Geometric realizations of representations for PSL (2, Fₚ) 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 (ₚ) -rational representations ₚ: PSL (2, Fₚ) Aut (L (X (p) ) ) of PSL (2, Fₚ), and Q (ₚ) -rational Galois representations ₚ: Gal (Q/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 \'etale cohomology of modular curves over Q, i. e. , Grothendieck motives (-adic system), automorphic representations of GL (2, Q) and -adic representations.