Lemma 58 ([Projected Kummer lines]). At each v=w, w there is a projected Kummer condition U_v , a split free line in H¹(K_v,M)[-1] (placed in degree one) mapping into RΓ(K_v,M) , over A . Here H¹(K_v,M) and the full complex refer to the …

We construct the lines and prove exact orthogonality before defining their logarithmic coordinates. local Shapiro tower. Put F_v=K_v=ℚ_2 , and let D_j/F_v here denote the unramified extension of degree 2^j (field notation at v ). In the present case the local θ -field P_v/F_v is cyclic totally ramified. The u -action is unramified with Frobenius value (1+u)^b for v_2(b)=1 . Use first the Shapiro scalar Θ[Gal(P_v/F_v)] [[Gal(D_∞/F_v)]] with tautological character, 1+s_0 the Frobenius generator in the unramified factor. Local cohomology is inverse cohomology over H_j^ loc =P_vD_j with corestriction. Project the finite factor by [σ]↦θ(σ) (idempotent splitting after inverting 2) and substitute 1+s_0=(1+u)^b , a finite flat change by Weierstrass or a cyclic-generator change. This computes exactly the desired diagram after these changes: fixed local groups/actions are available by stabilization at each finite precision, and Shapiro, free coefficient models and the fixed projector denominator apply there and in compact inverse limits. More generally for several conductor variables unramified at v , split θ if needed locally as a totally ramified cyclic-field character θ_P (on P_v ) times a finite unramified θ_U , by the abelian local description. Use the finite factor for θ_P , and substitute for 1+s_0 the product of θ_U(Fr_v) and the conductor-variable Frobenius powers. Provided some exponent is nonzero, the change is again flat, finite flat after adjoining the other variables. bounds for logarithmic lattices. At local finite levels the integral Kummer sequence on H^1 with Tate coefficients has ends A(H_j^ loc )^∧_2 and Hom_ ℤ_2 ( A^∨(H_j^ loc )^∧_2,ℤ_2) for the abelian variety A=E , or on full modules before a newform projection; identify via polarization allowing fixed denominators for other varieties. Transitions are norm and dual restriction. Compactness permits exact limits. After finite-character projection the log and dual lattices compare to corresponding additive lattices in the Lie spaces and their trace duals up to bounded 2-powers uniformly in j , and projected log torsion kernel has bounded exponent (one can use the unnormalized projector). Both θ_P^ ±1 have this property. Here and below, ``bounded'' means independent of the unramified degree 2^j . The constants may depend on the fixed character θ , its ramified field P_v , the fixed extension used for semiabelian reduction, the abelian variety, and its coefficient projector. Uniformity as θ varies is not needed for these characteristic-zero tests. Indeed a uniform small lattice exponentiates by fixed ramification, using a model first over P_v , then its unramified extensions. For the upper and torsion bounds extend to the compositum with a fixed Galois L/F_v of semiabelian reduction. Component exponents there are bounded since after LP_v the extensions are unramified. An inertia element ν over L , chosen with θ_P(ν)≠1 , acts trivially on geometric points of the special identity component there. To see this last fact one can use the special-fiber maps from the Néron model over L . On identity components they are surjective on geometric points: on auxiliary odd-adic Tate modules rationally they identify inertia invariants (Néron reduction over maximal unramified fields is surjective with odd-divisible formal kernel without odd torsion, and components are finite), unchanged under finite base extension by semistable unipotent inertia. Both components are semiabelian by the semistable reduction criterion, so these Tate images suffice for surjectivity. Thus ν indeed fixes the special points by functoriality from L . After a common component multiple, (ν-1) of any point therefore enters the formal group, and a uniform further multiple enters the small log range. This proves both upper and torsion bounds on applying the finite projectors (their eigenvalue difference from 1 costs only a fixed denominator). The dual log comparison uses the dual of the torsion-free log lattice, and the different denominator is bounded. bases and free local cohomology. Now O_ H_j^ loc = O_ P_v ⊗_ ℤ_2 O_ D_j . Use trace-compatible integral normal bases on the unramified tower. More explicitly, choose β_0=1 and choose successively β_j∈ O_ D_j with Tr_ D_j/D_ j-1 (β_j)=β_ j-1 . The unramified trace is surjective, so this is possible. The total trace of β_j is a unit. Modulo 2 , the trace operator for the cyclic group of order 2^j is the top power of its generator minus one. Nonzero total trace therefore makes β_j a generator of the residue-field module over the cyclic group algebra. Lifting gives an integral normal basis, and the chosen bases are trace-compatible. Use also a rational normal basis on the fixed finite factor. The projected norm-limit Kummer terms on both ends therefore become free via log and its dual after inverting 2 . On the dual end, use the trace-dual lattice; a self-dual normal basis is unnecessary. The transition dual to inclusion is again trace in additive coordinates, and the different of the fixed ramified extension contributes only a fixed denominator. Sandwiching the lattices between fixed multiples is compatible with these limits by compactness. Degree zero of the full cohomology upstairs vanishes, and the projected degree-two inverse cohomology vanishes after inverting 2 by local duality and the uniform projected torsion bounds. For an elliptic plane the two free ends have rank one each. The same holds on each Hecke-field plane and differential embedding for its modular abelian variety, by characteristic-zero projection (all projector, isogeny and polarization losses there are fixed). Thus H^1 is itself split free with a direct Kummer line: its two free rank-one ends form an exact sequence, and the quotient is projective. orthogonality. Cup on conjugately paired Kummer lines vanishes, by Shapiro on finite group quotients using ordinary Kummer cups of translates, then on the inverse limits and under the scalar changes. Hence by ranks and perfect local duality this gives exact orthogonality, also on field tests. The restrictions from the split degree-one lines admit isotropy nullhomotopies, so this is an assertion about the local complexes and their comparison triangles, rather than just the dimensions of their generic cohomology.