Vectorial Drinfeld Modular Forms Theory

• Vectorial Drinfeld modular forms are non-Archimedean, Banach algebra-valued functions on Drinfeld period domains that transform under finite-dimensional representations of modular groups.
• They generalize scalar-valued Drinfeld modular forms while preserving characteristic‑p analytic features such as u‑expansions, Hecke correspondences, and rigid-analytic uniformizers.
• The theory employs explicit constructions like Eisenstein series, determinant forms, and τ‑recurrent sequences to connect analytic methods with deep arithmetic structures.

Vectorial Drinfeld modular forms are rigid-analytic functions on Drinfeld period domains that take values in finite-dimensional modules over positive-characteristic Banach algebras—most notably Tate algebras and their completions—and transform under finite-dimensional representations of Drinfeld modular groups. They extend scalar-valued Drinfeld modular forms for groups such as $\Gamma=\mathrm{GL}_2(\mathbb{F}_q[\theta])$ and, in higher rank, $\mathrm{GL}_r(A)$, while preserving the characteristic-$p$ analytic features of the theory: Carlitz and Drinfeld exponentials, $u$-expansions at cusps, Hecke correspondences, and rigid-analytic or algebro-geometric boundary conditions (Pellarin, 2019). In the literature, the subject develops along two closely related lines: the rank-two theory over Tate algebras initiated by Pellarin and further developed with Perkins [(Pellarin, 2011); (Pellarin et al., 2016)], and the broader analytic and higher-rank programs that incorporate Banach algebra coefficients, arbitrary-rank period domains, determinant constructions, and Hecke eigenforms [(Pellarin, 2019); (Perkins, 2014); (Gezmiş et al., 25 Sep 2025)].

1. Analytic origin and scalar background

The scalar theory underlying vectorial Drinfeld modular forms begins with the Carlitz module and its exponential

$\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$

an entire, surjective, non-Archimedean analytic function attached to the Carlitz lattice $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$ in $\mathbb{C}_\infty$. Its Weierstrass product expansion and functional equation provide the analytic model for later modular constructions, including the standard uniformizer

$u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$

at the cusp infinity in rank two (Pellarin, 2019).

Drinfeld $A$-modules generalize the Carlitz module through injective algebra morphisms

$$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$

with associated exponentials

$\mathrm{GL}_r(A)$0

for lattices $\mathrm{GL}_r(A)$1. This correspondence between lattices and Drinfeld modules is the analytic basis for scalar modular forms, Anderson $\mathrm{GL}_r(A)$2-modules, and the vectorial generalizations built from them (Pellarin, 2019).

For $\mathrm{GL}_r(A)$3 with $\mathrm{GL}_r(A)$4, the relevant rigid-analytic space is the Drinfeld upper half-plane

$\mathrm{GL}_r(A)$5

on which $\mathrm{GL}_r(A)$6 acts by homographies. A scalar Drinfeld modular form of weight $\mathrm{GL}_r(A)$7 is an analytic function $\mathrm{GL}_r(A)$8 satisfying

$\mathrm{GL}_r(A)$9

together with a cusp condition described by the $p$0-expansion (Pellarin, 2019). In arbitrary rank, the period domain becomes

$p$1

and the slash action is expressed through the factor $p$2 extracted from the last coordinate of $p$3 (Basson et al., 2018).

This scalar background is essential because the vectorial theory does not replace it; rather, it reinterprets modularity in representation-theoretic form and imports the same cusp-expansion machinery into Banach algebra-valued settings. A common misconception is that vectorial forms are merely tuples of scalar forms. The published constructions show that this is too restrictive: the theory admits genuinely vector-valued modular forms attached to nontrivial representations and to coefficient algebras where phenomena absent from the scalar theory occur (Pellarin, 2019).

2. Definitions, representations, and coefficient algebras

In the rank-two Tate algebra setting of Pellarin and Perkins, a vectorial modular form of weight $p$4, type $p$5, and representation $p$6 is a rigid-analytic map

$p$7

such that for every $p$8,

$p$9

supplemented by a regularity condition at infinity formulated in terms of $u$0 after a suitable twist (Pellarin et al., 2016). Here $u$1 is the Tate algebra, and $u$2 is induced by evaluation of matrix entries at the variable $u$3.

More generally, for a Banach $u$4-algebra $u$5 and a representation

$u$6

a vector-valued Drinfeld modular form of weight $u$7 is an analytic function

$u$8

satisfying

$u$9

The coefficient algebras in this framework include finite-dimensional complete ultrametric extensions and modules over Tate algebras, and the analytic structure is organized by the completed tensor product sheaf $\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$0 (Pellarin, 2019).

Pellarin’s analytic theory broadens the coefficient field further. For a finite subset $\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$1 and a representation

$\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$2

a vector-valued Drinfeld modular form of weight $\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$3 is a rigid-analytic function

$\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$4

satisfying

$\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$5

with the same type of cusp regularity (Pellarin, 2019).

A central class in that theory is formed by the “representations of the first kind.” These are finite-dimensional representations built, by direct sums, tensor products, symmetric powers, exterior powers, and contragredients, from basic evaluation-type representations. They include the arithmetic cases emphasized in later applications (Pellarin, 2019). For special subsets $\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$6, the Kronecker product representations $\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$7 play a distinguished role in explicit formulas and harmonic products (Pellarin, 2019).

In arbitrary rank, the 2025 work develops vectorial Drinfeld modular forms for a specific representation

$\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$8

where $\exp_C(z) = \sum_{i\geq 0} d_i^{-1} z^{q^i},$9 denotes reduction modulo $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$0 to $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$1. A vectorial Drinfeld modular form

$\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$2

then satisfies

$\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$3

together with a holomorphy-at-infinity condition formulated through $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$4-expansions and a matrix $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$5 (Gezmiş et al., 25 Sep 2025).

3. Expansions at infinity, uniformizers, and boundary behavior

The $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$6-expansion is the characteristic analogue of the classical $\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$7-expansion. In arbitrary rank scalar theory, one decomposes

$\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$8

and defines the expansion parameter

$\Lambda=\mathbb{F}_q[\theta]\tilde{\pi}$9

where $\mathbb{C}_\infty$0 is the Drinfeld exponential attached to the strongly discrete group $\mathbb{C}_\infty$1. Any $\mathbb{C}_\infty$2-invariant holomorphic function on $\mathbb{C}_\infty$3 admits a unique Laurent-type expansion

$\mathbb{C}_\infty$4

convergent on a neighborhood of infinity, and the coefficients $\mathbb{C}_\infty$5 are holomorphic on $\mathbb{C}_\infty$6 (Basson et al., 2018).

The decisive recursive feature is that these coefficients are themselves weak modular forms of lower rank: $\mathbb{C}_\infty$7 Holomorphicity at infinity is then defined by the condition that no negative powers occur, i.e.

$\mathbb{C}_\infty$8

The same recursion is one of the main reasons the scalar higher-rank theory is structurally relevant for vectorial extensions (Basson et al., 2018).

In the rank-two vectorial theory over Banach algebras, Pellarin introduces a “field of uniformisers” $\mathbb{C}_\infty$9, a valued field containing $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$0, described as non-discretely valued, algebraically closed, and wildly ramified over $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$1. Entrywise expansions of vectorial modular forms are uniquely represented as generalized formal series in this field (Pellarin, 2019). This is not merely a technical reformulation of the scalar $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$2-expansion: it is designed to control expansions whose coefficients lie in more complicated Banach-algebraic targets than $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$3.

Boundary expansions in higher-rank scalar theory further refine the analytic picture. For moduli varieties of rank $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$4, expansions along boundary divisors use local parameters $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$5 and $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$6, and the coefficients again reduce to lower-rank modular data. Product formulas for discriminant forms $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$7 and explicit vanishing orders are expressed in terms of partial zeta values at $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$8 (Gekeler, 2023). This suggests that any higher-rank vectorial theory compatible with compactification should inherit a recursive boundary calculus, although the cited paper notes that vectorial forms are not its central focus (Gekeler, 2023).

An adjacent algebro-geometric formulation comes from Satake compactification. For a fine open compact subgroup $u(z)=\frac{1}{\tilde{\pi}\exp_C(\tilde{\pi} z)}$9, scalar Drinfeld modular forms of weight $A$0 are global sections

$A$1

where $A$2 is the dual of the relative Lie algebra of the extended universal family over the Satake compactification $A$3 (Pink, 2010). The paper explicitly notes that this sheaf-theoretic, boundary-aware formalism is the natural framework for future vector-valued generalizations via higher-rank automorphic bundles (Pink, 2010).

4. Explicit constructions: Eisenstein series, deformations, and determinant forms

Vectorial Eisenstein series are the primary explicit examples. In the rank-two Tate algebra theory they are defined by

$A$4

and they satisfy the vectorial modular transformation law. Pellarin and Perkins show that $A$5 and $A$6 generate the corresponding modules of vectorial modular forms over scalar forms (Pellarin et al., 2016).

A broader Eisenstein framework appears in the Banach algebra-valued theory. For a morphism $A$7, vectorial Eisenstein series of the form

$A$8

are constructed, with convergence and analytic properties controlled by the ultrametric setting (Pellarin, 2019). Their relation to matrix-valued and Perkins-type series is encoded by expansions of functions such as

$A$9

whose coefficients recover Eisenstein series (Pellarin, 2019).

Pellarin’s earlier work on $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$0-recurrent sequences supplies a deformation-theoretic precursor to the modern vectorial theory. The family $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$1 is defined by the $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$2-linear recurrence

$$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$3

with $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$4 and $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$5, and interpolates between normalized Eisenstein series and para-Eisenstein series through the specializations

$$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$6

This deformation is expressed in terms of vectorial modular ingredients such as $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$7 and $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$8 and an Eisenstein-type expansion involving $$\phi : A \to \operatorname{End}_{\mathbb{F}_q}(G_a(\mathbb{C}_\infty)) \simeq \mathbb{C}_\infty[\tau],$$9 (Pellarin, 2011).

In arbitrary rank, determinant constructions organize vectorial Eisenstein series into scalar cusp forms. For

$\mathrm{GL}_r(A)$00

the matrix

$\mathrm{GL}_r(A)$01

has determinant $\mathrm{GL}_r(A)$02, which is a nowhere-vanishing, single-cuspidal deformation of Drinfeld modular forms in

$\mathrm{GL}_r(A)$03

(Perkins, 2014). The same paper constructs a nowhere-vanishing, single-cuspidal Drinfeld modular form for $\mathrm{GL}_r(A)$04 of weight $\mathrm{GL}_r(A)$05 and type $\mathrm{GL}_r(A)$06 through determinants of Anderson generating functions and rigid analytic trivializations (Perkins, 2014).

The 2025 arbitrary-rank vectorial theory refines this determinant method using twisted Eisenstein series

$\mathrm{GL}_r(A)$07

and defines

$\mathrm{GL}_r(A)$08

For $\mathrm{GL}_r(A)$09, the specialization $\mathrm{GL}_r(A)$10 is a Drinfeld cusp form of weight $\mathrm{GL}_r(A)$11 and type $\mathrm{GL}_r(A)$12, and it is a Hecke eigenform for all Hecke operators (Gezmiş et al., 25 Sep 2025).

5. Module structure, specialization, and operators

A central structural theorem in the rank-two Tate algebra setting states that for every positive integer $\mathrm{GL}_r(A)$13 and $\mathrm{GL}_r(A)$14,

$\mathrm{GL}_r(A)$15

Thus the module of vectorial modular forms is free of rank two over the module of scalar Drinfeld modular forms, generated by explicit vectorial Eisenstein series (Pellarin et al., 2016).

The arbitrary-rank 2025 paper proves an analogous decomposition for the modules $\mathrm{GL}_r(A)$16 of vectorial Drinfeld modular forms: $\mathrm{GL}_r(A)$17 where the $\mathrm{GL}_r(A)$18 are explicit vectorial modular forms constructed from Anderson generating functions (Gezmiş et al., 25 Sep 2025). In the more general Banach-algebra setting, the corresponding module statement is phrased as

$\mathrm{GL}_r(A)$19

with $\mathrm{GL}_r(A)$20 the scalar modular ring and $\mathrm{GL}_r(A)$21 the target vector space, under the hypotheses of the theory (Pellarin, 2019).

Specialization at roots of unity is one of the most distinctive arithmetic features of vectorial Drinfeld modular forms over Tate algebras. If $\mathrm{GL}_r(A)$22 is a root of an irreducible polynomial $\mathrm{GL}_r(A)$23 and $\mathrm{GL}_r(A)$24, then $\mathrm{GL}_r(A)$25 yields a modular form for congruence subgroups such as $\mathrm{GL}_r(A)$26 or $\mathrm{GL}_r(A)$27 with character induced by evaluation at $\mathrm{GL}_r(A)$28 (Pellarin et al., 2016). Hyperdifferentiation in $\mathrm{GL}_r(A)$29 and subsequent specialization extend this interpolation to higher prime-power levels (Pellarin et al., 2016). The same paper states an equivalent characterization of the growth condition at infinity in terms of modularity of infinitely many specializations (Pellarin et al., 2016).

The analytic theory for representations of the first kind establishes finite dimensionality: $\mathrm{GL}_r(A)$30 with vanishing in negative weight and further weight-one bounds (Pellarin, 2019). This result uses both specialization at roots of unity and the field of uniformisers (Pellarin, 2019). It shows that vectoriality enlarges the coefficient category and representation theory without destroying the finiteness properties expected of automorphic objects.

Hecke operators persist throughout the theory. In rank two over Tate algebras, the Hecke action preserves the submodule of vectorial modular forms, and the vectorial Eisenstein series $\mathrm{GL}_r(A)$31 are Hecke eigenforms with eigenvalue $\mathrm{GL}_r(A)$32 (Pellarin et al., 2016). In Pellarin’s analytic theory, generalized Hecke operators are defined for representations of the first kind by double coset formulas such as

$\mathrm{GL}_r(A)$33

and differential operators

$\mathrm{GL}_r(A)$34

provide vectorial analogues of Serre-type higher derivatives (Pellarin, 2019).

6. Higher-rank developments and arithmetic structure

Higher-rank scalar theory supplies several ingredients that have become structural for vectorial work. The paper on arbitrary-rank analytic theory establishes the $\mathrm{GL}_r(A)$35-expansion, recursion to lower rank, and the definition of modular and cusp forms at all cusps (Basson et al., 2018). The companion examples paper constructs Eisenstein series, coefficient forms, discriminant forms, and Hecke operators in arbitrary rank, and in the case $\mathrm{GL}_r(A)$36 shows that the ring $\mathrm{GL}_r(A)$37 is generated by certain weight one Eisenstein series, while $\mathrm{GL}_r(A)$38 and $\mathrm{GL}_r(A)$39 are generated by coefficient forms and discriminant forms (Basson et al., 2018). These scalar generation theorems are repeatedly used as the background algebra over which vectorial modules are built.

Hecke theory in higher rank exhibits explicit interaction with $\mathrm{GL}_r(A)$40-expansions. For simple Hecke operators $\mathrm{GL}_r(A)$41, the action on

$\mathrm{GL}_r(A)$42

is given by a formula involving both scaled terms $\mathrm{GL}_r(A)$43 and Goss polynomials attached to finite lattices (Basson, 2023). The same paper states that Hecke operators preserve modular forms, cusp forms, and double cusp forms, and highlights the relevance of coefficient forms, viewed there as inherently vector-valued components of higher-rank modular data (Basson, 2023). It also proves complete multiplicativity for a natural class of Hecke operators and determines the eigenvalue of the discriminant function $\mathrm{GL}_r(A)$44 (Basson, 2023).

Boundary theory adds a global arithmetic layer. For rank $\mathrm{GL}_r(A)$45, expansions of discriminant forms $\mathrm{GL}_r(A)$46 along boundary divisors admit product formulas analogous to Jacobi’s classical formula, and the orders of vanishing are expressed by values of partial zeta functions at $\mathrm{GL}_r(A)$47 (Gekeler, 2023). The same work proves linear independence of Eisenstein series $\mathrm{GL}_r(A)$48 and obtains decompositions

$\mathrm{GL}_r(A)$49

for suitable weights (Gekeler, 2023). Although vectorial forms are not the main subject there, the paper states that this expansion theory provides the analytic underpinning for vector-valued Drinfeld modular forms (Gekeler, 2023).

A longer-term extension of the field is visible in the 2025 family of Hecke eigenforms. That paper explicitly describes itself as developing the theory of vectorial Drinfeld modular forms for arbitrary rank by using a particular representation and constructing determinant Hecke eigenforms from twisted Eisenstein series (Gezmiş et al., 25 Sep 2025). This suggests a transition from an originally rank-two theory toward a coherent arbitrary-rank representation-theoretic framework.

7. Scope, misconceptions, and open directions

Several points clarify the current scope of the subject. First, vectorial Drinfeld modular forms are not simply an alternative notation for several scalar modular forms written in a column. The cited works emphasize representations, Banach algebra coefficients, and specializations that produce modular forms for congruence subgroups with character; these features go beyond coordinatewise repetition of scalar theory (Pellarin et al., 2016, Pellarin, 2019).

Second, the theory is not confined to rank two, but the mature parts of the literature remain unevenly distributed. Rank two is the setting of the complete structure theorem over Tate algebras, explicit Hecke stability, root-of-unity specialization, and much of the deformation theory [(Pellarin et al., 2016); (Pellarin, 2011)]. Arbitrary-rank scalar theory is already extensive [(Basson et al., 2018); (Basson et al., 2018); (Pink, 2010)], while arbitrary-rank vectorial theory is developed more recently through determinant constructions, particular representations, and Hecke eigenforms [(Perkins, 2014); (Gezmiş et al., 25 Sep 2025)].

Third, some parts of the theory remain conjectural. Pellarin’s analytic theory proves a harmonic product formula for special representations $\mathrm{GL}_r(A)$50 and, together with three conjectures on the structure of an $\mathrm{GL}_r(A)$51-algebra of $\mathrm{GL}_r(A)$52-periodic multiple sums, derives conjectural formulas for Eisenstein series; some of these formulas can be proved (Pellarin, 2019). This is a controlled conjectural zone rather than a gap in the basic definitions.

The published literature identifies several directions of development. One direction is harmonic cocycle theory and explicit descriptions of modular forms as sections over analytic quotients; another is connection with $\mathrm{GL}_r(A)$53-functions, Galois representations, and Iwasawa-theoretic questions, as indicated in the course notes on the transition from the Carlitz exponential to vector modular forms (Pellarin, 2019). Another is the sheaf-theoretic extension from line bundles to higher-rank automorphic bundles on compactified moduli spaces, foreshadowed in the Satake-compactification framework (Pink, 2010). A further direction is the systematic production of Hecke eigenforms in arbitrary rank through determinants of twisted Eisenstein series and their relation to Anderson generating functions and the Anderson–Thakur function (Gezmiş et al., 25 Sep 2025).

Taken together, the literature presents vectorial Drinfeld modular forms as a characteristic-$\mathrm{GL}_r(A)$54, non-Archimedean analogue of vector-valued automorphic forms, but one with distinctive analytic mechanisms: Tate algebra coefficients, Frobenius-twisted recurrences, wild ramification in the field of uniformisers, specialization at roots of unity, and determinant constructions tied to Drinfeld and Anderson modules [(Pellarin, 2019); (Pellarin, 2011)]. The subject is therefore both an extension of classical modular ideas and a specifically function-field theory with its own representation-theoretic and arithmetic geometry.