Drinfeld-Hecke Eigenforms Overview

• Drinfeld-Hecke eigenforms are Drinfeld modular forms that act as simultaneous eigenfunctions for Hecke operators on function-field analogues of modular curves.
• They utilize techniques such as u-expansions, Goss polynomials, and twisted Eisenstein series to derive explicit eigenvalues and structural properties.
• The theory exposes novel features like the failure of multiplicity one and natural Hecke-stable subquotients, emphasizing deep connections with representation theory.

Drinfeld-Hecke eigenforms are Drinfeld modular forms on function-field analogues of modular curves or higher-rank period domains that are simultaneous eigenvectors for the relevant Hecke operators. In the contemporary literature, the subject spans rank-$2$ Drinfeld-Goss modular forms, higher-rank forms on $\Omega^r$, forms described by $u$-, $t$-, or $A$-expansions, vectorial Drinfeld modular forms built from twisted Eisenstein series, and $\wp$-adic finite-slope families. Across these settings, the theory combines explicit double-coset operators, Goss polynomials, product formulas, harmonic-cochain models, and overconvergent or Hida-theoretic methods; it also exhibits structural features not captured by a naive classical analogy, including failure of multiplicity one, nontrivial semisimple Hecke action modulo $T$, and natural Hecke-stable subquotients arising from representation theory (Basson, 2023, Petrov, 2012, Boeckle et al., 2021, Joshi et al., 2013).

1. Analytic setting and Hecke correspondences

For higher-rank Drinfeld modular forms in the sense of (Basson, 2023), the basic ground ring is

$$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$

with $C_\infty$ the completion of an algebraic closure of $F_\infty$. The Drinfeld period domain $\Omega^r$0 is the complement of all $\Omega^r$1-rational hyperplanes in $\Omega^r$2, and a point is written

$\Omega^r$3

There is an action of $\Omega^r$4 on $\Omega^r$5, and the weight-$\Omega^r$6 slash operator is

$\Omega^r$7

where $\Omega^r$8 is the normalization factor given by the last entry of $\Omega^r$9. A weak Drinfeld modular form of rank $u$0, weight $u$1, for a congruence subgroup $u$2, is a holomorphic function $u$3 satisfying $u$4 for $u$5; it is a modular form when it is holomorphic at the cusps, formulated via a $u$6-expansion at infinity (Basson, 2023).

Hecke operators are defined by double cosets. For congruence subgroups $u$7 and $u$8,

$u$9

where $t$0 runs through representatives of $t$1. A basic operator is attached to

$t$2

for an irreducible polynomial $t$3. The paper gives explicit representatives $t$4 for the double coset $t$5, and in rank $t$6 recovers the familiar formula

$t$7

up to normalization conventions used there (Basson, 2023).

In the rank-$t$8 setting of $t$9-expansions, the Hecke operator at a prime ideal $A$0 with monic generator $A$1 is written

$A$2

where $A$3. This formula already makes visible the close interaction between Hecke indices and the polynomial arithmetic of $A$4 (Petrov, 2012).

2. Expansion theories, Goss polynomials, and coefficient extraction

The higher-rank theory in (Basson, 2023) is organized around the $A$5-parameter attached to the lattice generated by the last $A$6 coordinates. If $A$7 is the relevant lattice, then

$A$8

where $A$9 is the exponential of the lattice $\wp$0. Every weak modular form admits a $\wp$1-expansion

$\wp$2

with uniquely determined coefficient functions $\wp$3. The Hecke action on these expansions is governed by Goss polynomials $\wp$4, which in the paper are recalled to satisfy: $\wp$5 is monic of degree $\wp$6; if $\wp$7, then $\wp$8; if $\wp$9, then $T$0; and every nonzero exponent in $T$1 is congruent to $T$2. Theorem 3.6 gives the central “Hecke-on-$T$3-expansion” formula: the simple Hecke operator sends

$T$4

to a sum whose first term replaces $T$5 by $T$6, while the remaining coset representatives contribute Goss polynomials $T$7. A basic corollary is

$T$8

The same paper proves that Hecke operators map modular forms to modular forms, cusp forms to cusp forms, and double cusp forms to double cusp forms; it also proves complete multiplicativity for the natural operators

$T$9

namely $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$0 and $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$1 for coprime $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$2 (Basson, 2023).

Petrov’s $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$3-expansions replace the usual indexing by natural numbers with indexing by monic polynomials in $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$4. An $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$5-expansion of exponent $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$6 has the form

$$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$7

where $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$8 is the set of monic polynomials in $$A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})),$$9, $C_\infty$0, and $C_\infty$1 is the $C_\infty$2-th Goss polynomial attached to the Carlitz module. For fixed exponent $C_\infty$3, the $C_\infty$4-expansion is unique. Its importance for Hecke theory is that if $C_\infty$5 has an $C_\infty$6-expansion with exponent $C_\infty$7 and is an eigenform for $C_\infty$8, then

$C_\infty$9

and more generally, if $F_\infty$0, then

$F_\infty$1

The Hecke eigenvalue is therefore exactly $F_\infty$2, and the eigensystem is encoded directly in the expansion index (Petrov, 2012).

For level-$F_\infty$3 double-cuspidal Drinfeld-Goss eigenforms with degree-one Hecke eigenvalues of power type,

$F_\infty$4

the coefficient behavior is subtler than in the classical normalized-eigenform setting. The paper (El-Guindy, 2017) isolates coefficients $F_\infty$5 and proves the closed formula

$F_\infty$6

together with the exact vanishing criterion

$F_\infty$7

This gives a large explicit family of coefficients determined by the Hecke eigensystem, while also showing that the relation between eigenvalues and expansion coefficients is indirect and highly structured in the Drinfeld setting (El-Guindy, 2017).

3. Canonical eigenforms and explicit eigenvalues

Several papers identify concrete Drinfeld-Hecke eigenforms with explicit eigenvalues. The literature studies different Hecke operators, including the simple operator attached to $F_\infty$8 and the higher-rank family $F_\infty$9, so the eigenvalue formulas are attached to the chosen operator.

Under the simple Hecke operator of (Basson, 2023), the rank-$\Omega^r$00 Drinfeld discriminant function $\Omega^r$01 is an eigenform with eigenvalue

$\Omega^r$02

The proof uses the one-dimensionality of cusp forms of weight $\Omega^r$03, the product formula for $\Omega^r$04, and the $\Omega^r$05-expansion formula. The same paper proves that the coefficient forms $\Omega^r$06 are also eigenforms for the simple $\Omega^r$07: for $\Omega^r$08, the eigenvalue is again $\Omega^r$09, while for $\Omega^r$10 it is

$\Omega^r$11

The paper also notes that each coefficient form associated to the discriminant is an eigenform as well (Basson, 2023).

The 2025 higher-rank paper (Gekeler, 3 Nov 2025) studies modular forms for $\Omega^r$12 and the operators $\Omega^r$13. It recalls that the ring of modular forms is generated by

$\Omega^r$14

with $\Omega^r$15, and proves that $\Omega^r$16 are eigenforms for all $\Omega^r$17, with

$\Omega^r$18

where $\Omega^r$19 is the monic generator of $\Omega^r$20. In particular,

$\Omega^r$21

The same paper proves that

$\Omega^r$22

It further studies the growth of the $\Omega^r$23-expansion coefficients of $\Omega^r$24 and shows that the product expansion of $\Omega^r$25 and the $\Omega^r$26-expansion of each modular form converge on the natural fundamental domain (Gekeler, 3 Nov 2025).

A different higher-rank construction appears in (Gezmiş et al., 25 Sep 2025), where determinants of twisted Eisenstein series produce a family of cusp forms $\Omega^r$27, $\Omega^r$28, of weight

$\Omega^r$29

and type $\Omega^r$30. These are Hecke eigenforms for $\Omega^r$31 with eigenvalue

$\Omega^r$32

independent of $\Omega^r$33 once $\Omega^r$34 (Gezmiş et al., 25 Sep 2025).

Form	Hecke operator	Eigenvalue
$\Omega^r$35	simple $\Omega^r$36	$\Omega^r$37
$\Omega^r$38 ($\Omega^r$39)	simple $\Omega^r$40	$\Omega^r$41
$\Omega^r$42	simple $\Omega^r$43	$\Omega^r$44
$\Omega^r$45	$\Omega^r$46	$\Omega^r$47
$\Omega^r$48	$\Omega^r$49	$\Omega^r$50
$\Omega^r$51	$\Omega^r$52	$\Omega^r$53
$\Omega^r$54	$\Omega^r$55	$\Omega^r$56
$\Omega^r$57	$\Omega^r$58	$\Omega^r$59

4. Constructive families and twisted eigensystems

A major source of explicit Drinfeld-Hecke eigenforms is Petrov’s infinite family

$\Omega^r$60

defined for integers $\Omega^r$61 satisfying that $\Omega^r$62 is a positive multiple of $\Omega^r$63 and $\Omega^r$64. The paper proves that

$\Omega^r$65

and that

$\Omega^r$66

A notable special subfamily is

$\Omega^r$67

whose members are cuspidal, not double-cuspidal, and occur in the quotient $\Omega^r$68, where the eigenforms all have eigenvalues $\Omega^r$69. Petrov also proves a restrictive multiplicity-one statement inside the class of forms with $\Omega^r$70-expansions: if an eigenform $\Omega^r$71 has an $\Omega^r$72-expansion with exponent $\Omega^r$73, then

$\Omega^r$74

so within that class the weight and the eigensystem determine the form (Petrov, 2012).

Twisting by Dirichlet characters produces another explicit mechanism. For square-free monic $\Omega^r$75 and a Dirichlet character $\Omega^r$76, the projection operator

$\Omega^r$77

sends forms to forms with character and preserves cuspidality. Its central Hecke property is

$\Omega^r$78

for monic irreducible $\Omega^r$79 coprime to the relevant level, so if $\Omega^r$80 is a Hecke eigenform with $\Omega^r$81 and $\Omega^r$82, then

$\Omega^r$83

The same paper emphasizes that the effect on $\Omega^r$84-expansions and on Petrov’s $\Omega^r$85-expansions is more complicated than coefficientwise twisting, because the translated arguments $\Omega^r$86 are related to $\Omega^r$87 by nontrivial rational expressions involving the Carlitz exponential. It also constructs Eisenstein series with character $\Omega^r$88 for irreducible level $\Omega^r$89, proves that they and their Fricke transforms are Hecke eigenforms, and gives the eigenvalue formulas

$\Omega^r$90

for $\Omega^r$91 (Perkins, 2017).

The determinant construction of (Gezmiş et al., 25 Sep 2025) extends this constructive program to arbitrary rank by assembling twisted Eisenstein series into a matrix $\Omega^r$92, extracting the $\Omega^r$93-cofactor $\Omega^r$94, and embedding it into a vectorial Drinfeld modular form $\Omega^r$95. The resulting specializations $\Omega^r$96 provide a higher-rank tower of scalar Hecke eigenforms with explicit weight, type, and eigenvalue data (Gezmiş et al., 25 Sep 2025).

5. Rigidity, failure phenomena, and Hecke-stable building blocks

A recurring theme is that Drinfeld-Hecke eigenforms do not satisfy a naive analogue of classical multiplicity one. Petrov emphasizes that forms such as

$\Omega^r$97

can share the same eigensystem, so multiplicity one fails in general. The restrictive theorem for $\Omega^r$98-expansion forms therefore applies only inside a very special class, and the same paper explicitly notes that there are eigenforms that do not seem to admit any $\Omega^r$99-expansion (Petrov, 2012).

The literature also gives direct non-eigenform criteria. In higher rank, (Basson, 2023) proves that if $u$00 and $u$01, then $u$02 is not an eigenform for a certain Hecke operator $u$03. Modulo $u$04, the paper (Joshi et al., 2013) shows that many powers of the normalized cuspidal form $u$05 cannot be eigenforms: if $u$06 has degree $u$07 and $u$08, then

$u$09

is not an eigenform for $u$10. It also proves that for

$u$11

the form $u$12 is an eigenform modulo $u$13 precisely when $u$14. These statements exhibit explicit arithmetic obstructions to eigenform status (Basson, 2023, Joshi et al., 2013).

The Hecke algebra itself behaves differently from the classical mod-$u$15 picture. For the Hecke algebra $u$16 acting on $u$17, the semisimple quotient modulo $u$18 is described explicitly by

$u$19

with $u$20. Thus the semisimple part modulo $u$21 is nonzero, in sharp contrast with the classical nilpotence phenomenon cited in the paper. The same work conjectures that for each $u$22 there exist large $u$23 and suitable $u$24 such that

$u$25

which would imply that the Hecke algebra is not smooth in large weight (Joshi et al., 2013).

A more structural explanation for the failure of a naive Maeda-type picture appears in (Boeckle et al., 2021). Using an adelic extension of Teitelbaum’s isomorphism,

$u$26

together with

$u$27

the paper proves that the functor $u$28 is exact on finite-dimensional $u$29-modules. Any filtration of the coefficient representation $u$30 therefore induces a Hecke-stable filtration on cusp forms, and the associated graded pieces are harmonic-cochain spaces for the simple composition factors $u$31. Numerical data for $u$32 then show a conjecturally infinite supply of $u$33-rational eigenforms in weights $u$34, even after the obvious representation-theoretic obstructions are removed (Boeckle et al., 2021).

6. $u$35-adic families, finite slope, and classicity

The finite-slope theory studies Drinfeld-Hecke eigenforms varying with the weight. In (Hattori, 2019), for spaces

$u$36

the key hypotheses are a degree-one prime factor condition, one-dimensionality of the generalized slope-$u$37 eigenspace, and a sufficiently small slope bound. Under these assumptions, fixing a slope-$u$38 Hecke eigenform $u$39, the paper constructs a family

$u$40

of Hecke eigenforms $u$41 of the same slope $u$42. For any $u$43, their Hecke eigenvalues satisfy the congruence estimate

$u$44

The construction uses weight-reduction maps on coefficient modules, local constancy of slope dimensions, and perturbation lemmas for Hecke matrices (Hattori, 2019).

A broader higher-rank framework is developed in (Nicole et al., 2018). There, Hida theory is built for rank-$u$45 Drinfeld modular forms of slope zero for a suitably defined $u$46-operator. The ordinary projector is

$u$47

and the ordinary part is finite over the relevant weight algebra. In the finite-slope setting, the paper constructs a Fredholm determinant

$u$48

a spectral variety

$u$49

and a Hecke variety $u$50 parametrizing Hecke eigensystems on finite-slope overconvergent forms. It also proves a classicity criterion: if an overconvergent Drinfeld modular form of weight $u$51 is a $u$52-eigenform with

$u$53

then $u$54 is classical. This establishes a function-field analogue of the ordinary and finite-slope eigenform machine familiar from classical $u$55-adic modular forms, but in a distinctly Drinfeld setting (Nicole et al., 2018).