[Paper Review] Vectorial Drinfeld modular forms over Tate algebras

Federico Pellarin, Rudolph Perkins|arXiv (Cornell University)|Mar 25, 2016

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper develops the theory of vectorial Drinfeld modular forms with values in Tate algebras for a specific two-dimensional representation of GL₂(𝔽ₚ[θ]), establishing the complete module structure, characterizing specializations at roots of unity, and proving that these forms are stable under Hecke operators. It further shows that specializations yield Hecke eigenforms for congruence subgroups, with eigenvalues determined by Goss polynomials and hyperdifferential operators.

ABSTRACT

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of Γ := GL 2 (Fq[$theta$])). Among several results that we prove here, we determine the complete structure of the modules of these forms, we describe their specializations at roots of unity and their connection with Drinfeld modular forms for congruence subgroups of Γ and we prove that the modules generated by these forms are stable under the actions of Hecke operators.

Motivation & Objective

• To develop the theory of vectorial modular forms with values in Tate algebras for a specific representation of GL₂(𝔽ₚ[θ]).
• To determine the complete module structure of these forms over the Tate algebra.
• To study their specializations at roots of unity and relate them to Drinfeld modular forms for congruence subgroups.
• To prove that the modules generated by these forms are stable under Hecke operators.
• To establish connections between hyperdifferential operators, A-expansions, and Hecke eigenforms in the function field setting.

Proposed method

• Constructs vectorial modular forms using hyperdifferential operators and Anderson generating functions in rank 2.
• Uses the τ-difference equation for the Eisenstein series 𝒟₁ to analyze its structure and transformation properties.
• Applies evaluation maps at roots of unity to specialize forms and relate them to classical Drinfeld modular forms.
• Employs hyperdifferential operators in t to generate new forms and study their Hecke eigenvalue behavior.
• Utilizes A-expansions and Goss polynomials to characterize coefficients and eigenvalues of Hecke operators.
• Applies Petersson slash operators and congruence subgroup theory to analyze modularity and holomorphy at cusps.

Experimental results

Research questions

• RQ1 How are vectorial Drinfeld modular forms with values in Tate algebras structured as modules over the Tate algebra?
• RQ2 What is the behavior of these forms under specialization at roots of unity, and how do they relate to Drinfeld modular forms for congruence subgroups?
• RQ3 Do the modules generated by these forms remain invariant under the action of Hecke operators?
• RQ4 How do hyperdifferential operators in t interact with Hecke eigenforms and A-expansions in this setting?
• RQ5 To what extent do the coefficients in A-expansions of specializations reflect Hecke eigenvalues, and how do they differ from classical modular forms?

Key findings

• The module of vectorial Drinfeld modular forms for the given representation is completely determined and shown to be free over the Tate algebra.
• Specializations of the vectorial Eisenstein series 𝒟₁ at t = ζ (a root of unity) yield Hecke eigenforms for congruence subgroups Γ₁(𝔮ⁿ), with eigenvalues 𝔭ᵏ for primes 𝔭 not dividing the level 𝔮.
• The Hecke operators preserve regularity at infinity and stabilize the modules generated by these forms.
• For weight k ≡ 1 mod (q−1), the specializations ev_ζ(𝒟ₜ⁽ⁿ⁻¹⁾[ℰₖ]₁) are simultaneous Hecke eigenforms for the family {Tₚ : 𝔭(ζ) ≠ 0} with eigenvalues {𝔭ᵏ}.
• The A-expansion coefficient of u(𝑧) vanishes for n ≥ 2, indicating a key difference from classical modular forms where such coefficients determine eigenvalues.
• The Hecke eigenvalues are fully determined by the Goss polynomial Gₖ appearing in the A-expansion, and the coefficient of u(𝔭𝑧) is a function of 𝔭 and the level, not directly of the eigenvalue.