Comparing local constants of elliptic curves in dihedral extensions

TL;DR

This paper investigates the relationship between analytic and arithmetic local constants of elliptic curves in dihedral extensions, providing new calculations and extending bounds on p-Selmer rank growth.

Contribution

It establishes the connection between local constants and root numbers for a broad class of elliptic curves and extends results on p-Selmer rank growth in dihedral extensions.

Findings

Confirmed the link between arithmetic and analytic local constants in various cases.

Extended the class of elliptic curves with known lower bounds for p-Selmer rank growth.

Calculated arithmetic local constants in new cases, supporting the parity conjecture.

Abstract

In this paper, we study the theories of analytic and arithmetic local constants of elliptic curves, with the work of Rohrlich, for the former, and the work of Mazur and Rubin, for the latter, as a basis. With the Parity Conjecture as motivation, one expects that the arithmetic local constants should be the algebraic additive counterparts to ratios of local analytic root numbers. We calculate the constants on both sides in various cases, establishing this connection for a substantial class of elliptic curves. By calculating the arithmetic constants in some new cases, we also extend the class of elliptic curves for which one can determine lower bounds for the growth of p-Selmer rank in dihedral extensions of number fields.