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On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points

Yen-Tsung Chen, Oğuz Gezmiş

TL;DR

The paper develops a higher-rank theory of meromorphic arithmetic Drinfeld modular forms and analyzes their special CM-values. It establishes a Shimura-type relation between CM values and CM periods in arbitrary rank, and proves algebraic independence results for periods and quasi-periods of CM Drinfeld modules using t-motives and torus Galois groups. The approach combines integrality properties of arithmetic forms, CM theory, and Tannakian methods to relate modular form values at CM points to CM periods, yielding precise transcendence degrees under natural endomorphism-field hypotheses. This extends rank-two results of Chang to arbitrary rank, enriching the transcendence theory in function-field arithmetic and advancing understanding of CM phenomena in Drinfeld modular forms.

Abstract

In the present paper, we introduce meromorphic Drinfeld modular forms of arbitrary rank equipped with a particular arithmeticity property. We also study their special values at CM points and show the algebraic independence of these values under some conditions. Our results may be seen as a generalization of Chang's results on the special values of arithmetic Drinfeld modular forms in the rank two setting.

On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points

TL;DR

The paper develops a higher-rank theory of meromorphic arithmetic Drinfeld modular forms and analyzes their special CM-values. It establishes a Shimura-type relation between CM values and CM periods in arbitrary rank, and proves algebraic independence results for periods and quasi-periods of CM Drinfeld modules using t-motives and torus Galois groups. The approach combines integrality properties of arithmetic forms, CM theory, and Tannakian methods to relate modular form values at CM points to CM periods, yielding precise transcendence degrees under natural endomorphism-field hypotheses. This extends rank-two results of Chang to arbitrary rank, enriching the transcendence theory in function-field arithmetic and advancing understanding of CM phenomena in Drinfeld modular forms.

Abstract

In the present paper, we introduce meromorphic Drinfeld modular forms of arbitrary rank equipped with a particular arithmeticity property. We also study their special values at CM points and show the algebraic independence of these values under some conditions. Our results may be seen as a generalization of Chang's results on the special values of arithmetic Drinfeld modular forms in the rank two setting.

Paper Structure

This paper contains 15 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Elliptic modular forms at CM points
  3. Drinfeld modular forms at CM points
  4. Periods and quasi-periods of non-isogenous CM Drinfeld modules
  5. Preliminaries and Background
  6. Drinfeld modules and quasi-periodic functions
  7. Drinfeld modules with complex multiplication
  8. Rigid analytic trivialization and dual $t$-motives
  9. Drinfeld modular forms
  10. Arithmetic Drinfeld modular forms
  11. Arithmetic Drinfeld modular forms with level $N$
  12. Special CM values of arithmetic Drinfeld modular forms
  13. $t$-motivic Galois groups and algebraic independence
  14. Properties of algebraic tori
  15. Tannakian formalism for rigid analytically trivial pre-$t$-motives

Key Result

Theorem 1.2.1

Let $F$ be a meromorphic arithmetic Drinfeld modular form of non-zero weight $k$ for a congruence subgroup $\Gamma$ of $\mathop{\mathrm{GL}}\nolimits_r(A)$. For any CM point $\boldsymbol{\omega}\in\Omega^r$, if $F$ is defined at $\boldsymbol{\omega}$, then we have where $\lambda_{\boldsymbol{\omega}}\in\mathbb{C}_\infty^\times$ is a non-zero period of a CM Drinfeld module $\psi^{\boldsymbol{\omeg

Theorems & Definitions (33)

  • Theorem 1.2.1
  • Remark 1.2.2
  • Theorem 1.3.1
  • Remark 1.3.2
  • Remark 1.3.3
  • Theorem 1.3.4
  • Lemma 2.2.2
  • proof
  • Definition 2.4.1
  • Definition 2.4.3
  • ...and 23 more