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

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 ${\pi }_p:\text{PSL}(2,{\mathbb{F}}_p)\to \text{Aut}(\mathcal{L}(X(p)))$ of $\text{PSL}(2,{\mathbb{F}}_p)$, and $\mathbb{Q}({\zeta }_p)$-rational Galois representations ${\rho }_p:\text{Gal}(\bar{\mathbb{Q}}\mathrm{/}\mathbb{Q})\to \text{Aut}(\mathcal{L}(X(p)))$ 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 étale 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.