On the Jacobians of Curves Defined by the Generalized Laguerre Polynomials

ABSTRACT

Let $L_n^{⟨\alpha ⟩}\left(x\right)={\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+\alpha }{n-j}\right)\frac{{(-x)}^j}{j!}$ be the nth Generalized Laguerre Polynomial. In this paper we study the arithmetic of the algebraic curves $\mathcal{L}\left(n\right)/\mathbf{Q}$ defined by L^⟨α⟩_n(x) = 0, viewed as a two-variable polynomial over Q, and their Jacobians $\mathcal{J}\left(n\right)/\mathbf{Q}$. We introduce a conjecture for the endomorphism ring, Mordell–Weil group, and image of the ℓ-adic representations of the $\mathcal{J}\left(n\right)$ for all n ⩾ 4.

KEYWORDS:

• algebraic curve
• galois representation
• orthogonal polynomial

AMS Subject Classification:

• 11G10
• 11G20

Acknowledgments

This project was started at the 2015 ICERM “Modular Forms and Curves of Low Genus: Computational Aspects.” We would like to thank the organizers of that conference for their generous support. We would also like to thank Farshid Hajir and Siman Wong for helpful conversations and feedback. Finally, we would like to thank the referee for a careful and thorough review of the manuscript and for providing us with the proof of Proposition 6.