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

TL;DR

This paper proves a parity result for p-Selmer groups of elliptic curves over certain number field extensions, using local root number calculations and congruences, advancing understanding of the p-parity conjecture.

Contribution

The main novelty is applying Deligne's epsilon factor congruences to determine local root numbers in complex cases, extending parity results to D_{2p^n}-extensions.

Findings

Established p-parity results for elliptic curves in D_{2p^n}-extensions.

Utilized epsilon factor congruences for local root number calculations.

Applied results to the broader p-parity conjecture using Dokchitser's machinery.

Abstract

In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) 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 $Gal(L/K)=D_{2p^{n}},$ and $X_{p}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $% \epsilon_{0}$-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).