# Advanced analytic number theory by Carl Ludwig Siegel

By Carl Ludwig Siegel

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The cokernel of this map is isomorphic to H 2 Δη , H 0 (K∞,η , E v [p∞ ]) = H 2 Δη , E v (kη )[p∞ ] . This cohomology group obviously vanishes if E v (kη )[p] = 0 or, equivalently, as we pointed out in the introduction, if v is non-anomalous for E/K. Under that assumption, it follows that βv is surjective. Σ 3. PROJECTIVITY OR QUASI-PROJECTIVITY OF XE 0 (K∞ ). 34 The map βv will also be surjective if we assume that p ev (K/F ). To see this, let Υη denote the inertia subgroup of Δη , whose order will also be prime to p.

If those primes are real, then the primes η of K∞ lying above v are also real and hence v splits completely in K∞ /F . Thus, Hv (K∞ , E) is isomorphic to H 1 (K∞,η , E[p∞ ]) ⊗Zp Zp [Δ], and this is cohomologically trivial. H. Vanishing of H i Δ, Hv (K∞ , E) for v ∈ Σp and i ≥ 1. We need to assume that either v is non-anomalous for E/K or that p ev (K∞ /F∞ ). Again, Δ permutes the primes of K∞ above v and so one must prove that H 1 Δη , H 1 (K∞,η , E v [p∞ ]) = 0 for any η|v. Just as before, one can use the Hochschild-Serre spectral sequence 36 Σ 3.

PROJECTIVE AND QUASI-PROJECTIVE MODULES. The following result gives a direct relationship between the cohomological triviality (with respect to Δ) of a ΛG -module (or its dual) and the projective dimension of the module (as a ΛG -module), speciﬁcally whether or not that projective dimension is 1. A rather diﬀerent proof of such a relationship (at least in the case where G is abelian) can be found in a paper by Greither. 4 in [Gre]. 1. Suppose that X is a ﬁnitely-generated ΛG -module. Assume that X is a torsion Λ-module and that X has no nonzero, ﬁnite Λ-submodules.