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On the partial derivatives of Drinfeld modular forms of arbitrary rank

Yen-Tsung Chen, Oğuz Gezmiş

TL;DR

This work introduces a higher-rank generalization of the Serre derivation for Drinfeld modular forms by defining a multilinear differential operator $D_{(k_1,\dots,k_{r-1})}$ as a determinant of rank-$(r-1)$ differential components, extending Gekeler's rank-two framework to arbitrary rank $r$. The authors develop the analytic and algebraic toolkit (Tate algebra, Anderson generating functions, and Gauss–Manin/Kodaira–Spencer considerations) to study partial derivatives of coefficient forms $g_i$ and the cusp form $h_r$, establishing explicit formulas such as $\partial_j(h_r)=-h_r E_r^{[j]}$ and $\partial_j(g_i)=E_r^{[j]}g_i+\tilde{\pi}^{q+\cdots+q^{r-1}}h_r L_{j(i+1)}$, with $E_r^{[j]}=-\partial_j(h_r)/h_r$. A central result proves that $D_{(k_1,\dots,k_{r-1})}$ is a $\mathbb{C}_\infty$-multilinear derivation that sends $(f_1,\dots,f_{r-1})$ into $M_{k_1+\cdots+k_{r-1}+r}^{m_1+\cdots+m_{r-1}+1}$, and the paper constructs a finitely generated algebra $\mathcal{M}_r$ generated by $g_i$ and $h_rL_{ij}$ that is closed under partial derivatives and contains the Drinfeld modular forms for $\mathrm{GL}_r(A)$. The results illuminate the differential structure of higher-rank Drinfeld modular forms and point toward a robust higher-rank quasi-modular theory, with implications for the arithmetic of CM points and transcendence phenomena of special values.

Abstract

In this paper, we obtain an analogue of the Serre derivation acting on the product of spaces of Drinfeld modular forms which generalizes the differential operator introduced by Gekeler in the rank two case. We further introduce a finitely generated algebra $\mathcal{M}_r$ containing all the Drinfeld modular forms for the full modular group and show its stability under the partial derivatives.

On the partial derivatives of Drinfeld modular forms of arbitrary rank

TL;DR

This work introduces a higher-rank generalization of the Serre derivation for Drinfeld modular forms by defining a multilinear differential operator as a determinant of rank- differential components, extending Gekeler's rank-two framework to arbitrary rank . The authors develop the analytic and algebraic toolkit (Tate algebra, Anderson generating functions, and Gauss–Manin/Kodaira–Spencer considerations) to study partial derivatives of coefficient forms and the cusp form , establishing explicit formulas such as and , with . A central result proves that is a -multilinear derivation that sends into , and the paper constructs a finitely generated algebra generated by and that is closed under partial derivatives and contains the Drinfeld modular forms for . The results illuminate the differential structure of higher-rank Drinfeld modular forms and point toward a robust higher-rank quasi-modular theory, with implications for the arithmetic of CM points and transcendence phenomena of special values.

Abstract

In this paper, we obtain an analogue of the Serre derivation acting on the product of spaces of Drinfeld modular forms which generalizes the differential operator introduced by Gekeler in the rank two case. We further introduce a finitely generated algebra containing all the Drinfeld modular forms for the full modular group and show its stability under the partial derivatives.
Paper Structure (11 sections, 22 theorems, 163 equations)

This paper contains 11 sections, 22 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Drinfeld modular forms
  4. Partial derivatives of Drinfeld modular forms
  5. The Tate algebra and Anderson generating functions
  6. Partial derivatives of the coefficient forms and the $h$-function
  7. Certain functional equations and identities
  8. The differential operator $\mathop{\mathrm{D}}\nolimits_{(k_1,\dots,k_{r-1})}$
  9. Estimates on the exponential function
  10. Proof of Theorem \ref{['T:I2']}
  11. The $\mathbb{C}_{\infty}$-algebra $\mathcal{M}_r$

Key Result

Theorem 1.1

For each $1\leq i \leq r-1$, let $\Gamma_i \leqslant \mathop{\mathrm{GL}}\nolimits_r(A)$ be a congruence subgroup, $k_i\in \mathbb{Z}_{\geq 0}$ and $m_i\in \mathbb{Z}/(q-1)\mathbb{Z}$. Consider the operator $\mathop{\mathrm{D}}\nolimits_{(k_1,\dots,k_{r-1})}$ on $M_{k_1}^{m_1}(\Gamma_1)\times \cdots Then the following statements hold.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Example 3.4
  • Proposition 4.4
  • Lemma 4.6
  • proof
  • ...and 37 more