Invariance of the parity conjecture for p-Selmer groups of elliptic curves in a D₂ₚₙ-extension

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In section 2, we show a p-parity result in a D₂ₚⁿ-extension of number fields L/K (p≥ 5) for the twist 1⊕ η⊕ τ: W(E/K,1⊕ η⊕ τ)=(-1)< ¹⊕η⊕ τ, ˣₚ⁽ᵉ/ˡ⁾>, where E is an elliptic curve over K, η and τ are respectively the quadratic character and an irreductible representation of degree 2 of Gal(L/K)=D₂ₚⁿ, and Xₚ(E/L) is the p-Selmer group. The main novelty is that we use a congruence result between % ε₀-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the p-parity conjecture (using the machinery of the Dokchitser brothers).