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 \(\mathscr{L}(E,\chi)\) at \(s=1\) is an algebraic integer. For instance, for semistable curves \(\mathscr{L}(E,\chi)\) 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.

11G40, 11G05, 11F67

elliptic curves, \(L\)-functions, modular symbols

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University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

University of Nottingham, School of Mathematical Sciences, University Park, Nottingham, NG7 2RD, UK