L-values of elliptic curves twisted by cubic characters

Abstract

Given a rational elliptic curve $E$ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $\chi$ of order $q$, and in many cases its associated central algebraic L-value $\mathcal{L}(E,\chi )$ is known to be integral. This paper derives some arithmetic consequences from a congruence between $\mathcal{L}(E,1)$ and $\mathcal{L}(E,\chi )$ arising from this integrality, with an emphasis on cubic characters $\chi$. These include $q$-adic valuations of the denominator of $\mathcal{L}(E,1)$, determination of $\mathcal{L}(E,\chi )$ in terms of Birch--Swinnerton-Dyer invariants, and asymptotic densities of $\mathcal{L}(E,\chi )$ modulo $q$ by varying $\chi$.

20 pages