Multiplicative Hecke Operators

• Multiplicative Hecke operators are defined via explicit product formulas that capture local factor structures in classical modular form theory.
• They extend to nonlinear settings by acting on meromorphic modular forms, identifying eta quotients as eigenforms and linking to Borcherds products and divisor maps.
• In vector-valued and noncommutative frameworks, these operators satisfy modified product identities and recurrences, unveiling deeper arithmetic and geometric insights.

Multiplicative Hecke operators are a family of constructions in which the multiplicative structure of Hecke theory is made explicit either in the operator algebra, in the target on which the operators act, or in the resulting eigenvalue relations. In the classical linear setting, this appears through prime-by-prime factorization and commuting local Hecke actions; in a genuinely nonlinear setting, one replaces additive Hecke sums by products and lets the operators act on multiplicative groups of meromorphic modular forms; and in several recent generalizations the same theme reappears in noncommutative, vector-valued, geometric, and derived contexts (Walling, 2014, Kim et al., 2024, Hain, 2023).

1. Classical multiplicativity and local Hecke structure

In the standard theory of modular and Siegel modular forms, multiplicativity is encoded by commuting local operators and by the organization of the Hecke algebra prime by prime. For Siegel Eisenstein series of arbitrary degree $n$, weight $k$, level $N$, and character $\chi$, Walling gives explicit formulas for the action of $T(p)$ and $T_j(p^2)$, diagonalizes the Eisenstein space for all primes $p\nmid N$, and, in square-free level, also diagonalizes the operators at primes dividing the level. On the averaged basis $E_{\sigma,\psi}$, the good-prime eigenvalue of $T(p)$ is

$$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$

and for the modified operators $k$0 one gets

$k$1

These product formulas are the clearest local manifestation of multiplicativity in that setting; in square-free level they lead to simultaneous diagonalization and a multiplicity-one theorem (Walling, 2014).

In half-integral weight Siegel theory, the relevant operators are $k$2, not $k$3: Walling states that “the half-integral weight Hecke operator $k$4 is $k$5 for any prime $k$6.” The local Hecke algebra is therefore organized around the square-indexed operators, and alternate generators $k$7 are introduced precisely because they simplify the Fourier-coefficient formulas and expose the local multiplicative structure. For Eisenstein series of level $k$8 with $k$9 odd and square-free, the resulting eigenvalues become explicit product expressions, for example

$N$0

and the odd-prime Hecke operators commute, allowing simultaneous diagonalization and multiplicity-one on the relevant Eisenstein subspace. A closely related reorganization of the local half-integral weight Hecke algebra replaces the standard generators $N$1 by $N$2, turning twisted Gauss-sum formulas into representation numbers and making the local structure compatible with theta series and Jacobi theory (Walling, 2016, Walling, 2011).

These results do not identify a single universal notion of “multiplicative Hecke operator,” but they show that classical Hecke theory is already governed by local product formulas, commuting prime-indexed operators, and prime-power recursions. This suggests that later nonlinear or noncommutative versions are best understood as extensions of a prime-by-prime local paradigm rather than as completely unrelated constructions.

2. Genuine multiplicative Hecke operators on meromorphic modular forms

A literal multiplicative Hecke theory is developed for the multiplicative group $N$3 of integral weight meromorphic modular forms for $N$4, with unitary multiplier system, integer Fourier coefficients, and leading coefficient $N$5. If $N$6 has weight $N$7, then for a prime $N$8 and $N$9 the multiplicative Hecke operator $\chi$0 is defined by a product of slash transforms, with different formulas according to whether $\chi$1 or $\chi$2, and $\chi$3 is the identity. For general

$\chi$4

If

$\chi$5

then $\chi$6 again has a product expansion, with weight and order transformed by

$\chi$7

where $\chi$8. For $\chi$9, the exponent formula simplifies at $T(p)$0 to

$T(p)$1

The core multiplicative relation is the product identity

$T(p)$2

so in particular $T(p)$3 when $T(p)$4. For $T(p)$5, the prime-power recursion is

$T(p)$6

while for $T(p)$7,

$T(p)$8

In this framework, a multiplicative Hecke eigenform is defined by

$T(p)$9

and the classification theorem states that for $T_j(p^2)$0 the following are equivalent: $T_j(p^2)$1 is a multiplicative Hecke eigenform; $T_j(p^2)$2 has no zeros or poles on $T_j(p^2)$3; and $T_j(p^2)$4 is an eta quotient. Thus eta quotients are exactly the multiplicative Hecke eigenforms in this setting (Kim et al., 2024).

The same paper emphasizes that this theory is adapted to infinite-product exponents $T_j(p^2)$5, divisors, and Borcherds products rather than to additive Fourier coefficients. A plausible implication is that “multiplicative Hecke operator” is most precise when the natural algebraic structure on the target is multiplication rather than addition.

3. Borcherds products, logarithmic derivatives, and divisors

The nonlinear Hecke formalism becomes especially effective when paired with Borcherds lifts and divisor maps. For a fundamental discriminant $T_j(p^2)$6 and $T_j(p^2)$7, the generalized Borcherds product of type $T_j(p^2)$8

$T_j(p^2)$9

sends a vector-valued harmonic weak Maass form to a meromorphic modular form for $p\nmid N$0 with unitary character, possibly of infinite order. On the additive side the relevant Hecke operator is $p\nmid N$1, and the equivariance theorem states that for primes $p\nmid N$2,

$p\nmid N$3

This extends Guerzhoy’s multiplicative Hecke operator to the generalized Borcherds setting attached to $p\nmid N$4 and the Weil representation (Jeon et al., 2022).

In the meromorphic-modular setting of Kim and Shin, the Borcherds product and the logarithmic derivative are Hecke equivariant for all $p\nmid N$5 coprime to the level and discriminant. If $p\nmid N$6 denotes the Borcherds product and

$p\nmid N$7

then

$p\nmid N$8

when $p\nmid N$9, and

$E_{\sigma,\psi}$0

when $E_{\sigma,\psi}$1. These identities are proved prime-power first and then extended by multiplicativity, and they place product expansions, divisor data, and additive Hecke theory in a single compatible formalism (Kim et al., 2024).

The divisor-theoretic version is even more direct. On

$E_{\sigma,\psi}$2

the multiplicative Hecke action attached to a double coset $E_{\sigma,\psi}$3 is

$E_{\sigma,\psi}$4

The divisor map to $E_{\sigma,\psi}$5 is then Hecke equivariant: for $E_{\sigma,\psi}$6,

$E_{\sigma,\psi}$7

This yields the divisor-sum identity

$E_{\sigma,\psi}$8

and is applied to Bruinier–Kohnen–Ono type formulas, Rohrlich-type divisor sums, and recurrences for polyharmonic Maass forms. In this context the paper explicitly frames additive and multiplicative Hecke operators as two representations of the same Hecke algebra $E_{\sigma,\psi}$9, with divisors naturally compatible with the multiplicative one (Jeon et al., 3 May 2025).

4. Nonabelian and noncommutative multiplicativity

A different generalization appears in Hain’s nonabelian Hecke theory on loops and conjugacy classes. Starting from a finite unramified covering $T(p)$0, one has pushforward and pullback on free homotopy classes, and the classical Hecke correspondences $T(p)$1 on the modular orbifold induce operators on

$T(p)$2

These operators preserve elliptic, parabolic, and hyperbolic classes, and the coprime multiplicativity relation survives: $T(p)$3 At a fixed prime $T(p)$4, however, the classical commutative local Hecke algebra is replaced by a noncommutative one. The auxiliary operator

$T(p)$5

enters the prime-power relation

$T(p)$6

which replaces the scalar correction term of the classical formula. On parabolic classes,

$T(p)$7

and

$T(p)$8

These formulas imply

$T(p)$9

so the algebra generated by the $$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$0 is not commutative. Hain packages the local structure as

$$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$1

with

$$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$2

and recovers the classical Hecke algebra as the quotient obtained by imposing $$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$3. Dually, the opposite algebra acts on the ring of class functions on the relative completion of $$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$4, hence on conjugation-invariant iterated Shimura integrals, preserving mixed Hodge structures and commuting with the absolute Galois action (Hain, 2023).

This construction shows that multiplicativity need not imply commutativity. Here coprime factorization survives intact, but prime-power structure becomes operator-valued and noncentral.

5. Vector-valued, analytic, and formal variants

The expression “multiplicative Hecke operators” also appears in several nonclassical analytic settings. In rational conformal field theory, Hecke operators are defined on vector-valued modular functions of conductor $$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$5. For a prime $$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$6 with $$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$7,

$$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$8

and the transformed modular representation is

$$\lambda_{\sigma,\psi}(p)= \psi_1(p)\psi_2(p^n)\prod_{i=1}^n(\psi_2\chi(p)p^{k-i}+1),$$9

These operators satisfy

$k$00

and the prime-power recursion

$k$01

At weight $k$02, the correction term is $k$03. The paper interprets this as an extension of Galois symmetry from modular data to full RCFT characters (Harvey et al., 2018).

For the Lerch zeta function, Lagarias and Li define two-variable operators

$k$04

on twisted-periodic real-analytic function spaces. The exact semigroup law

$k$05

holds on the twisted-periodic space, and for each $k$06 there is a two-dimensional space $k$07 generated by Lerch zeta functions on which every $k$08 acts by the scalar $k$09. Under twisted-periodicity and integrability hypotheses, this $k$10 is the maximal simultaneous eigenspace for the family $k$11 (Lagarias et al., 2015).

On formal power series and hypergeometric functions, the operators

$k$12

satisfy

$k$13

On the set of hypergeometric functions, the spectrum of $k$14 is

$k$15

and the principal eigenfunctions are polylogarithms and the rational functions $k$16. Simultaneous eigenfunctions for all $k$17 are exactly

$k$18

which is equivalent to complete multiplicativity of the coefficient sequence (Moll et al., 2010).

Taken together, these examples show that multiplicative Hecke structures extend beyond scalar modular forms to vector-valued, two-variable, and purely formal settings. The common feature is an operator family indexed by positive integers whose composition law is encoded directly in multiplication.

6. Geometric and arithmetic extensions

Recent work has pushed multiplicative Hecke ideas into geometric representation theory, derived Hecke theory, and Archimedean harmonic analysis. In the Jacquet–Rallis relative trace formula over local function fields, the spherical Hecke module on the symmetric side and the spherical Hecke algebra on the unitary side are compared by a canonical transfer

$k$19

and the fundamental lemma is proved using multiplicative Hitchin fibrations. For matching strongly regular semisimple elements $k$20 and matching spherical test functions $k$21, the main identity is

$k$22

At the level of Satake functions, the corresponding orbital integrals are realized as point counts on affine Jacquet–Rallis fibers. In this setting, “multiplicative” refers primarily to multiplicative Hitchin fibrations and multiplicative affine Springer fibers rather than to product-valued Hecke operators (Wang et al., 2024).

For the multiplicative group $k$23 over a number field, the derived Hecke algebra acquires an explicit local description: $k$24 and the resulting operators act on cohomology by

$k$25

When $k$26, the paper proves non-vanishing of the degree-one derived Hecke action under mild assumptions. If $k$27 does not divide the order of $k$28, then the map

$k$29

is an isomorphism of $k$30-modules. The proof uses the Grunwald–Wang theorem to show that finitely many local mod-$k$31 unit characters detect the global unit characters (Kim et al., 2024).

At the Archimedean place, the Hecke–Baxter operator for $k$32 is the $k$33-biinvariant kernel

$k$34

an element of the spherical Hecke algebra. On spherical principal series it acts by scalar multiplication with the local Archimedean $k$35-factor, and the paper reinterprets $k$36 as a generalized Whittaker function for an extension of $k$37 by a Heisenberg Lie group. It is then lifted to a corresponding extension of $k$38. This gives a representation-theoretic realization of an Archimedean multiplicative Hecke kernel as a matrix coefficient (Gerasimov et al., 2024).

These developments suggest that multiplicative Hecke operators are no longer confined to a single formalism. They appear as nonlinear product operators on meromorphic modular forms, as local commuting or twisted semigroups on analytic function spaces, as noncommutative operators on loops and conjugacy classes, and as geometric or derived actions built from spherical Hecke categories, Heisenberg extensions, and arithmetic tori.