Research proof index

For any subgroup H of G, the connecting hommorphism in the splitting module long exact sequence H¹(H,aug) ⟶ H²(H,M) is an isomorphism.

Given a 2-cocycle σ, the image of σ in the splitting module of σ is equal to the coboundary of τ σ.

To check that a finite group has trivial Tate cohomology, it's enough to show it has trivial cohomology and trivial homology, and that the 0-th and -1st Tate cohomology groups are trivial.

The analytic form of the main theorem (Silverman, Advanced topics, Theorem V.1.1(a)): for 0 < ‖q‖ < ‖u‖ < 1, Yₐ² + XₐYₐ = Xₐ³ - 5s₃(q)Xₐ - (5s₃(q) + 7s₅(q))/12. Proof sketch: the hypotheses ensure u ∉ qᶻ, and we may choose z, τ with e z = u, e τ = q, 0 < im z < im τ (so z ∉ Λ_τ). Substitute the four q-expansions into the differential...

The q-expansion of ℘' (Silverman, Advanced topics, Theorem I.6.2): under the hypotheses of weierstrassP_q_expansion, ℘'(z; Λ_τ) = (2πi)³ (Xₐ(e z, e τ) + 2Yₐ(e z, e τ)). Proof: as for weierstrassP_q_expansion, but simpler: group the absolutely convergent sum ℘'(z) = -2∑_ω (z - ω)⁻³ into rows ω = nτ + m (no regularising terms are needed here...

The descent lemma: a formal power series F ∈ ℚ(u)⟦q⟧ vanishes provided that, for infinitely many u₀ : ℂ, the evaluated series ∑ₙ Fₙ(u₀)q₀ⁿ converges with sum 0 for all sufficiently small nonzero q₀. Proof sketch: fix u₀. The function q₀ ↦ ∑ₙ Fₙ(u₀)q₀ⁿ is analytic on ‖q₀‖ < r (a power series converging pointwise on a disc is analytic there)...