Meromorphic Drinfeld Modular Forms

• Meromorphic Drinfeld modular forms are defined on Drinfeld upper half-spaces and extend classical modular forms by integrating analytic, arithmetic, and geometric perspectives.
• They are characterized by u-expansions at cusps, Satake compactification frameworks, and an arithmeticity condition that supports robust structure and special value theorems.
• Their study leverages period relations and t-motivic Galois theory to establish algebraic independence of CM values and deepen insights into function field arithmetic.

Meromorphic Drinfeld modular forms are central objects in the function field arithmetic of positive characteristic, unifying analytic, arithmetic, and algebro-geometric perspectives on moduli of Drinfeld modules and their compactifications. These forms are defined on Drinfeld upper half-spaces of arbitrary rank and admit a rich theory paralleling, yet extending, classical modular forms, with arithmetic properties, expansion principles, and structural theorems about special values at CM points. This article provides a detailed overview, emphasizing the foundational definitions, the Satake compactification framework, the arithmeticity property, special value and algebraic independence theorems, connections to t-motivic Galois theory, and comparison with both the classical and rank-two function field frameworks.

1. Definition and Analytic Foundations

Let $A = \mathbb{F}_q[\theta]$, $K = \mathbb{F}_q(\theta)$, $K_\infty$ its completion at $\infty$, and $\mathbb{C}_\infty$ the completed algebraic closure of $K_\infty$. For $r \geq 2$, the rank-$r$ Drinfeld upper half-space is defined as

$$\Omega^r = \mathbb{P}^{r-1}(\mathbb{C}_\infty) \setminus \{ K_\infty\text{-rational hyperplanes} \},$$

with points normalized as column vectors $\omega = (w_1, \ldots, w_r)^\top$ with $w_r = 1$ and entries $K_\infty$-linearly independent.

Given a congruence subgroup $\Gamma \subset \mathrm{GL}_r(A)$, a weak Drinfeld modular form of weight $k \in \mathbb{Z}$ and type $m \in \mathbb{Z}/(q-1)\mathbb{Z}$ is a rigid analytic function $f : \Omega^r \to \mathbb{C}_\infty$ such that, for all $\gamma \in \Gamma$,

$$(f|_{k,m}\gamma)(\omega) = \det(\gamma)^m j(\gamma;\omega)^{-k} f(\gamma \cdot \omega),$$

where $j(\gamma;\omega)$ is the last entry of $\gamma \omega$. The space of such forms is denoted $\mathcal{W}^r_{k,m}(\Gamma)$, and $$\mathcal{W}^r(\Gamma) = \bigoplus_{k,m} \mathcal{W}^r_{k,m}(\Gamma)$$.

A crucial analytic feature is the existence of the parameter at infinity $u_\Gamma : \Omega^r \to \mathbb{C}_\infty$, which, near cusps, enables a $u_\Gamma$-expansion: $$f(\omega) = \sum_{n \in \mathbb{Z}} f_n(\widetilde{\omega}) u_\Gamma(\omega)^n,\qquad \widetilde{\omega} \in \Omega^{r-1}$$ with each $f_n$ itself being a modular form (or weak form) for a smaller subgroup in rank $r-1$.

A function is a (holomorphic) Drinfeld modular form if all its $u_\Gamma$-expansions have only nonnegative powers at all cusps.

2. Satake Compactification and Algebro-Geometric Perspective

The Satake compactification is essential for the algebro-geometric treatment of Drinfeld modular forms of arbitrary rank. For a “fine” open compact $K \subset \mathrm{GL}_r(\mathbb{A}_f)$, the coarse moduli scheme $M_K$ of Drinfeld modules admits a normal, projective Satake compactification $M_K^*$, unique up to isomorphism. The universal Drinfeld module extends over $M_K^*$ as a "weakly separating" family. The dual of the relative Lie algebra, denoted by $\omega$, is an ample line bundle on $M_K^*$.

For any $k \geq 0$, the space of (algebraic) Drinfeld modular forms of weight $k$ is

$M_k(M_K^*) := H^0(M_K^*, \omega^{\otimes k}),$

with $R(M_K^*) = \bigoplus_{k \geq 0} M_k(M_K^*)$ a finitely generated, normal, integral graded $F$-algebra. The compactification satisfies $M_K^* \cong \operatorname{Proj} R(M_K^*)$.

Meromorphic Drinfeld modular forms of weight $k$ are identified with rational sections of $\omega^{\otimes k}$, that is,

$$f \in H^0(M_K^*, \omega^{\otimes k} \otimes \mathcal{O}_{M_K^*}(mD)),$$

where $D = M_K^* \setminus M_K$ is the boundary Cartier divisor, and $m \geq 0$. The divisor records the pole orders at each boundary component.

For $r = 2$, the boundary consists of cusps; in higher rank, it stratifies into Drinfeld modular varieties of smaller rank. Meromorphic forms are described locally by $u$-expansions in terms of cusp parameters.

3. Arithmeticity and Structure of Meromorphic Forms

In the rank-$2$ theory, arithmetic Drinfeld modular forms have Fourier coefficients in an algebraic closure $\overline{K}$ of $K$. For $r \geq 3$, the arithmeticity criterion follows the inductive Basson–Sugiyama process. Define $AW^1(\Gamma) = K$. For $r \geq 2$, a function $f \in \mathcal{W}^r(\Gamma)$ is arithmetic if in its $u_\Gamma$-expansion $\sum_{n \in \mathbb{Z}} f_n X^n$, each $f_n$ lies in $AW^{r-1}(\tilde{\Gamma})$ for some congruence subgroup $\tilde{\Gamma} \subset \mathrm{GL}_{r-1}(A)$. The $K$-algebra $AW^r(\Gamma)$ consists of all such forms.

Arithmetic modular forms are $$AM^r(\Gamma) = AW^r(\Gamma) \cap \mathcal{M}^r(\Gamma)$$, and meromorphic arithmetic forms are defined via ratios: $$F = \frac{f}{g},\quad f \in AM^r_{k+\ell}(\Gamma),\ g \in AM^r_\ell(\Gamma),\ \ell > 0$$ with $AF^r_k(\Gamma)$ denoting the $\mathbb{C}_\infty$-span of such forms of weight $k$.

Key structural results include:

• For $f \in AM^r_k(\Gamma(N))$, there is a monic polynomial relation over $K$ of the form

$$f^n + Q_{n-1}(E_u^{\mathrm{ari}}) f^{n-1} + \cdots + Q_0(E_u^{\mathrm{ari}}) = 0,$$

with $Q_i$ polynomials in the weight-$1$ arithmetic Eisenstein series $E_u^{\mathrm{ari}}$. Thus, any arithmetic modular form is integral over the ring generated by these Eisenstein series (Chen et al., 4 Dec 2025).

4. Special Values at CM Points

A point $\omega \in \Omega^r$ is called a CM point if its lattice $\Lambda_\omega$ satisfies $\operatorname{End}(\Lambda_\omega)$ is an $A$-order of rank $r$. For all $u$ and CM $\omega$,

$$E_u(\omega) = \widetilde{\pi} \cdot \alpha,\ \alpha \in K,$$

where $\widetilde{\pi}$ is a period of the Carlitz module, and $$E_u^{\mathrm{ari}}(\omega) \sim \lambda_\omega / \widetilde{\pi}$$ for some CM period $\lambda_\omega$.

The special-value theorem states:

Theorem (Special-value period relation).

Let $F \in AF^r_k(\Gamma)$ be a nonzero-weight meromorphic arithmetic Drinfeld modular form, and $\omega$ a CM point at which $F$ is defined. Then

$$F(\omega) \sim \left( \frac{\lambda_\omega}{\widetilde{\pi}} \right)^k$$

where $\lambda_\omega$ is a period of a $K$-rational CM Drinfeld module with lattice homothetic to $\Lambda_\omega$ (Chen et al., 4 Dec 2025).

In particular, the special value is algebraic up to a $k$th power of the period.

5. Algebraic Independence and t-Motivic Galois Theory

The transcendence and algebraic independence results for special values at CM points are obtained via t-motivic Galois techniques. For each Drinfeld module $\phi_i$ of rank $r_i$ with endomorphism algebra $\mathcal{K}_i$ (a Galois extension of $\mathbb{F}_q(t)$), its dual t-motive $\mathcal{M}_i$ has a rigid analytic trivialization. The associated Galois group is

$$\Gamma_{\mathcal{M}_i} \cong \mathrm{Res}_{\mathcal{K}_i/\mathbb{F}_q(t)}(\mathbb{G}_m).$$

For the direct sum motive $\mathcal{M} = \bigoplus_{i=1}^n \mathcal{M}_i$, the Galois group sits inside the subtorus

$$T_n = \left\{ (\gamma_1, \ldots, \gamma_n) \in G_1 \times \cdots \times G_n \mid \det \gamma_1 = \cdots = \det \gamma_n \right\}$$

with $$\dim \Gamma_{\mathcal{M}} = (r_1 + \cdots + r_n) - (n-1)$$ under the assumption that the $\mathcal{K}_i$ are linearly disjoint.

Applying Papanikolas’s main theorem,

$$\operatorname{tr.deg}_K K(\Psi(\theta)) = \dim \Gamma_{\mathcal{M}},$$

this yields

Theorem (Algebraic independence of periods).

With $\lambda_i$ periods of corresponding CM Drinfeld modules,

$$\operatorname{tr.deg}_K K(\lambda_1 / \widetilde{\pi}, \ldots, \lambda_n / \widetilde{\pi}) = n.$$

Combining with the period relation for special values, one has

Theorem (Main algebraic-independence of CM-values).

For CM points $\omega_1, \ldots, \omega_n$ with linearly disjoint Galois endomorphism algebras and $F \in AF^r_k(\Gamma)$, $F(\omega_i) \neq 0$,

$$\operatorname{tr.deg}_K K(F(\omega_1), \ldots, F(\omega_n)) = n,$$

i.e., the $n$ special values are algebraically independent over $K$ (Chen et al., 4 Dec 2025).

6. Examples, Expansions, and Basis Constructions

In the rank-$2$ setting for $A = \mathbb{F}_q[T]$ and level $\Gamma_0(T)$, explicit canonical bases of the spaces of weakly holomorphic (meromorphic at genus-zero) Drinfeld modular forms are constructed. For $k \equiv 2l \pmod{q-1}$, the forms $f_{k,l,i}(z)$ in $M^!_{k,l}(\Gamma_0(T))$ are indexed by pole order at infinity and satisfy principal part expansions.

Generating functions for these bases satisfy rational function identities, and the action of the Serre-Ramanujan operator (Drinfeld Theta operator) on these forms is explicitly computable as a recurrence depending on divisors of the meromorphic modular forms (Dalal, 2023). These canonical bases underpin the structure and arithmetic of $q$-expansions, zeros, and congruences.

Explicit meromorphic Drinfeld modular forms include ratios of Eisenstein series and modular discriminants, with special reference to the $h$-function for $q+1$ and type $1$, and modular functions of nonzero type constructed via powers and ratios of $g(z)$ and $h(z)$ (Breuer, 2016).

7. Generalization and Comparison with Previous Theories

The results described generalize Chang’s rank-$2$ theorems on special values of arithmetic Drinfeld modular forms (Chen et al., 4 Dec 2025). In rank two, the lattice periods and values of modular forms at CM points satisfy analogous period relations and algebraic independence results, but are limited by the quadratic nature of endomorphism algebras.

The present framework supports arbitrary rank ($r \geq 2$), arbitrary (meromorphic) weight, and allows for any finite number of pairwise linearly disjoint CM points, relying critically on the recursive arithmeticity definition and properties of algebraic tori within t-motivic Galois groups.

In higher rank, the boundary of the compactified moduli space is stratified by lower-rank Drinfeld modular varieties, and the analytic and algebraic description aligns with classical $r=2$ theory, but with the essential novel complexity of stratified boundary components (Pink, 2010).

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Key References:

• Chen–Gezmiş, "On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points" (Chen et al., 4 Dec 2025)
• Pink, "Compactification of Drinfeld modular varieties and Drinfeld Modular Forms of Arbitrary Rank" (Pink, 2010)
• Gekeler, "A note on Gekeler’s h-function" (Breuer, 2016)
• Basson–Sugiyama arithmeticity, Papanikolas’s t-motivic Galois theory
• Explicit basis constructions: "A Basis for the space of weakly holomorphic Drinfeld modular forms of level $T$" (Dalal, 2023)