Let \(E/\mathbb{Q}\) be an elliptic curve and let \(\chi\) be a primitive Dirichlet character of conductor \(m\) and Gauss sum denoted by \(G(\chi)\). Set \(\varepsilon=\chi(-1) \in \{ \pm 1 \} \) and let \(\Omega_{\varepsilon}(E)\) be the \(\pm\)-periods of \(E/\mathbb{Q}\) depending on whether \(\varepsilon\) is positive or negative. The algebraic \(L\)-value at \(s=1\), which is the main object of study in this work, is defined as \[\mathscr{L}(E, \chi)= \frac{L(E, \chi, 1) \cdot m}{G(\chi) \cdot \Omega_{\varepsilon} (E)}.\]
Except for being interesting in its own right, the algebraic \(L\)-value also has connections to the Birch and Swinnerton-Dyer conjecture. In this article, the authors show, under mild assumptions on \(E/\mathbb{Q}\) and \(\chi\), that \(\mathscr{L}(E, \chi)\) is an algebraic integer. More precisely, their main result is the following:
Theorem. Let \(E/\mathbb{Q}\) be an elliptic curve. Then the following two statements are true.

1.

Assume that the Stevens’s Manin constant \(c_1(E)\) of \(E/\mathbb{Q}\) is equal to \(1\) (which is conjecturally true). Then \(\mathscr{L}(E, \chi) \in \mathbb{Z}[\zeta_d]\) for every non-trivial primitive Dirichlet character \(\chi\) of order \(d\) whose conductor \(m\) is not divisible by a prime of bad reduction for \(E/\mathbb{Q}\).

2.

Assume that the Manin constant \(c_0(E)\) of \(E/\mathbb{Q}\) is equal to \(1\). Then \(\mathscr{L}(E, \chi) \in \mathbb{Z}[\zeta_d]\) for every non-trivial primitive Dirichlet character \(\chi\) of order \(d\) whose conductor \(m\) is not divisible by a prime of additive reduction for \(E/\mathbb{Q}\).

It follows from the above theorem that \(\mathscr{L}(E, \chi) \in \mathbb{Z}[\zeta_d]\) for every semistable \(X_0(N)\)-optimal curve \(E/\mathbb{Q}\) and every \(\chi\) of order \(d\) (as in this case it is known that \(c_0(E)=1\)). In addition, as a corollary of their main theorem, the authors prove that there are only finitely many \(\chi\) such that \(\mathscr{L}(E, \chi)\) is non-integral when \(c_1(E)=1\) and \(E/\mathbb{Q}\) admits a semistable quadratic twist over \(\mathbb{Q}\).
In the final section, the authors present examples showing that their theorems are sharp in the sense that their assumptions cannot be weakened. A table containing all examples of non-integral \(L\)-values for elliptic curves with conductor less than \(100\) is also presented.