Drinfeld Cusp Forms

• Drinfeld cusp forms are rigid-analytic functions defined on the Drinfeld upper half-plane that vanish at cusps, serving as the function field analog of classical modular forms.
• Their theory integrates rigid analytic geometry, non-Archimedean harmonic analysis, and representation theory to elucidate structures in arithmetic geometry.
• Key techniques involve Fourier-Jacobi expansions, Hecke operator analysis, and harmonic cocycle realizations, which underpin applications in the Langlands program.

A Drinfeld cusp form is a rigid-analytic function on the Drinfeld upper half-plane for a global function field, which transforms according to a specified factor of automorphy with respect to a given arithmetic subgroup and exhibits prescribed vanishing at the cusps (boundary divisors) of the modular curve or higher rank Drinfeld moduli space. Drinfeld cusp forms are the analogues, over function fields, of classical cusp forms in the theory of modular forms over number fields. Their theory intertwines rigid analytic geometry, the arithmetic of moduli spaces of Drinfeld modules, non-Archimedean harmonic analysis, and representation theory of p-adic and finite matrix groups. Fundamental objects in the Langlands program for function fields, Drinfeld cusp forms serve as the basic source of eigenforms for Hecke algebras in positive characteristic.

1. Definition and Basic Properties

Let $A$ be the ring of functions regular away from a fixed place $\infty$ of a global function field $F$ of characteristic $p$, with constant field $\mathbb{F}_q$. For instance, $A=\mathbb{F}_q[T]$. The Drinfeld upper half-plane $\Omega$ is the complement in $\mathbb{P}^1(C_\infty)$ of $\mathbb{P}^1(F_\infty)$, where $C_\infty$ is the completion of an algebraic closure of $F_\infty$.

Given an arithmetic subgroup $\Gamma\subset\mathrm{GL}_r(A)$ (e.g., $\Gamma_0(\mathfrak{n})$), weight $k\in\mathbb{Z}$ and type $m\in\mathbb{Z}/(q-1)$, a Drinfeld modular form $f:\Omega^r\rightarrow C_\infty$ satisfies the transformation property:

$$f(\gamma\cdot z) = \det(\gamma)^m \, j(\gamma,z)^k f(z), \qquad \forall \gamma\in\Gamma,$$

where $j(\gamma,z)$ is the automorphy factor (e.g., $cz+d$ for $r=2$). Holomorphy at the cusps requires that $f$ be analytic at each boundary divisor. The subspace of forms vanishing at each cusp defines the space of Drinfeld cusp forms $S_{k,m}(\Gamma)$ (Valentino, 2020, Gekeler, 2023, Vries, 2024, Bandini et al., 2018).

Cusp forms admit Fourier-Jacobi expansions (u-expansions at the boundary), and the vanishing of the constant term in all such local expansions is the hallmark condition:

$$f(z) = \sum_{n=0}^\infty a_n u(z)^n, \qquad \text{cuspidality} \Leftrightarrow a_0=0.$$

(Gekeler, 2023, Pellarin, 2019, Petrov, 2012).

The vector space $S_{k,m}(\Gamma)$ is always finite-dimensional; explicit dimension formulas involve representation theory and arithmetic geometry (Boeckle et al., 24 Feb 2025, Pink, 2019, Vries, 2024).

2. Fourier Expansions, Boundary Behavior, and Compactification

A key technical apparatus is provided by expansions of modular forms at the boundary. For $A=\mathbb{F}_q[T]$, the uniformizing parameter at the standard cusp $\infty$ is given by $u(z) = e_A(z)^{-1}$, with $e_A$ the Carlitz exponential. General coordinates near boundary divisors for higher rank involve additional parameters, and expansions take the form:

$f(w) = \sum_{n\ge n_0} b_n(w') u(w)^n,$

where $w'$ varies on a lower rank Drinfeld half-plane (Gekeler, 2023). The cusp criterion states that $f$ is a cusp form if and only if all coefficients $b_n(w')=0$ for $n\le 0$, for all choices of local boundary parameter (Gekeler, 2023).

The algebraic geometric perspective realizes Drinfeld cusp forms as global sections vanishing on the boundary of a (Satake or $A$-reciprocal) compactification of the moduli space of Drinfeld modules of rank $r$ with level $N$. In the $A$-reciprocal compactification, cusp forms correspond to the graded ideal generated by "cusp monomials," and explicit formulas for dimensions and bases can be deduced via module-theoretic and group-theoretic arguments (Pink, 2019).

Product expansions for distinguished forms such as discriminants $\Delta_n$ generalize Jacobi's product for the elliptic discriminant and encode vanishing orders at cusps in terms of special values of partial zeta functions (Gekeler, 2023).

3. Hecke Operators, Trace Formulas, and Ramanujan Bounds

Drinfeld cusp forms carry natural actions of Hecke operators $T_\mathfrak{p}$ for primes $\mathfrak{p}$ of $A$:

$$(T_\mathfrak{p} f)(z) = P^{k-1} f(Pz) + \sum_{\deg b<\deg P} P^{-1} f\left(\frac{z+b}{P}\right).$$

Hecke operators preserve cuspidality and endow $S_{k,m}(\Gamma)$ with a commuting family of endomorphisms (Vries, 2024, Valentino, 2020, Bandini et al., 2018). Oldforms, newforms, and Atkin-Lehner involutions can be defined by analogy with the classical theory, using explicit degeneracy maps and trace operators (Valentino, 2020, Bandini et al., 2019).

Closed-form expressions for traces of Hecke operators for degree $\leq2$ primes have been established, and these traces inform explicit and effective Ramanujan bounds for eigenvalues and slopes of Hecke eigenforms (Vries, 2024). The Ramanujan bound asserts that the absolute value (suitably normalized) of the eigenvalues is bounded by a function of the weight, type, and degree of the prime.

Trace formulas in terms of isogeny classes of Drinfeld modules and their characteristic polynomials provide arithmetic interpretations analogous to Eichler–Selberg trace formulae (Vries, 2024).

4. Oldforms, Newforms, and Atkin-Lehner Theory

The oldform/newform framework for Drinfeld cusp forms, particularly at prime level, is developed via degeneracy maps $\delta_1,\delta_\mathfrak{p}$ from level $1$ to level $\mathfrak{p}$, and the newform subspace is characterized as the intersection of the kernels of certain trace operators (including twists by Fricke involutions) (Valentino, 2020, Bandini et al., 2018, Bandini et al., 2019, Vries, 2024). The direct sum decomposition into old and new subspaces is governed by the invertibility of a specific operator and is directly verified in cases of dimension one, see (Valentino, 2020, Bandini et al., 2019).

A critical result: $$S_{k,l}(\Gamma_0(\mathfrak{p})) = S^{\text{old}} \oplus S^{\text{new}}$$ if and only if $\pm P^{(k-2)/2}$ is not a Hecke eigenvalue at level $1$ (Vries, 2024). In odd characteristic, Atkin-Lehner theory yields a commutative algebra of Hecke operators and involutions simultaneously diagonalizable on the new subspace (Valentino, 2020, Bandini et al., 2017).

5. Dimension Formulae and Harmonic Cocycle Realization

Teitelbaum's isomorphism realizes Drinfeld cusp forms as $\Gamma$-invariant harmonic cocycles on the Bruhat-Tits tree with values in certain algebraic representations (Boeckle et al., 2021, Boeckle et al., 24 Feb 2025). This concrete realization interprets cuspidality as vanishing of cocycles at the boundary of the tree, and harmonic conditions correspond to modularity and vanishing at the cusps.

Dimension formulas for $S_{k,l}(\mathrm{SL}_2(A))$ are given by

$$d_{k,q} = \dim_{F} C_{\mathrm{har}}(L_k)^\Gamma = \frac{1}{2}\left(\frac{1}{q-1} \sum_{\zeta\in\mu_{q-1}} \psi_{\overline{L}_k}(A_\zeta) - \frac{1}{q+1} \sum_{\xi\in\mu_{q+1}} \psi_{\overline{L}_k}(A_\xi)\right),$$

where $\psi_{\overline{L}_k}$ is the Brauer character of the appropriate highest-weight representation (Boeckle et al., 24 Feb 2025). Asymptotically, the dimension of cusp forms grows proportionally to the dimension of the irreducible representation, with explicit ratios depending on $p$ and $q$. For $A = \mathbb{F}_q[t]$ and small $q$, combinatorial formulas are available (Boeckle et al., 24 Feb 2025, Boeckle et al., 2021).

6. Special Constructions, A-Expansions, and Vector-Valued Forms

Drinfeld cusp forms with $A$-expansions, as introduced by Petrov, admit expansions indexed by $A_+$ (monic polynomials) with coefficients related by explicit congruence and Hecke eigenvalue recursion; this structure produces infinite families of explicit eigenforms and sharp congruence relations (Petrov, 2012). Within the $A$-expansion subspace, a restricted form of the multiplicity-one theorem holds.

Vector-valued Drinfeld modular forms, with targets in representations of "the first kind" on Banach algebras, admit q-expansions and eigen-theory generalizing the scalar-valued case. Poincaré series constructions are available for both scalar and vector-valued cusp forms, and dimension formulas rely on the representation-theoretic data of the coefficient module (Pellarin, 2019).

In higher rank, the Legendre determinant form yields nowhere-vanishing, single-cuspidal Drinfeld modular forms of least possible weight (Perkins, 2014).

7. Open Questions and Structural Directions

The analytic and algebraic geometry underlying Drinfeld cusp forms provides finer invariants than in the classical setting, such as p-adic variation of slopes and potential for nontrivial Jordan block structure among Hecke operators. For even characteristic, the diagonalizability of Atkin $U$-operators can fail, and slope patterns display periodicity and congruence phenomena reminiscent of Gouvêa–Mazur conjectures (Bandini et al., 2017, Bandini et al., 2018).

Geometric constructions involving compactifications (Satake and $A$-reciprocal), moduli interpretations, and boundary vanishing ideals furnish explicit presentations and dimension counts for spaces of cusp forms (Pink, 2019, Gekeler, 2023).

Potential further developments include extensions of the algebraic and representation-theoretic dimension formulas to higher-level subgroups, twisted forms, and for groups beyond $\mathrm{GL}_2$; p-adic interpolation phenomena; and a deeper understanding of Hecke algebra semisimplicity and rationality patterns for eigenvalues (Boeckle et al., 24 Feb 2025, Boeckle et al., 2021).

---

Summary Table: Key Structures in Drinfeld Cusp Form Theory

Structure/Operator	Description, Role
$S_{k,m}(\Gamma)$	Space of cusp forms of weight $k$, type $m$
$T_\mathfrak{p}$	Hecke operator at prime $\mathfrak{p}$
$U_\mathfrak{p}$	Atkin operator at level divisible by $\mathfrak{p}$
Oldforms/Newforms	Decomposition via degeneracy/trace maps
Harmonic cocycle realization	Teitelbaum isomorphism, basis for arithmetic
A-expansion	Expansion in $t_a$ indexed by monic $a \in A_+$
Dimension/slope formula	Via Brauer characters, asymptotics
Compactification, ideal $I$	Geometric/algebro-coho. definition of cusp forms

---

References: (Valentino, 2020, Boeckle et al., 24 Feb 2025, Boeckle et al., 2021, Gekeler, 2023, Bandini et al., 2018, Bandini et al., 2017, Vries, 2024, Pink, 2019, Pellarin, 2019, Petrov, 2012, Perkins, 2014, Bandini et al., 2019).