Invariance of the parity conjecture for p-Selmer groups of elliptic curves in a $D_{2p^n}$-extension

In section 2, we show a $p$-parity result in a $D_{2p^n}$-extension of number fields $L\mathrm{/}K$ ($p\ge 5$) for the twist $1\oplus \eta \oplus \tau$: W(E/K,1\oplus \eta \oplus \tau)=(-1)^{< 1\oplus\eta \oplus \tau, X_{p}(E/L)>}, where $E$ is an elliptic curve over $K,$ $\eta$ and $\tau$ are respectively the quadratic character and an irreductible representation of degree 2 of $Gal(L\mathrm{/}K)=D_{2p^n},$ and $X_p(E\mathrm{/}L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $$-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).