Integrality of twisted $L$-values of elliptic curves

Abstract

Under suitable, fairly weak hypotheses on an elliptic curve $E/\mathbb{Q}$ and a primitive non-trivial Dirichlet character $\chi$, we show that the algebraic $L$-value $\mathcal{L}(E,\chi )$ at $s=1$ is an algebraic integer. For instance, for semistable curves $\mathcal{L}(E,\chi )$ is integral whenever $E$ admits no isogenies defined over $\mathbb{Q}$. Moreover we give examples illustrating that our hypotheses are necessary for integrality to hold.