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 $\mathbb{Q}$, which gives rise to a Rosetta stone for geometric representations of $\text{PSL}(2, \mathbb{F}_p)$. The defining ideal of a modular curve, i.e., an anabelian counterpart of the Eisenstein ideal, is the anabelianization of the Jacobian of this modular curve and is a reification of the fundamental group $\pi_1$. 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. It is an anabelian counterpart of the global Langlands correspondence for $\text{GL}(2, \mathbb{Q})$ by the \'{e}tale cohomology of modular curves as well as an anabelian counterpart of Artin's conjecture, Serre's modularity conjecture and the Fontaine-Mazur conjecture. It is an ideal theoretic (i.e. nonlinear) counterpart of Grothendieck's section conjecture and an ideal theoretic (i.e. nonlinear) reification of ``arithmetic theory of $\pi_1$'' expected by Weil for modular curves. It is also an anabelian counterpart of the theory of Kubert-Lang and Mazur-Wiles on the cuspidal divisor class groups and the Eisenstein ideals of modular curves.