Geometric realizations of representations for $\text{PSL}(2, \mathbb{F}_p)$ and Galois representations arising from defining ideals of modular curves

Abstract

We construct a geometric realization of representations for $\text{PSL}(2, \mathbb{F}_p)$ by the defining ideals of rational models $\mathcal{L}(X(p))$ of modular curves $X(p)$ over $\mathbb{Q}$. Hence, for the irreducible representations of $\text{PSL}(2, \mathbb{F}_p)$, whose geometric realizations can be formulated in three different scenarios in the framework of Weil's Rosetta stone: number fields, curves over $\mathbb{F}_q$ and Riemann surfaces. In particular, we show that there exists a correspondence among the defining ideals of modular curves over $\mathbb{Q}$, reducible $\mathbb{Q}(\zeta_p)$-rational representations $\pi_p: \text{PSL}(2, \mathbb{F}_p) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ of $\text{PSL}(2, \mathbb{F}_p)$, and $\mathbb{Q}(\zeta_p)$-rational Galois representations $\rho_p: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ as well as their modular and surjective realization. This leads to a new viewpoint on the last mathematical testament of Galois by Galois representations arising from the defining ideals of modular curves, which leads to a connection with Klein's elliptic modular functions. It is a nonlinear and anabelian counterpart of the global Langlands correspondence among the $\ell$-adic \'{e}tale cohomology of modular curves over $\mathbb{Q}$, i.e., Grothendieck motives ($\ell$-adic system), automorphic representations of $\text{GL}(2, \mathbb{Q})$ and $\ell$-adic representations.