Taelman Class Groups in Function Field Arithmetic

• Taelman class groups are finite A-modules attached to Drinfeld modules that serve as the function-field analogues of classical ideal class groups.
• They play a central role in class number formulas by linking the exponential maps on Drinfeld modules with special L-values and regulators.
• They admit equivariant, modulus-modified, and P-adic extensions that enrich their structure and application in Iwasawa theory and multivariable deformations.

Taelman class groups, also called class modules, are finite $A$-modules attached to Drinfeld modules over global function fields and, in later extensions, to Anderson $t$-modules. In the basic Drinfeld-module setting, for a global $Q$-field $K$ with integer ring $O_K$, a place $\infty$, and a Drinfeld $A$-module $E$, they are defined by

$$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$

where $$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$ and $t$0 is the Drinfeld exponential. These groups were introduced as function-field analogues of ideal class groups, and the modern literature treats them simultaneously as class-module terms in class number formulas, as equivariant $t$1-modules in finite Galois extensions, as objects admitting ray-class-like refinements by moduli, and as inputs to $t$2-adic and multivariable special-value formulas (Frutos-Fernández et al., 2023, Kataoka et al., 8 Sep 2025, Lucas, 4 Apr 2025).

1. Definition and basic structure

The ambient arithmetic setting in the surveyed literature begins with a global function field $t$3 with constant field $t$4, a fixed place $t$5, and the ring

$t$6

of elements regular outside $t$7. The model example is

$t$8

A global $t$9-field $Q$0 is equipped with an injective $Q$1-algebra map $Q$2, and its integer ring is the integral closure $Q$3 of $Q$4 in $Q$5. For a Drinfeld $Q$6-module $Q$7 over $Q$8, written as

$Q$9

the exponential series induces maps

$K$0

for each infinite place $K$1, hence globally

$K$2

Its kernel is

$K$3

and the associated invariant is

$K$4

The standard Taelman class group is

$K$5

The corresponding unit-type modules are

$K$6

and

$K$7

They fit into the exact sequence

$K$8

A central structural theorem is that $K$9 is finite, while $O_K$0 is finitely generated of rank

$O_K$1

In the convention used in the Iwasawa-theoretic work, the usual nontriviality condition on a Drinfeld module is not imposed; if $O_K$2, then the exponential map is the identity and the Taelman class group is trivial (Kataoka et al., 8 Sep 2025).

2. Analogy with ideal class groups and the class number formula

Taelman class groups are presented explicitly as the function-field/Drinfeld-module analogue of the ideal class group $O_K$3. The analogy is formalized by comparing the classical number-field sequence

$O_K$4

with the Drinfeld-module sequence

$O_K$5

This identifies $O_K$6 as the finite defect between global integral points and the image of the exponential, exactly in the way that the ideal class group measures failure of principal generation (Kataoka et al., 8 Sep 2025).

Number fields	Function fields / Drinfeld modules
$O_K$7	$O_K$8
$O_K$9	$\infty$0
ray class groups $\infty$1	Taelman class groups with moduli $\infty$2
$\infty$3-ramified class groups $\infty$4	$\infty$5

The analytic role of the class module is fixed by Taelman’s class number formula. In the scalar setting, the special $\infty$6-value is expressed as a regulator or covolume term multiplied by the characteristic polynomial or Fitting generator of the finite class module. One formulation recalled in the literature is that the ratio of co-volumes in $\infty$7 of the $\infty$8-lattices $\infty$9 and $A$0, multiplied by the characteristic polynomial of $A$1, equals the special $A$2-value (Frutos-Fernández et al., 2023).

For Anderson $A$3-modules, the same pattern is written in Fitting-ideal language. With

$A$4

and unit lattice

$A$5

the $A$6-adic class formula is

$A$7

The same paper recalls the Stark-unit identity

$A$8

where

$A$9

Accordingly, the class module is not an auxiliary construction: it is one of the two terms, together with the unit/regulator side, that determine the special value (Lucas, 4 Apr 2025).

3. Equivariant and non-abelian refinements

For a finite Galois extension $E$0 of global function fields with group

$E$1

the equivariant theory tracks the $E$2-action on Taelman class groups. In the setup

$E$3

a Drinfeld module $E$4 over the integral closure $E$5 yields the usual objects

$E$6

and

$E$7

Because wild ramification obstructs good $E$8-module behavior of $E$9, the equivariant refinement introduces a taming module $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$0, an $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$1-submodule such that $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$2 is $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$3-projective and $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$4 is finite and supported only at wildly ramified primes. If $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$5 is tamely ramified, one may take $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$6 (Frutos-Fernández et al., 2023).

The natural equivariant replacement for the ordinary class group is then

$$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$7

It appears as part of the cohomology of the $$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$8-modified complex of units

$$H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)},$$9

with

$$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$0

After extension of scalars to $$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$1, the natural identification yields

$$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$2

and hence a refined Euler characteristic

$$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$3

The local Euler factors are no longer expressed by determinants alone. For a finite $$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$4-module $$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$5, one defines the characteristic class

$$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$6

using the endomorphism $$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$7 on

$$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$8

For finite $$K_\infty=Q_\infty\otimes_Q K=\prod_{v\mid\infty}K_v$$9-cohomologically trivial modules, these classes are multiplicative in short exact sequences, and in the abelian case their determinants generate Fitting ideals. This leads to the $t$00-modified equivariant $t$01-value

$t$02

whose convergence is proved in $t$03.

The refined class number formula is the identity

$t$04

in $t$05. This lifts Taelman’s scalar formula to relative algebraic $t$06-theory and encodes the full $t$07-module structure of the Taelman class group. It recovers Taelman’s original formula when $t$08, and in the abelian case it recovers the Ferrara–Green–Higgins–Popescu equivariant Tamagawa number formula (Frutos-Fernández et al., 2023).

Under the hypothesis that $t$09 does not divide the order of the commutator subgroup of $t$10, the group rings split as sums of matrix algebras over commutative rings and reduced determinants exist. One then obtains non-commutative Fitting ideals

$t$11

with

$t$12

for finite $t$13-cohomologically trivial $t$14-modules $t$15. The reduced determinant of $t$16,

$t$17

acts as a non-abelian Stickelberger element. The resulting regulator-normalized ideals lie in

$t$18

and therefore in the central annihilator of $t$19. In the tame case $t$20, Corollary 3.13 gives an exact equality with $t$21, yielding a strong Galois-structure statement for the Taelman class group (Frutos-Fernández et al., 2023).

4. Moduli and Iwasawa-theoretic growth

A major extension of the theory introduces Taelman class groups with moduli. For a nonzero ideal $t$22 of $t$23, regarded as a modulus

$t$24

the Taelman unit group with modulus $t$25 and the Taelman class group with modulus $t$26 are the kernel and cokernel of the diagonal map

$t$27

Thus there is an exact sequence

$t$28

The trivial modulus recovers the original objects: $t$29 If $t$30, then

$t$31

In particular, every $t$32 is finite, and $t$33 has the same rank as $t$34, namely

$t$35

Passing to inverse limits over moduli supported in a finite set $t$36 of finite places gives

$t$37

where

$t$38

These $t$39-modifications satisfy

$t$40

for $t$41. The construction is explicitly described as the Drinfeld-module analogue of passing from ordinary class groups to ray class groups and then to $t$42-ramified variants (Kataoka et al., 8 Sep 2025).

This modification is necessary for Iwasawa theory. In a $t$43-extension

$t$44

ordinary Taelman class groups have the expected descent property only when the tower is unramified at finite places. In the ramified case, one replaces them by $t$45, with $t$46 containing the relevant ramified places. Then the modified groups satisfy descent: $t$47 provided $t$48. For finite Galois extensions $t$49, if $t$50 contains the places outside which the extension is tamely ramified, then

$t$51

The associated Iwasawa modules are

$t$52

and for a prime $t$53,

$t$54

is a compact $t$55-module. Finite generation over $t$56 is proved.

The main asymptotic theorem states that if $t$57 contains all ramified $t$58-adic places, then there exist integers $t$59 and $t$60 such that

$t$61

with

$t$62

In the unramified case, this simplifies to

$t$63

When the tower is unramified at all finite places,

$t$64

The absence of a $t$65-term is explained algebraically by the characteristic-$t$66 identity

$t$67

The same source emphasizes that Taelman Iwasawa modules are generally not torsion over $t$68, so the theorem must isolate the maximal finite $t$69-submodule of the finite-level groups (Kataoka et al., 8 Sep 2025).

5. $t$70-adic and multivariable extensions

The $t$71-adic theory extends the class-module formalism from Drinfeld modules to general Anderson $t$72-modules. With

$t$73

a finite extension $t$74, integral closure $t$75, and a fixed monic irreducible polynomial $t$76, an Anderson $t$77-module $t$78 of dimension $t$79 over $t$80 is an $t$81-algebra homomorphism

$t$82

with $t$83 for each $t$84. The global Taelman class module is defined exactly as

$t$85

and its $t$86-deformations are

$t$87

and

$t$88

The paper stresses that the class-module formalism remains fundamentally global and $t$89-adic in the definition of $t$90: the $t$91-adic theory does not introduce a new quotient-defined local class module (Lucas, 4 Apr 2025).

The associated local factors are

$t$92

and the $t$93-adic $t$94-series is obtained by deleting the Euler factor at $t$95: $t$96 A $t$97-adic logarithmic regulator is then defined from a basis of the unit lattice $t$98: $t$99 The main $Q$00-adic class formula is

$Q$01

Equivalently,

$Q$02

The omitted Euler factor reappears as a local correction term, and the usual covolume regulator is replaced by a $Q$03-adic logarithmic regulator. This is described as a genuine $Q$04-adic refinement or variant of Taelman’s class number formula (Lucas, 4 Apr 2025).

The same paper develops a multivariable extension in the style of Pellarin. For

$Q$05

the unit and class modules become

$Q$06

and

$Q$07

The multivariable $Q$08-adic class formula is

$Q$09

Thus the Taelman class module persists unchanged in spirit under both the passage to Anderson $Q$10-modules and the passage to a multivariable setting, with $Q$11-Fitting generators replaced by $Q$12-Fitting generators (Lucas, 4 Apr 2025).

6. Special-value deformations and limits of the current formalism

Work on Taelman $Q$13-values for Drinfeld modules over Tate algebras occupies the same arithmetic circle but stops short of constructing full global Taelman class modules in that setting. For a Drinfeld module

$Q$14

and its Tate-algebra deformation

$Q$15

the Taelman $Q$16-value is defined by the Euler product

$Q$17

The local Fitting factor is computed explicitly: $Q$18 The principal special-value identity is

$Q$19

which identifies the Taelman value with a generalized Pellarin series. Under a range condition on $Q$20, one also has

$Q$21

The $Q$22-deformed value

$Q$23

satisfies the log-algebraicity statement

$Q$24

The same source is explicit about scope: it does not define global Taelman class modules, unit modules, or regulators for the Tate-algebra deformations $Q$25 and $Q$26. Instead, it develops the Euler-product side, the local finite-module and Fitting-ideal side, and the logarithmic special-value side. This corrects a common overstatement in the subject: not every Taelman-style special-value formula in a deformed setting comes with a full class-module formalism. The Tate-algebra paper is therefore best understood as extending the special-value technology rather than as constructing new Taelman class groups (Gezmiş, 2018).

Taken together, the literature gives a coherent picture. Taelman class groups begin as finite $Q$27-modules

$Q$28

attached to Drinfeld modules. They then admit equivariant refinements $Q$29 that encode full $Q$30-module structure, modulus-modified refinements $Q$31 and $Q$32 required for ramified Iwasawa theory, and $Q$33-adic and multivariable appearances in which the same global class module enters class formulas through its Fitting generator. Across these developments, the persistent structural theme is that Taelman class groups are the finite class-module term governing special values, regulators, annihilators, and growth laws in characteristic-$Q$34 arithmetic.