Comparing local constants of elliptic curves in dihedral extensions

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.

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