On the Jacobians of curves defined by the generalized Laguerre polynomials. (English) Zbl 1470.11160
Summary: Let \(L^{\langle\alpha\rangle}_n (x)\sum^n_{j=0}\begin{pmatrix} n+\alpha \\ n-j\end{pmatrix} \frac{(-x)^j}{j!}\) be the \(n\)th Generalized Laguerre Polynomial. In this paper we study the arithmetic of the algebraic curves \(\mathcal{L} (n)/\mathbb{Q}\) defined by \(L^{\langle\alpha\rangle}_n (x)=0\), viewed as a two-variable polynomial over \(\mathbb{Q}\), and their Jacobians \(\mathcal{J} (n)/\mathbb{Q}\). We introduce a conjecture for the endomorphism ring, Mordell-Weil group, and image of the \(\ell\)-adic representations of the \(\mathcal{J}(n)\) for all \(n\geq 4\).
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