Taelman Class Groups with Moduli

• Taelman class groups with moduli are function-field analogues of ray class groups that integrate local congruence conditions in Drinfeld and Anderson t-modules.
• Different approaches—including exact-sequence methods, z-deformations with Stark units, and P-adic corrections—modify Euler factors or local units to impose modular constraints.
• Iwasawa theory and equivariant K-theory further extend the framework by providing precise control over asymptotic growth and refined class number formulas.

Searching arXiv for the cited papers and closely related work on Taelman class groups with moduli. Taelman class groups with moduli are function-field analogues of ray class groups formed in the setting of Drinfeld modules and, more broadly, Anderson $t$-modules. In the recent literature, the phrase “with moduli” is realized in several technically distinct but closely related ways: by imposing local ray-type conditions in exact sequences defining $U_f(E/O_K)$ and $H_f(E/O_K)$; by using $z$-deformations and Stark-unit lattices whose specialization at $z=1$ encodes congruence conditions; by removing Euler factors at a fixed finite prime $P$ and inserting explicit local correction factors in $P$-adic class formulas; and by truncating Euler products in equivariant $K$-theoretic refinements (Kataoka et al., 8 Sep 2025, Anglès et al., 2015, Lucas, 4 Apr 2025, Frutos-Fernández et al., 2023). Across these formulations, the guiding principle is that a modulus is represented by prescribed local conditions at finite places, or equivalently by a controlled modification of local Euler factors.

1. Classical Taelman modules and the role of local conditions

The unmodified Taelman class group is the baseline object from which modular variants are defined. For a global function field $Q$ with constant field $\mathbb F_q$, a distinguished place $U_f(E/O_K)$0, and coefficient ring $U_f(E/O_K)$1 of functions regular outside $U_f(E/O_K)$2, let $U_f(E/O_K)$3 be a global $U_f(E/O_K)$4-field with integer ring $U_f(E/O_K)$5. If $U_f(E/O_K)$6 is a Drinfeld $U_f(E/O_K)$7-module over $U_f(E/O_K)$8, with exponential map

$U_f(E/O_K)$9

then for the product over infinite places $H_f(E/O_K)$0 one defines

$H_f(E/O_K)$1

The corresponding unit groups are

$H_f(E/O_K)$2

There is an exact sequence

$H_f(E/O_K)$3

and also

$H_f(E/O_K)$4

Moreover, $H_f(E/O_K)$5 is finite over $H_f(E/O_K)$6, while $H_f(E/O_K)$7 is finitely generated of rank $H_f(E/O_K)$8, where

$H_f(E/O_K)$9

for $z$0 (Kataoka et al., 8 Sep 2025).

For Anderson $z$1-modules the same architecture persists. If $z$2, $z$3, $z$4 is finite with integral closure $z$5, and $z$6 is an Anderson $z$7-module over $z$8, then one has

$z$9

$z=1$0

Demeslay’s finiteness and lattice results yield that $z=1$1 is an $z=1$2-lattice in $z=1$3 and $z=1$4 is finite. The associated archimedean Euler factors $z=1$5 define a convergent product

$z=1$6

and the class formula reads

$z=1$7

(Lucas, 4 Apr 2025).

These classical formulas make clear why moduli enter naturally: the Taelman class group is built from global points, infinite-place exponential data, and local Euler factors. A modulus modifies one or more of these ingredients.

Framework	Modulus mechanism	Resulting object
Drinfeld-module Iwasawa theory	local conditions $z=1$8	$z=1$9, $P$0
$P$1-deformation/Stark units	specialization at $P$2, modified Euler factors	$P$3, ray-class-type modules
$P$4-adic Anderson $P$5-modules	remove Euler factor at $P$6, insert $P$7	corrected $P$8-adic $P$9-series
Equivariant $P$0-theory	$P$1-truncated Euler products	$P$2-modified complexes and class modules

2. Exact-sequence definitions of Taelman class groups with moduli

A direct analogue of the ray class construction appears in the Iwasawa-theoretic study of Drinfeld modules. Let $P$3 be a nonzero ideal of $P$4, viewed as a modulus

$P$5

with $P$6 ranging over finite places of $P$7. The unit group $P$8 and class group $P$9 with modulus $K$0 are defined by the exact sequence

$K$1

The diagonal map $K$2 encodes both the infinite-place contribution and the local ray condition at each finite $K$3 through the subgroup $K$4 (Kataoka et al., 8 Sep 2025).

This definition is functorial in the modulus. If $K$5, then there is a natural exact sequence

$K$6

In particular,

$K$7

The modular groups retain the same finiteness profile as the classical ones: for any modulus $K$8, $K$9 is finite as an $Q$0-module, and $Q$1 is finitely generated of rank $Q$2 (Kataoka et al., 8 Sep 2025).

The same paper notes that one can equivalently work with $Q$3-integers $Q$4 for a finite $Q$5 containing the support of $Q$6. This is structurally parallel to the distinction between ray class groups and $Q$7-ray class groups in number fields. The analogy is explicit: the construction mirrors the number field ray class setting where local conditions at $Q$8 enforce congruences modulo $Q$9, and the trivial modulus $\mathbb F_q$0 recovers Taelman’s original class group (Kataoka et al., 8 Sep 2025).

A common misconception is that all recent work on moduli defines a separate object $\mathbb F_q$1 in this exact-sequence sense. That is correct for the Iwasawa-theoretic framework above, but not for every other approach.

3. $\mathbb F_q$2-deformations, Stark units, and ray-class-type quotients

A second realization of moduli uses deformation in an auxiliary variable $\mathbb F_q$3. For a rank-one Drinfeld $\mathbb F_q$4-module $\mathbb F_q$5 over $\mathbb F_q$6, the canonical $\mathbb F_q$7-deformation $\mathbb F_q$8 is defined by

$\mathbb F_q$9

The associated deformed class module is

$U_f(E/O_K)$00

which is a finitely generated torsion $U_f(E/O_K)$01-module and specializes at $U_f(E/O_K)$02 back to $U_f(E/O_K)$03 (Anglès et al., 2015).

The Stark-unit submodule is defined by evaluation at $U_f(E/O_K)$04: $U_f(E/O_K)$05 where

$U_f(E/O_K)$06

The fundamental relation is that the quotient on the unit side is controlled by the $U_f(E/O_K)$07 fiber of the deformed class module: $U_f(E/O_K)$08 The paper further proves

$U_f(E/O_K)$09

In this sense, the $U_f(E/O_K)$10 specialization measures a ray-class-type defect on the unit side (Anglès et al., 2015).

The modular interpretation becomes explicit when Euler factors at primes dividing an ideal $U_f(E/O_K)$11 are removed or altered. Fixing a finite set of finite primes $U_f(E/O_K)$12, or equivalently an ideal $U_f(E/O_K)$13, one defines partial or modified $U_f(E/O_K)$14-series by omitting Euler factors at primes dividing $U_f(E/O_K)$15. In the non-equivariant language quoted in the paper,

$U_f(E/O_K)$16

In the equivariant setting over a finite abelian extension $U_f(E/O_K)$17 of degree prime to $U_f(E/O_K)$18, one similarly defines

$U_f(E/O_K)$19

The resulting Stark-unit module with modulus satisfies

$U_f(E/O_K)$20

and one obtains the class number formula with modulus

$U_f(E/O_K)$21

The exact sequence built from the “difference” map

$U_f(E/O_K)$22

shows that the deformed module $U_f(E/O_K)$23 records the ray-class-type quotient on the unit side (Anglès et al., 2015).

This deformation-theoretic approach is especially effective for the Carlitz module and its multivariable deformations. It underlies the paper’s “discrete analogues” of Greenberg’s pseudo-cyclicity and pseudo-nullity conjectures, where the modulus is encoded through $U_f(E/O_K)$24 and through the $U_f(E/O_K)$25-derivative at $U_f(E/O_K)$26 (Anglès et al., 2015).

4. $U_f(E/O_K)$27-adic class formulas and the modulus as a local correction factor

For Anderson $U_f(E/O_K)$28-modules, the $U_f(E/O_K)$29-adic viewpoint realizes the modulus not by a separate class module $U_f(E/O_K)$30, but by modifying the Euler product. Let $U_f(E/O_K)$31 be a fixed monic prime. The paper defines the $U_f(E/O_K)$32-adic partial Euler product

$U_f(E/O_K)$33

where $U_f(E/O_K)$34 is the Tate algebra over $U_f(E/O_K)$35 in the variable $U_f(E/O_K)$36. The product converges in $U_f(E/O_K)$37 (Lucas, 4 Apr 2025).

The local factor at the modulus is

$U_f(E/O_K)$38

and similarly in the $U_f(E/O_K)$39-twisted setting. The corrected, modular $U_f(E/O_K)$40-adic $U_f(E/O_K)$41-series is then

$U_f(E/O_K)$42

The paper states explicitly that in this $U_f(E/O_K)$43-adic framework the role of a modulus is implemented by removing Euler factors at primes dividing the modulus and inserting explicit local correction factors on the left-hand side, rather than defining a separate class module $U_f(E/O_K)$44 (Lucas, 4 Apr 2025).

The corresponding regulator is $U_f(E/O_K)$45-adic. After constructing $U_f(E/O_K)$46, $U_f(E/O_K)$47, and a normalized extended logarithm

$U_f(E/O_K)$48

one defines a determinant regulator $U_f(E/O_K)$49, independent of the choice of $U_f(E/O_K)$50-bases. The main class formula is

$U_f(E/O_K)$51

and, after evaluation at $U_f(E/O_K)$52,

$U_f(E/O_K)$53

The same theorem identifies this quantity with the regulator of the Stark-unit lattice $U_f(E/O_K)$54 (Lucas, 4 Apr 2025).

This framework extends to several variables in the sense of Pellarin. For $U_f(E/O_K)$55, with $U_f(E/O_K)$56 and $U_f(E/O_K)$57 an Anderson $U_f(E/O_K)$58-module over $U_f(E/O_K)$59, the multivariable class formula becomes

$U_f(E/O_K)$60

and the treatment of the modulus is again by removing Euler factors at $U_f(E/O_K)$61 and inserting $U_f(E/O_K)$62 on the left-hand side (Lucas, 4 Apr 2025).

Two additional structural points are explicit in the paper. First, no ordinarity or good reduction at $U_f(E/O_K)$63 is assumed. Second, there is a vanishing criterion: if $U_f(E/O_K)$64 is not injective on $U_f(E/O_K)$65, then $U_f(E/O_K)$66; the converse is conjectured under an $U_f(E/O_K)$67-Leopoldt-type rank condition on the unit image $U_f(E/O_K)$68 (Lucas, 4 Apr 2025).

5. Equivariant refinements, $U_f(E/O_K)$69-truncation, and non-abelian structure

The equivariant refinement of Taelman’s class number formula for Drinfeld modules over finite Galois extensions is formulated in relative algebraic $U_f(E/O_K)$70-theory. Let $U_f(E/O_K)$71 be a finite Galois extension with group $U_f(E/O_K)$72, $U_f(E/O_K)$73 a Drinfeld $U_f(E/O_K)$74-module over $U_f(E/O_K)$75, and $U_f(E/O_K)$76 a taming module. The $U_f(E/O_K)$77-modified complex of units

$U_f(E/O_K)$78

has cohomology

$U_f(E/O_K)$79

The principal identity is

$U_f(E/O_K)$80

where $U_f(E/O_K)$81 is the $U_f(E/O_K)$82-modified Euler product in $U_f(E/O_K)$83 and $U_f(E/O_K)$84 is a refined Euler characteristic in $U_f(E/O_K)$85 (Frutos-Fernández et al., 2023).

For the topic of moduli, the crucial point is that this paper does not define ray class modules $U_f(E/O_K)$86 explicitly. Instead, Section 5 develops an $U_f(E/O_K)$87-truncated refined trace formula. For a finite set $U_f(E/O_K)$88 of places, removing Euler factors at $U_f(E/O_K)$89 modifies the global power-series class exactly by the local factor $U_f(E/O_K)$90, and after evaluation at $U_f(E/O_K)$91 this becomes

$U_f(E/O_K)$92

Thus deleting Euler factors at $U_f(E/O_K)$93 is the precise equivariant mechanism corresponding to a modulus supported on $U_f(E/O_K)$94 (Frutos-Fernández et al., 2023).

The paper states that this strongly suggests a modulus variant obtained by omitting local factors at $U_f(E/O_K)$95 and imposing corresponding local congruence conditions on the global complex. It further states that the proofs of the refined class number formula and the Fitting ideal bounds adapt mutatis mutandis to this setting, with taming corrections at wildly ramified primes handled by $U_f(E/O_K)$96. However, those modular statements are not themselves formally asserted as theorems in the paper (Frutos-Fernández et al., 2023).

This distinction matters conceptually. In the exact-sequence framework of modular Taelman groups, the modulus is an actual parameter in the definition of $U_f(E/O_K)$97. In the equivariant $U_f(E/O_K)$98-theoretic framework, the modular structure is present through $U_f(E/O_K)$99-truncated Euler products and modified local conditions, but the paper stops short of introducing a separate ray class module notation.

6. Iwasawa theory, asymptotic growth, and comparative perspective

The Iwasawa-theoretic use of moduli is developed for a $H_f(E/O_K)$00-extension $H_f(E/O_K)$01 of global $H_f(E/O_K)$02-fields with finite layers $H_f(E/O_K)$03 and $H_f(E/O_K)$04. For a finite set $H_f(E/O_K)$05 of finite places, one defines inverse limits

$H_f(E/O_K)$06

together with their $H_f(E/O_K)$07-localizations. These are compact modules over $H_f(E/O_K)$08 and $H_f(E/O_K)$09, respectively (Kataoka et al., 8 Sep 2025).

Control and descent are expressed by exact sequences. If $H_f(E/O_K)$10, then

$H_f(E/O_K)$11

If $H_f(E/O_K)$12 contains the ramified places $H_f(E/O_K)$13, then

$H_f(E/O_K)$14

for all $H_f(E/O_K)$15. Moreover, for ramified $H_f(E/O_K)$16 one has

$H_f(E/O_K)$17

so enlarging $H_f(E/O_K)$18 by ramified places does not change the Iwasawa module; in fact

$H_f(E/O_K)$19

This is one of the main structural advantages of the modular formalism (Kataoka et al., 8 Sep 2025).

The asymptotic formula is then purely $H_f(E/O_K)$20-type. If $H_f(E/O_K)$21 is a prime of $H_f(E/O_K)$22 and $H_f(E/O_K)$23, there exist integers $H_f(E/O_K)$24 and $H_f(E/O_K)$25 such that

$H_f(E/O_K)$26

with

$H_f(E/O_K)$27

When $H_f(E/O_K)$28 is unramified at all finite places, this yields

$H_f(E/O_K)$29

The paper emphasizes that there is no $H_f(E/O_K)$30-term: in characteristic $H_f(E/O_K)$31, the specialization $H_f(E/O_K)$32 forces linear-in-$H_f(E/O_K)$33 growth without an additional $H_f(E/O_K)$34-term (Kataoka et al., 8 Sep 2025).

Several standard families of $H_f(E/O_K)$35-extensions are listed: constant field towers $H_f(E/O_K)$36, Carlitz $H_f(E/O_K)$37-cyclotomic towers containing many $H_f(E/O_K)$38-extensions with $H_f(E/O_K)$39, and Artin–Schreier–Witt towers that are totally ramified at $H_f(E/O_K)$40 and unramified at all finite places. In each case, the modular formalism is used to obtain descent and asymptotic control (Kataoka et al., 8 Sep 2025).

From a comparative viewpoint, the recent literature supports three precise conclusions. First, “Taelman class groups with moduli” do not yet have a single universal definition; the exact-sequence, deformation-theoretic, $H_f(E/O_K)$41-adic, and equivariant formulations emphasize different aspects of the same ray-class phenomenon. Second, the strongest structural theory with an explicit modular class group $H_f(E/O_K)$42 is presently Iwasawa-theoretic (Kataoka et al., 8 Sep 2025), whereas the strongest analytic class formulas with a modulus appear in the $H_f(E/O_K)$43-deformation and $H_f(E/O_K)$44-adic settings (Anglès et al., 2015, Lucas, 4 Apr 2025). Third, extending Taelman’s analytic class number formula to a fully general modular setting remains open in some frameworks: the Iwasawa paper explicitly states that it does not prove a modular class number formula, and the equivariant refined paper states only that its $H_f(E/O_K)$45-truncated formalism strongly suggests such extensions (Kataoka et al., 8 Sep 2025, Frutos-Fernández et al., 2023).

These developments position moduli as a unifying device for local control in Taelman theory. Depending on context, the modulus may be imposed through local subgroups $H_f(E/O_K)$46, encoded by the fiber at $H_f(E/O_K)$47, inserted as a correction factor $H_f(E/O_K)$48 in a $H_f(E/O_K)$49-adic Euler product, or realized through $H_f(E/O_K)$50-truncation in relative $H_f(E/O_K)$51-theory. The underlying arithmetic content is consistent: the modulus records local constraints, and the resulting global object measures the failure of units, exponentials, and Euler factors to satisfy those constraints simultaneously.