---
title: Taelman Class Groups with Moduli
url: https://www.emergentmind.com/topics/taelman-class-groups-with-moduli
type: topic
---

# Taelman Class Groups with Moduli

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 [2509.06633], [1506.06286], [2504.03430], [2309.17256]. 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 \(\infty\), and coefficient ring \(A\subset Q\) of functions regular outside \(\infty\), let \(K\) be a global \(Q\)-field with integer ring \(O_K\). If \(E\) is a Drinfeld \(A\)-module over \(O_K\), with exponential map
\[
\exp_E(X)=X+\sum_{i=1}^{\infty} e_i X^{q^i}\in K[[X]],
\]
then for the product over infinite places \(K_\infty\) one defines
\[
H(E/O_K):=\frac{E(K_\infty)}{E(O_K)+\exp_E(K_\infty)}.
\]
The corresponding unit groups are
\[
U(E/O_K)=\Ker\!\left(E(O_K)\longrightarrow \frac{E(K_\infty)}{\exp_E(K_\infty)}\right),
\qquad
U'(E/O_K)=\Ker\!\left(\Lie_E(K_\infty)\overset{\exp_E}{\longrightarrow}\frac{E(K_\infty)}{E(O_K)}\right).
\]
There is an exact sequence
\[
0\to U(E/O_K)\to E(O_K)\to \frac{E(K_\infty)}{\exp_E(K_\infty)}\to H(E/O_K)\to 0,
\]
and also
\[
0\to \Lambda_E(K_\infty)\to U'(E/O_K)\overset{\exp_E}{\to}U(E/O_K)\to 0.
\]
Moreover, \(H(E/O_K)\) is finite over \(A\), while \(U(E/O_K)\) is finitely generated of rank \([K:Q]-r_E(K)\), where
\[
r_E(K)=\rank_A \Lambda_E(K_\infty)\le r\cdot \#S_\infty(K)
\]
for \(r=\operatorname{rank}(E)\) [2509.06633].

For Anderson \(t\)-modules the same architecture persists. If \(A=\mathbb F_q[\theta]\), \(K=\mathbb F_q(\theta)\), \(L/K\) is finite with integral closure \(O\), and \(E\) is an Anderson \(t\)-module over \(O\), then one has
\[
U(E;O):=\{x\in \operatorname{Lie}_E(L_\infty)\mid \exp_E(x)\in E(O)\},
\]
\[
H(E;O):=\frac{E(L_\infty)}{E(O)+\exp_E(\operatorname{Lie}_E(L_\infty))}.
\]
Demeslay’s finiteness and lattice results yield that \(U(E;O)\) is an \(A\)-lattice in \(\Lie_E(L_\infty)\) and \(H(E;O)\) is finite. The associated archimedean Euler factors \(z_Q(E/O)\) define a convergent product
\[
L(E/O)=\prod_Q z_Q(E/O),
\]
and the class formula reads
\[
L(E/O)=\big[\operatorname{Lie}_E(O):U(E;O)\big]_A\cdot \big[H(E;O)\big]_A
\]
[2504.03430].

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 \(E(fO_{K,v})\) | \(U_f(E/O_K)\), \(H_f(E/O_K)\) |
| \(z\)-deformation/Stark units | specialization at \(z=1\), modified Euler factors | \(U_{\mathrm{St}}^{(\mathfrak m)}\), ray-class-type modules |
| \(P\)-adic Anderson \(t\)-modules | remove Euler factor at \(P\), insert \(z_P\) | corrected \(P\)-adic \(L\)-series |
| Equivariant \(K\)-theory | \(S\)-truncated Euler products | \(S\)-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 \(f\) be a nonzero ideal of \(O_K\), viewed as a modulus
\[
f=\prod_v v^{\operatorname{ord}_v(f)}
\]
with \(v\) ranging over finite places of \(K\). The unit group \(U_f(E/O_K)\) and class group \(H_f(E/O_K)\) with modulus \(f\) are defined by the exact sequence
\[
0 \to U_{f}(E/O_K)
\to E(K)
\to \frac{E(K_{\infty})}{\exp_E(K_{\infty})}\oplus \bigoplus_v \frac{E(K_v)}{E(f O_{K, v})}
\to H_{f}(E/O_K)
\to 0.
\]
The diagonal map \(E(K)\to (\cdots)\) encodes both the infinite-place contribution and the local ray condition at each finite \(v\) through the subgroup \(E(fO_{K,v})\subset E(K_v)\) [2509.06633].

This definition is functorial in the modulus. If \(g\mid f\), then there is a natural exact sequence
\[
0 \to U_{f}(E/O_K)
\to U_{g}(E/O_K)
\to \bigoplus_v \frac{E(g O_{K, v})}{E(f O_{K, v})}
\to H_{f}(E/O_K)
\to H_{g}(E/O_K)
\to 0.
\]
In particular,
\[
U_{(1)}(E/O_K)=U(E/O_K), \qquad H_{(1)}(E/O_K)=H(E/O_K).
\]
The modular groups retain the same finiteness profile as the classical ones: for any modulus \(f\), \(H_f(E/O_K)\) is finite as an \(A\)-module, and \(U_f(E/O_K)\) is finitely generated of rank \([K:Q]-r_E(K)\) [2509.06633].

The same paper notes that one can equivalently work with \(S\)-integers \(O_{K,S}\) for a finite \(S\) containing the support of \(f\). This is structurally parallel to the distinction between ray class groups and \(S\)-ray class groups in number fields. The analogy is explicit: the construction mirrors the number field ray class setting where local conditions at \(v\) enforce congruences modulo \(fO_{K,v}\), and the trivial modulus \(f=(1)\) recovers Taelman’s original class group [2509.06633].

A common misconception is that all recent work on moduli defines a separate object \(H(E/O_K;\mathfrak m)\) in this exact-sequence sense. That is correct for the Iwasawa-theoretic framework above, but not for every other approach.

## 3. \(z\)-deformations, Stark units, and ray-class-type quotients

A second realization of moduli uses deformation in an auxiliary variable \(z\). For a rank-one Drinfeld \(A\)-module \(\phi\) over \(O_L\), the canonical \(z\)-deformation \(\phi\mapsto \phi^e\) is defined by
\[
\phi^e_\theta=\sum_{j=0}^r z^j\alpha_j\tau^j
\quad \text{if}\quad
\phi_\theta=\sum_{j=0}^r \alpha_j\tau^j.
\]
The associated deformed class module is
\[
H(\phi/O_L[z]):=\frac{L_\infty[z]}{O_L[z]+\exp_{\phi^e}(L_\infty[z])},
\]
which is a finitely generated torsion \(\mathbb F_q[z]\)-module and specializes at \(z=1\) back to \(H(\phi/O_L)\) [1506.06286].

The Stark-unit submodule is defined by evaluation at \(z=1\):
\[
U_{\mathrm{St}(\phi/O_L)}:=\operatorname{ev}_{z=1}\big(U(\phi^e/O_L[z])\big)\subset U(\phi/O_L),
\]
where
\[
U(\phi^e/O_L[z])=\{x\in L_\infty[z]: \exp_{\phi^e}(x)\in O_L[z]\}.
\]
The fundamental relation is that the quotient on the unit side is controlled by the \(z=1\) fiber of the deformed class module:
\[
U(\phi/O_L)/U_{\mathrm{St}(\phi/O_L)}\simeq H(\phi/O_L[z])[z-1].
\]
The paper further proves
\[
[H(\phi/O_L)]_A=[U(\phi/O_L)/U_{\mathrm{St}(\phi/O_L)}]_A,
\qquad
L(\phi/O_L)=[O_L:U_{\mathrm{St}(\phi/O_L)}]_A.
\]
In this sense, the \(z=1\) specialization measures a ray-class-type defect on the unit side [1506.06286].

The modular interpretation becomes explicit when Euler factors at primes dividing an ideal \(\mathfrak m\) are removed or altered. Fixing a finite set of finite primes \(S\), or equivalently an ideal \(\mathfrak m\subset O_L\), one defines partial or modified \(L\)-series by omitting Euler factors at primes dividing \(\mathfrak m\). In the non-equivariant language quoted in the paper,
\[
L(\phi/O_L;\mathfrak m,z):=\prod_{P\nmid \mathfrak m}\frac{f_P(z)}{N_{L/K}(P)}\in T_z(K_\infty)^\times.
\]
In the equivariant setting over a finite abelian extension \(E/L\) of degree prime to \(p\), one similarly defines
\[
L(\phi/(O_E/A),G;\mathfrak m)
:=\prod_{P\nmid \mathfrak m}\frac{[O_E/P]_{A[z][G]}}{[\phi(O_E[z]/PO_E[z])]_{A[z][G]}}
\in T_z(K_\infty)[G]^\times.
\]
The resulting Stark-unit module with modulus satisfies
\[
U_{\mathrm{St}^{(\mathfrak m)}(\phi/O_E)}
=
L(\phi/(O_E/A),G;\mathfrak m)\big|_{z=1}\cdot O_E,
\]
and one obtains the class number formula with modulus
\[
[O_E:U_{\mathrm{St}^{(\mathfrak m)}(\phi/O_E)}]_{A[G]}
=
L(\phi/(O_E/A),G;\mathfrak m)\big|_{z=1}.
\]
The exact sequence built from the “difference” map
\[
\alpha(x)=\frac{\exp_{\phi^e}(x)-\exp_\phi(x)}{z-1}
\]
shows that the deformed module \(H(\phi/O_E[z])[z-1]\) records the ray-class-type quotient on the unit side [1506.06286].

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 \(z\) and through the \(z\)-derivative at \(z=1\) [1506.06286].

## 4. \(P\)-adic class formulas and the modulus as a local correction factor

For Anderson \(t\)-modules, the \(P\)-adic viewpoint realizes the modulus not by a separate class module \(H(E;O;P)\), but by modifying the Euler product. Let \(P\in A=\mathbb F_q[\theta]\) be a fixed monic prime. The paper defines the \(P\)-adic partial Euler product
\[
L_P(\widetilde{E}/\widetilde{O})
:=
\prod_{Q\neq P} z_Q(\widetilde{E}/\widetilde{O})
\in \mathcal T_z(K_P),
\qquad
L_P(E/O):=\operatorname{ev}_{z=1}L_P(\widetilde{E}/\widetilde{O})\in K_P,
\]
where \(\mathcal T_z(K_P)\) is the Tate algebra over \(K_P\) in the variable \(z\). The product converges in \(\mathcal T_z(K_P)\) [2504.03430].

The local factor at the modulus is
\[
z_P(E/O)
=
\frac{[\operatorname{Lie}_E(O/PO)]_A}{[E(O/PO)]_A},
\]
and similarly in the \(z\)-twisted setting. The corrected, modular \(P\)-adic \(L\)-series is then
\[
z_P(\widetilde{E}/\widetilde{O})\cdot L_P(\widetilde{E}/\widetilde{O})\in \mathcal T_z(K_P),
\qquad
z_P(E/O)\cdot L_P(E/O)\in K_P.
\]
The paper states explicitly that in this \(p\)-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 \(H(E;O;P)\) [2504.03430].

The corresponding regulator is \(P\)-adic. After constructing \(\log_{E,P}\), \(\exp_{E,P}\), and a normalized extended logarithm
\[
\mathrm{Log}_{E,P}(x):=\frac{1}{g(1)}\,\log_{E,P}\big(E_{g(1)}(x)\big),
\]
one defines a determinant regulator \(R_P(U(E;O))\in K_P\), independent of the choice of \(A\)-bases. The main class formula is
\[
z_P(\widetilde{E}/\widetilde{O})\cdot L_P(\widetilde{E}/\widetilde{O})
=
R_P\big(U(\widetilde{E};\widetilde{O})\big),
\]
and, after evaluation at \(z=1\),
\[
z_P(E/O)\cdot L_P(E/O)
=
R_P\big(U(E;O)\big)\cdot [H(E;O)]_A.
\]
The same theorem identifies this quantity with the regulator of the Stark-unit lattice \(U_{\mathrm{St}(E;O)}=\operatorname{ev}_{z=1}U(\widetilde E;O[z])\) [2504.03430].

This framework extends to several variables in the sense of Pellarin. For \(s\ge 1\), with \(A_s=k[\theta]\) and \(E\) an Anderson \(A_s\)-module over \(\mathscr O_{L,s}=kO\), the multivariable class formula becomes
\[
z_P(E/\mathscr O_{L,s})\cdot L_P(E/\mathscr O_{L,s})
=
R_P\big(U(E;\mathscr O_{L,s})\big)\cdot [H(E;\mathscr O_{L,s})]_{A_s},
\]
and the treatment of the modulus is again by removing Euler factors at \(P\) and inserting \(z_P(E/\mathscr O_{L,s})\) on the left-hand side [2504.03430].

Two additional structural points are explicit in the paper. First, no ordinarity or good reduction at \(P\) is assumed. Second, there is a vanishing criterion: if \(\exp_E\) is not injective on \(L_\infty^d\), then \(L_P(E/O)=0\); the converse is conjectured under an \(A_P\)-Leopoldt-type rank condition on the unit image \(\mathcal U(E;PO)\) [2504.03430].

## 5. Equivariant refinements, \(S\)-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 \(K\)-theory. Let \(K/k\) be a finite Galois extension with group \(G\), \(E\) a Drinfeld \(A\)-module over \(O_k\), and \(M\) a taming module. The \(M\)-modified complex of units
\[
C^M_{E,K/k}: K_\infty \longrightarrow (E(K)/E(M))\oplus M
\]
has cohomology
\[
H^1(C^M_{E,K/k})=U_{E,K},\qquad H^2(C^M_{E,K/k})=M,\qquad H^0(C^M_{E,K/k})=H(E/M).
\]
The principal identity is
\[
\delta_G(\Omega^M_{E,K/k})=-\chi_G(C^M_{E,K/k},\lambda_{M,E,K/k}),
\]
where \(\Omega^M_{E,K/k}\) is the \(M\)-modified Euler product in \(K_1(F[G])\) and \(\chi_G\) is a refined Euler characteristic in \(K_0(A[G],F[G])\) [2309.17256].

For the topic of moduli, the crucial point is that this paper does **not** define ray class modules \(H(E/K,\mathfrak m)\) explicitly. Instead, Section 5 develops an \(S\)-truncated refined trace formula. For a finite set \(S\supset S_\infty\) of places, removing Euler factors at \(v\notin S\) modifies the global power-series class exactly by the local factor \([1+\sigma\mid M/p_vM]\), and after evaluation at \(Z=t^{-1}\) this becomes
\[
c_G(M/pM)^{-1}\cdot c_G(E(M/pM)).
\]
Thus deleting Euler factors at \(S\) is the precise equivariant mechanism corresponding to a modulus supported on \(S\) [2309.17256].

The paper states that this strongly suggests a modulus variant obtained by omitting local factors at \(S\) 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 \(M\). However, those modular statements are not themselves formally asserted as theorems in the paper [2309.17256].

This distinction matters conceptually. In the exact-sequence framework of modular Taelman groups, the modulus is an actual parameter in the definition of \(H_f(E/O_K)\). In the equivariant \(K\)-theoretic framework, the modular structure is present through \(S\)-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 \(\mathbb Z_p\)-extension \(K_\infty/K\) of global \(Q\)-fields with finite layers \(K_n\) and \(\Gamma=\operatorname{Gal}(K_\infty/K)\simeq \mathbb Z_p\). For a finite set \(S\) of finite places, one defines inverse limits
\[
\hat U_S(E/O_{K_\infty})=\varprojlim_n \hat U_S(E/O_{K_n}),
\qquad
H_S(E/O_{K_\infty})=\varprojlim_n H_S(E/O_{K_n}),
\]
together with their \(A_p\)-localizations. These are compact modules over \(\hat A[[\Gamma]]\) and \(A_p[[\Gamma]]\), respectively [2509.06633].

Control and descent are expressed by exact sequences. If \(T\subset S\), then
\[
0 \to \hat{U}_{S}(E/O_{K_\infty})
\to \hat{U}_{T}(E/O_{K_\infty})
\to \bigoplus_{v \in S \setminus T} \varprojlim_n E(O_{K_n, v})
\to H_{S}(E/O_{K_\infty})
\to H_{T}(E/O_{K_\infty})
\to 0.
\]
If \(S\) contains the ramified places \(S_{\mathrm{ram}}(K_\infty/K)\), then
\[
H_S(E/O_{K_\infty})_{\Gamma^{p^n}}\simeq H_S(E/O_{K_n})
\]
for all \(n\ge 0\). Moreover, for ramified \(v\) one has
\[
\varprojlim_n E(O_{K_n,v})=0,
\]
so enlarging \(S\) by ramified places does not change the Iwasawa module; in fact
\[
H_{S\cup S_{\mathrm{ram}}}(E/O_{K_\infty})\simeq H_S(E/O_{K_\infty}).
\]
This is one of the main structural advantages of the modular formalism [2509.06633].

The asymptotic formula is then purely \(\mu\)-type. If \(p\) is a prime of \(A\) and \(S\supset S_{\mathrm{ram}}(K_\infty/K)\cap S_p\), there exist integers \(\mu_p\ge 0\) and \(\nu_p\) such that
\[
\length_{A_p}\!\left(H_S(E/O_{K_n})_{p,\mathrm{fin}}\right)=\mu_p p^n+\nu_p
\quad (n\gg 0),
\]
with
\[
\mu_p=\mu^*\!\left(H_S(E/O_{K_\infty})_{p,\tors}\right).
\]
When \(K_\infty/K\) is unramified at all finite places, this yields
\[
\length_A(H(E/O_{K_n}))=\mu p^n+\nu
\quad (n\gg 0).
\]
The paper emphasizes that there is no \(\lambda\)-term: in characteristic \(p\), the specialization \((1+T)^{p^n}-1=T^{p^n}\) forces linear-in-\(p^n\) growth without an additional \(n\)-term [2509.06633].

Several standard families of \(\mathbb Z_p\)-extensions are listed: constant field towers \(K_n=\mathbb F_{q^{p^n}}(\theta)\), Carlitz \(p\)-cyclotomic towers containing many \(\mathbb Z_p\)-extensions with \(S_{\mathrm{ram}}=\{p\}\), and Artin–Schreier–Witt towers that are totally ramified at \(\infty\) and unramified at all finite places. In each case, the modular formalism is used to obtain descent and asymptotic control [2509.06633].

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, \(P\)-adic, and equivariant formulations emphasize different aspects of the same ray-class phenomenon. Second, the strongest structural theory with an explicit modular class group \(H_f\) is presently Iwasawa-theoretic [2509.06633], whereas the strongest analytic class formulas with a modulus appear in the \(z\)-deformation and \(P\)-adic settings [1506.06286], [2504.03430]. 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 \(S\)-truncated formalism strongly suggests such extensions [2509.06633], [2309.17256].

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 \(E(fO_{K,v})\), encoded by the fiber at \(z=1\), inserted as a correction factor \(z_P\) in a \(P\)-adic Euler product, or realized through \(S\)-truncation in relative \(K\)-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.

Source: https://www.emergentmind.com/topics/taelman-class-groups-with-moduli