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On nearly holomorphic Drinfeld modular forms for admissible coefficient rings

Oğuz Gezmiş, Sriram Chinthalagiri Venkata

TL;DR

The paper develops a function-field analogue of nearly holomorphic modular forms in the setting of Drinfeld modules over admissible coefficient rings. It constructs analytic notions as continuous, non-holomorphic functions on parts of the Drinfeld upper half-plane and provides a rigorous algebraic description via an extended de Rham sheaf on a compactified Drinfeld moduli space, with a precise comparison between the analytic and algebraic pictures. The authors prove transcendence results for CM-values of nearly holomorphic Drinfeld modular forms, establish a robust t-expansion framework at cusps through Tate-Drinfeld modules, and show an exact equivalence between spaces of nearly holomorphic forms and global sections of natural sheaves on the moduli space. This yields a complete algebraic description of these forms and recovers classical Drinfeld modular forms as a special case, advancing the understanding of function-field analogues of modular-analytic phenomena and their arithmetic properties.

Abstract

Let $X$ be a smooth projective and geometrically irreducible curve over the finite field $\mathbb{F}_q$ with $q$ elements and $K$ be its function field. Let $\infty$ be a fixed closed point on $X$ and $A$ be the ring of functions regular away from $\infty$. In the present paper, by generalizing the previous work of Chen and the first author, we introduce the notion of nearly holomorphic Drinfeld modular forms for congruence subgroups of $GL_2(K)$ as continuous but non-holomorphic functions on a certain subdomain of the Drinfeld upper half plane. By extending the de Rham sheaf to a compactification $\overline{M_I^2}$ of the Drinfeld moduli space $M_I^2$, we also describe such forms algebraically as global sections of an explicitly described sheaf on $\overline{M_I^2}$ as well as construct a comparison isomorphism between analytic and algebraic description of them. Furthermore, we show the transcendence of special values of nearly holomorphic Drinfeld modular forms at CM points and relate them to the periods of CM Drinfeld $A$-modules.

On nearly holomorphic Drinfeld modular forms for admissible coefficient rings

TL;DR

The paper develops a function-field analogue of nearly holomorphic modular forms in the setting of Drinfeld modules over admissible coefficient rings. It constructs analytic notions as continuous, non-holomorphic functions on parts of the Drinfeld upper half-plane and provides a rigorous algebraic description via an extended de Rham sheaf on a compactified Drinfeld moduli space, with a precise comparison between the analytic and algebraic pictures. The authors prove transcendence results for CM-values of nearly holomorphic Drinfeld modular forms, establish a robust t-expansion framework at cusps through Tate-Drinfeld modules, and show an exact equivalence between spaces of nearly holomorphic forms and global sections of natural sheaves on the moduli space. This yields a complete algebraic description of these forms and recovers classical Drinfeld modular forms as a special case, advancing the understanding of function-field analogues of modular-analytic phenomena and their arithmetic properties.

Abstract

Let be a smooth projective and geometrically irreducible curve over the finite field with elements and be its function field. Let be a fixed closed point on and be the ring of functions regular away from . In the present paper, by generalizing the previous work of Chen and the first author, we introduce the notion of nearly holomorphic Drinfeld modular forms for congruence subgroups of as continuous but non-holomorphic functions on a certain subdomain of the Drinfeld upper half plane. By extending the de Rham sheaf to a compactification of the Drinfeld moduli space , we also describe such forms algebraically as global sections of an explicitly described sheaf on as well as construct a comparison isomorphism between analytic and algebraic description of them. Furthermore, we show the transcendence of special values of nearly holomorphic Drinfeld modular forms at CM points and relate them to the periods of CM Drinfeld -modules.

Paper Structure

This paper contains 28 sections, 47 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Nearly holomorphic Drinfeld modular forms
  4. Special values at CM points
  5. Algebraic description of nearly holomorphic Drinfeld modular forms
  6. Outline of the paper
  7. Preliminaries and Background
  8. Drinfeld $A$-modules over an $A$-algebra
  9. Drinfeld-Hayes $A$-modules
  10. Drinfeld $A$-modules over $K$-schemes and their moduli spaces
  11. de Rham cohomology of Drinfeld $A$-modules
  12. The decomposition of the de Rham module
  13. Rigid analytic structure on $\Gamma \backslash \Omega$ and its compactification
  14. Sheaves on $\Gamma \backslash \Omega$
  15. Drinfeld $A$-modules over rigid analytic spaces
  16. ...and 13 more sections

Key Result

Theorem 1.5

Let $z_0\in \Omega$ be a CM point and let $F\in \mathcal{N}_{k}^{\leq r}(\Gamma)$ be an arithmetic nearly holomorphic Drinfeld modular form as above. Then for some $c\in \overline{K}$ and a period $w_{z_0}$ of a CM Drinfeld $A$-module. In particular, if $f$ is a Drinfeld modular form of weight $k$ for $\Gamma$, then for some $\tilde{c}\in \overline{K}$. Furthermore, if $F(z_0)$ and $\delta_k^r(f

Theorems & Definitions (111)

  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • Remark 2.3
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • ...and 101 more