ if j>k,

where exp∗ and log are the Bloch–Kato dual-exponential and logarithm maps, ε is the finite-order character εg,p ·η−1 of Γ, r 0 is the conductor of ε, and G(ε) =

a∈(Z/pr_Z)×ε(a)ζ_paris the Gauss sum.

Proof The construction of the mapL is immediate from (6.2.1). The content of the theorem is that the mapLrecovers the maps exp∗and log for the specialisations ofFand G; this follows from Nakamura’s construction of exp∗and log for (ϕ,Γ)-modules. Theorem 7.1.5 (Explicit reciprocity law)If the Viare sufficiently small, then we have

"

LcBF[₁F_,1,G]

,η_F⊗ω_G#

=(c2−c−(k+k−2j)ε_F(c)−1ε_G(c)−1)(−1)1+jλN(F)−1Lp(F,G,1+j).

Here, Lp(F,G,1+j)denotes Urban’s3-variable p-adic L-function as constructed in[32], andεF andεG are the characters by which the prime-to-p diamond operators act onF andG.

Proof The two sides of the desired formula agree at every (k, k, j) withk ∈ V1,k ∈V2 and 0jmin(k, k), by [18, Theorem 6.5.9]. These points are manifestly Zariski-dense,

and the result follows.

Remark 7.1.6 The construction ofω_Gand the proof of the explicit reciprocity law are also valid ifGis a Coleman family passing through ap-stabilisationg_αof ap-regular weight 1 form, as in Theorem4.7.2; the only difference is that one may need to replaceV2with a finite flat covering ˜V2. In this setting,gα is automatically ordinary, soGis in fact a Hida family, and one can use the construction ofω_Ggiven in [19, Proposition 10.12.2]. 8 Bounding Selmer groups

8.1 Notation and hypotheses

Letf, g be cuspidal modular newforms of weightsk+2, k+2, respectively, and levels

Nf, Ngprime top. Wedopermit here the casek= −1. We suppose, however, thatk>k,

so in particulark0, and we choose an integerjsuch thatk+1jk. Ifj= k+₂k+1, then we assume thatεfεgis not trivial, whereεf andεgare the characters off andg.

As usual, we letE be a finite extension ofQpwith ring of integersO, containing the

coefficients off andg. Our goal will be to bound the Selmer group associated with the Galois representationM_O(f⊗g)(1+j), in terms of theL-valueL(f, g,1+j); our hypotheses on (k, k, j) are precisely those required to ensure that thisL-value is acriticalvalue.

It will be convenient to impose the following local assumptions atp:

• (p-regularity) We haveαf =βf andαg =βg, whereαf,βf are the roots of the Hecke

polynomial off atp, and similarly forg. • (no local zero) None of the pairwise products

αfαg,αfβg,βfαg,βfβg

is equal topjorp1+j, so the Euler factor ofL(f, g, s) atpdoes not vanish ats = jor

s=1+j.

• (nobility off_α) Iff is ordinary, then eitherαf is the unit root of the Hecke polynomial,

orME(f)|GQpis not the direct sum of two characters (so the eigenformfαis noble in

the sense of4.6.3).

• (nobility ofg_α andg_β) Ifk 0, thenME(g)|GQp does not split as a direct sum of

characters, so bothp-stabilisationsg_αandg_βare noble.

Remark 8.1.1 (1) In our arguments we will use bothp-stabilisationsg_αandg_βofg, but only the onep-stabilisationf_α off; in particular, we do not require that the other

p-stabilisationf_βbe noble.

(2) Note that the “no local zero” hypothesis is automatic, for weight reasons, unlessk+k

is even andj= k+₂k orj= k+₂k +1 (so theL-valueL(f, g,1+j) is a “near-central” value).

Thep-regularity hypothesis implies that we have direct sum decompositions Dcris(ME(f)∗)=Dcris(ME(f)∗)αf ⊕Dcris(ME(f)∗)βf

where ϕ acts on the two direct summands as multiplication byα_f−1,β_f−1, respectively, and similarly forg. This induces a decomposition ofDcris(ME(f ⊗g)∗) into four direct summandsDcris(ME(f ⊗g)∗)αfαg _etc.

Definition 8.1.2 We write