Invariance of the parity conjecture for \(p\)-Selmer groups of elliptic curves in a \(D_{2p^n}\)-extension (Q2880704)

Let \(K\) be a number field and let \(E\) be an elliptic curve over \(K\). For a prime number \(p\), set \(X_{p}(E/K)=\text{Hom}_{{\mathbb Z}_{p}}(S(E/K, p^{\infty}), {\mathbb Q}_{p}/{\mathbb Z}_{p})\otimes_{{\mathbb Z}_{p}}{\mathbb Q}_{p}\), where \(S(E/K, p^{\infty})\) is the \(p^{\infty}\)-Selmer group of \(E/K\), and let \(\text{rank}_{p}(E/K)\) denote its dimension as a \({\mathbb Q}_{p}\)-vector space. The \textit{\(p\)-parity conjecture} is the assertion that \((-1)^{\text{rank}_{p}(E/K)}=W(E/K)\), where \(W(E/K)\) is the ``root number'' of \(E/K\). The latter is the product over all primes \(v\) of \(K\) of ``local root numbers'' \(W(\sigma_{E/K_{v}}^{\prime})\) attached to certain linear representations \(\sigma_{E/K_{v}}^{\prime}\) of \(G_{v}=\text{Gal}(\overline{K}_{v}/K_{v})\). The above conjecture admits the following equivariant form. Let \(L/K\) be a finite Galois extension and let \(\tau\) be a self-dual \(\overline{\mathbb Q}_{p}\)-representation of \(\text{Gal}(L/K)\). Then NEWLINE\[NEWLINE (-1)^{\langle \tau,X_{p}(E/L)\rangle}=W(E/K,\tau), NEWLINE\]NEWLINE where \(W(E/K,\tau)=\prod_{v}W(\sigma_{E/K_{v}}^{\prime}\otimes \text{res}_{G_{v}}\tau)\) and \(\langle \tau,X_{p}(E/L)\rangle\) is the usual representation-theoretic inner product of \(\tau\) and the complexification of \(X_{p}(E/L)\). The paper under review discusses this last conjecture in a particular case. More precisely, assume that \(\text{Gal}(L/K)\simeq D_{2p^{n}}\) is dihedral, where \(p\geq 5\), and let \(\eta\) be the quadratic character of \(D_{2p^{n}}\). Then the author shows that the \(p\)-parity conjecture for \(E/K\) tensored by \(1\otimes\eta\otimes\tau\) holds, i.e., NEWLINE\[NEWLINE (-1)^{\langle 1\otimes\eta\otimes\tau,X_{p}(E/L)\rangle}= W(E/K,1\otimes\eta\otimes\tau). NEWLINE\]NEWLINE The proof proceeds via a preliminary reduction to the case \(n=1\) (using the Galois invariance of root numbers due to D. Rohrlich) and then analyzing several subcases according to whether \(G_{v}\) is \(1, D_{2}, \mathbb Z/p\) or all of \(D_{2p}\). Regarding the actual proof, the author comments that ``the main novelty [of our proof] is that we use a congruence result between \(\varepsilon_{0}\)-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above \(2\) and \(3\))''. The paper also contains an Appendix, whose purpose is to make a small improvement upon Theorem 6.7 in: [\textit{T. Dokchitser} and \textit{V. Dokchitser}, ``Root numbers and parity of ranks of elliptic curves'', J. Reine Angew. Math. 658, 39--64 (2011; Zbl 1314.11041)].