Geometric realizations of representations for $PSL(2, F_p)$ and Galois representations arising from defining ideals of modular curves

We construct a geometric realization of representations for PSL(2, Fₚ) by the defining ideals of rational models L(X(p)) of modular curves $X(p)$ over Q. Hence, for the irreducible representations of PSL(2, Fₚ), whose geometric realizations can be formulated in three different scenarios in the framework of Weil's Rosetta stone: number fields, curves over Fq and Riemann surfaces. In particular, we show that there exists a correspondence among the defining ideals of modular curves over Q, reducible Q(ζₚ)-rational representations πₚ: PSL(2, Fₚ) → Aut(L(X(p))) of PSL(2, Fₚ), and Q(ζₚ)-rational Galois representations ρₚ: Gal(Q̄/Q) → Aut(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 -adic \'{e}tale cohomology of modular curves over Q, i.e., Grothendieck motives (-adic system), automorphic representations of GL(2, Q) and -adic representations.