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

2024/09/04 by Lei Yang, Yang, Lei
Mathematics · #11F80 #11G18 #11G32 #11R39 #13A50 #14C25 #14H37 #14H42 #14L30 #20C33 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2409.02589

openalex publication_date 2024/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a geometric realization of representations for PSL(2, \mathbbF_p) by the defining ideals of rational models L(X(p)) of modular curves X(p) over ℚ, which gives rise to a Rosetta stone for geometric representations of PSL(2, \mathbbF_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 π₁. We show that there exists a correspondence among the defining ideals of modular curves over ℚ, reducible ℚ(ζ_p)-rational representations π_p: PSL(2, \mathbbF_p) → Aut(L(X(p))) of PSL(2, \mathbbF_p), and ℚ(ζ_p)-rational Galois representations ρ_p: Gal(ℚ/ℚ) → Aut(L(X(p))) as well as their modular and surjective realization. It is an anabelian counterpart of the global Langlands correspondence for GL(2, 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 π₁'' 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.

Related

• Potentially GL₂-type Galois representations associated to noncongruence modular forms
• Moduli stacks of Galois representations and the p-adic local Langlands correspondence for GL₂(ℚ_p)
• A moduli approach to quadratic Q-curves realizing projective mod p Galois representations
• Computing Galois representations and equations for modular curves X_H(ℓ)
• Descent for projective twists of modular curves