---
title: 'Dedekind–Rademacher Cocycle: Cohomology & Computations'
url: https://www.emergentmind.com/topics/dedekind-rademacher-cocycle
type: topic
---

# Dedekind–Rademacher Cocycle: Cohomology & Computations

Searching arXiv for recent and foundational papers on the Dedekind–Rademacher cocycle and closely related generalizations.
The Dedekind–Rademacher cocycle is a cohomological refinement of the transformation law of $\log \Delta(z)$, where $\Delta$ is the weight-$12$ modular discriminant. In its classical form on $\mathrm{SL}_2(\mathbb Z)$, it appears through Dedekind’s function $\Phi$ and Rademacher’s modification $\Psi$, an integer-valued class-invariant homomorphism. In more recent formulations, it is realized as a measure-valued or distribution-valued $1$-cocycle, notably the class $\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$ with explicit cocycle representative $J_{DR}$, and as a rigid-analytic cocycle whose values at real multiplication points are global $p$-units related to Gross–Stark units [2307.00425] [2103.02490].

## 1. Classical cocycle on $\mathrm{SL}_2(\mathbb Z)$

For $\Gamma=\mathrm{SL}_2(\mathbb Z)$ and
$$
\Delta(z)=q\prod_{n\ge1}(1-q^n)^{24},\qquad q=e^{2\pi i z},
$$
Dedekind’s transformation law takes the form
$$
\log \Delta(\gamma z)-\log \Delta(z)
=12\,\mathrm{sgn}(c)^2\log\!\bigl((cz+d)/(i\,\mathrm{sgn}(c))\bigr)+2\pi i\,\Phi(\gamma),
$$
for $\gamma=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma$, with $\Phi:\Gamma\to\mathbb Z$ arising from the multivaluedness of $\log$ [2003.12354]. Dedekind showed that
$$
\Phi\!\begin{pmatrix}a&b\\ c&d\end{pmatrix}
=
\begin{cases}
\dfrac{a+d}{c}-12\,\mathrm{sgn}(c)\,s(a,|c|),& c\neq 0,\\[1ex]
\dfrac{b}{d},& c=0,
\end{cases}
$$
where $s(a,c)$ is the classical Dedekind sum [2003.12354].

Rademacher modified $\Phi$ by
$$
\Psi(\gamma)=\Phi(\gamma)-3\,\mathrm{sgn}(c(a+d)),
$$
and showed that $\Psi$ satisfies
$$
\Psi(\gamma)=\Psi(-\gamma)=-\Psi(\gamma^{-1})=\Psi(g^{-1}\gamma g)
$$
for all $g\in\Gamma$, together with the $1$-cocycle relation
$$
\Psi(\gamma_1\gamma_2)=\Psi(\gamma_1)+\Psi(\gamma_2).
$$
In particular, for hyperbolic $\gamma$ one has $\Psi(\gamma^n)=n\,\Psi(\gamma)$ [2003.12354].

The classical symbol also admits integral and continued-fraction descriptions. Writing
$$
E_2(z)=1-24\sum_{m\ge1}\sigma_1(m)q^m=(2\pi i)^{-1}(d/dz)\log \Delta(z),
$$
one has for hyperbolic $\gamma$
$$
\Psi(\gamma)=\lim_{n\to\infty}\int_{\gamma^n z_0}^{\gamma^{n+1}z_0}E_2(z)\,dz,
$$
for any $z_0$ on the axis $S_\gamma$; and if $\gamma$ is conjugate to a purely periodic continued fraction, then
$$
\Psi(\gamma)=\sum_{j=0}^{2n-1}(-1)^j a_j
$$
in the notation of the continued-fraction expansion [2003.12354].

This classical construction provides the prototype for later distribution-valued and rigid-analytic formulations. It also underlies the level-$N$ homomorphism $\phi_N:\Gamma_0(N)\to\mathbb Q$, defined by
$$
\phi_N(A)=\phi\!\begin{pmatrix}a&bN\\ c/N&d\end{pmatrix}-\phi(A),
$$
which appears as the period of the eta-quotient $\eta(Nz)/\eta(z)$ [2507.16671].

## 2. Distribution-valued cohomology class $\mu_{DR}$

A global formulation begins with $V=\mathbb Q^2$, viewed as row vectors with the topology generated by its $\mathbb Z$-lattices. A test function on $V$ is a compactly supported locally constant $\mathbb Z$-valued function,
$$
\mathcal S(V)=\{\,f:V\to\mathbb Z\mid f\text{ locally constant, compact support}\,\},
$$
and $\mathcal S(V)'\subset\mathcal S(V)$ consists of those supported away from $0$ [2307.00425].

Let $G=\mathrm{GL}_2(\mathbb Q)$ act on $V$ by
$$
\gamma\ast v=v\,\gamma^{-1},
$$
and on test functions by
$$
(f\mid\gamma)(v)=f(\gamma\ast v).
$$
For a right $G$-module $A$, one sets
$$
\mathrm{Dist}(V,A)=\mathrm{Hom}\bigl(\mathcal S(V),A\bigr),
$$
with left $G$-action
$$
(\gamma\cdot\mu)(f)=\mu(f\mid\gamma)\mid\gamma.
$$
This gives the ambient module for the Dedekind–Rademacher cohomology class [2307.00425].

Fix an integer $c>1$ with $(c,6)=1$, and let $S=\{\ell:\ell\mid c\}$. Scholl’s construction of Kato–Siegel units on the universal elliptic curve yields a $G$-invariant distribution
$$
\mu_{KS}\in H^0\bigl(G,\mathrm{Dist}(V,\mathcal O_{\mathcal H}^\times)\bigr),
$$
where $\mathcal O_{\mathcal H}^\times$ denotes the $\mathbb C$-valued holomorphic functions on $\mathcal H$ with no zeros, equipped with the usual $\mathrm{GL}_2(\mathbb Q)$-action by slash operators [2307.00425].

Applying the long exact sequence induced by
$$
0\longrightarrow\mathbb Z\longrightarrow\mathcal O_{\mathcal H}
\xrightarrow{e^{2\pi i(\,\cdot\,)}}\mathcal O_{\mathcal H}^\times\longrightarrow 1,
$$
one obtains the Dedekind–Rademacher cohomology class
$$
\mu_{DR}=\partial(\mu_{KS})\in H^1\bigl(G,\mathrm{Dist}(V,\mathbb Z)\bigr).
$$
This is the class whose evaluations at real-quadratic points were shown by Darmon, Pozzi, and Vonk to give Gross–Stark units up to a controlled torsion [2307.00425].

The significance of this formulation is that the classical symbol is recast as a global cohomology class built from modular units and distributions, rather than only as an explicit function on $\mathrm{SL}_2(\mathbb Z)$. That shift makes possible both explicit cocycle formulas and comparison with $p$-adic measure constructions.

## 3. Explicit cocycle representative $J_{DR}$

An explicit representative is obtained from a section
$$
E:\mathcal S(V)\longrightarrow\mathcal O_{\mathcal H},\qquad f\mapsto E(f;\tau),
$$
satisfying
$$
e^{2\pi i\,E(f;\tau)}=\mu_{KS}(f;\tau).
$$
Writing out the three factors in the $q$-expansion of the Kato–Siegel unit gives
$$
\mu_{KS}(f)
=
q^{2\,(B_2\otimes B_0)_c(f)}
\times
\zeta_c\,(\mathcal DCyc)_c(f)
\times
u_c(f),
$$
where $B_r$ is the $r$-th Bernoulli distribution on $\mathbb Q$, $\mathcal DCyc$ and its log-variants come from cyclotomic distributions on $\mathbb Q$, and
$$
u(f)=\prod_{y>0}(1-q^y\zeta^b)
$$
is the theta-unit distribution; all are $c$-smoothed by the group-algebra element $c[\mathrm{Id}]-[D(c)]$ [2307.00425].

The section is then defined by
$$
E(f;\tau)=
\underbrace{\tau\cdot(B_2\otimes B_0)_c(f)+\pi_c(f)}_{=:\rho(f;\tau)}
+\frac1{2\pi i}(\delta_0\otimes \mathrm{Clog})_c(f)
+\frac1{2\pi i}\int_{i\infty}^{\tau}d\log\,u_c(f),
$$
and the explicit Dedekind–Rademacher cocycle is
$$
J_{DR}=dE\in Z^1\!\bigl(G,\mathrm{Dist}(V,\mathbb Z)\bigr).
$$
Sim describes this as the fully explicit cocycle of Dedekind–Rademacher [2307.00425].

Because $\mathrm{GL}_2(\mathbb Q)$ is generated by the diagonal scalars $T_m=\mathrm{diag}(m,1)$ together with
$$
T=\begin{pmatrix}1&1\\ 0&1\end{pmatrix},
\qquad
S=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix},
$$
the cocycle is determined by its values on these generators. For every $m\in\mathbb Q^\times$,
$$
J_{DR}(\mathrm{Id},T_m)=0.
$$
For $f\in\mathcal S(V)$,
$$
J_{DR}(\mathrm{Id},T)(f)
=
-(B_2\otimes B_0)_c(f\mid T)
+\Psi(f\mid T-f),
$$
and
$$
J_{DR}(\mathrm{Id},S)(f)
=
(B_2\otimes B_1)_c(f)
+\Psi(f\mid S-f),
$$
where $\Psi$ is an explicit $\mathbb Q$-valued distribution. All other values are determined by cocycle and invariance relations [2307.00425].

The derivation rests on a period comparison:
$$
\int_{i\infty}^{\gamma(i\infty)}d\log\,u(f)
=
C(\alpha,\beta;a,b)-(\delta_0\otimes\mathcal R\mathrm{Clog})(f\mid\gamma-f),
$$
for $\gamma\in \mathrm{SL}_2(\mathbb Z)$ with $\gamma(i\infty)=\alpha/\beta$. One rewrites $d\log u$ as a sum of Eisenstein series of weight $2$, applies Mellin-transform formulas of Stevens, and tracks Bernoulli-sum identities. The terms $C(\alpha,\beta;\dots)$ yield the $B_2\otimes B_r$ summands, and the correction by $\mathcal R\mathrm{Clog}$ yields the $\Psi$ terms [2307.00425].

This explicit formula is central because it converts a cohomology class defined through Kato–Siegel units into a computable cocycle with concrete Bernoulli-tensor and correction terms.

## 4. Smoothing and comparison with Darmon–Dasgupta measures

Let $G$ be a group and
$$
\delta=\sum_{g\in G}n_g[g]\in\mathbb Z[G].
$$
For a left $G$-module $M$, the $\delta$-smoothing operator is
$$
m\mapsto m^\delta=\delta\ast m=\sum_g n_g\,g\cdot m.
$$
For any subgroup $H\subset G$, this induces
$$
\delta^\ast:H^i(G,M)\longrightarrow H^i(H,M^\delta).
$$
In particular, if
$$
\delta=\sum_{D\mid N}n_D[LR(D)],
\qquad
\sum_{D\mid N}n_DD=0,
$$
then one may form the $\delta$-smoothed class
$$
\mu_{DR}^\delta=\delta^\ast(\mu_{DR})
\in H^1\!\bigl(G,\mathrm{Dist}(V,\mathbb Z)\bigr)
\quad
(\text{away from level }N)
$$
[2307.00425].

Fix a prime $p\nmid Nc$. Darmon and Dasgupta constructed a family of period-of-Eisenstein measures
$$
\bigl\{\widetilde\mu^\delta_{\{r\to s\}}\in\mathrm{Dist}(V_p,\mathbb Z_p)\bigr\},
$$
for $r,s\in\Gamma_0(N)\backslash\mathbb P^1(\mathbb Q)$, which glue to a $1$-cocycle
$$
\mu_{DD,\delta}\in Z^1\Bigl(\mathrm{GL}_2(\mathbb Q_p),\mathrm{Dist}(V_p,\mathbb Z_p)\Bigr),
$$
characterized by the moment condition
$$
\int_{V_p}h(x,y)\,d\widetilde\mu^\delta_{\{r\to s\}}(x,y)
=
\Re(1-p^{k-2})\,\int_r^s h(z,1)\,E_k(z)\,dz
\qquad (\text{polynomial }h)
$$
[2307.00425].

After pulling $\mu_{DR}^\delta$ down to $p$-adics and twisting by $c$, Sim proves that in cohomology
$$
-\,\mu_{DD,\delta}^c
=
\tfrac1{12}\,\mu_{DR}^\delta
\in H^1\!\bigl(\mathrm{GL}_2(\mathbb Q_p),\mathrm{Dist}(V_p,\mathbb Z_p)\bigr).
$$
Equivalently, if $c>1$, $(c,6)=1$, $N$ is coprime to $c$, $p\nmid Nc$, and
$$
\delta=\sum_{D\mid N}n_D[LR(D)],\qquad \sum n_DD=0,
$$
then
$$
-\,\mu_{DD,\delta}^c=\tfrac1{12}\,\mu_{DR,p}^{\delta}
$$
in $H^1(\mathrm{GL}_2(\mathbb Q_p),\mathrm{Dist}(V_p,\mathbb Z_p))$ [2307.00425].

This comparison identifies the global Dedekind–Rademacher class with the $p$-adic measures of Darmon–Dasgupta up to the explicit scalar $\tfrac1{12}$. It also explains the smoothing formalism in a way that is compatible with explicit cocycle calculations.

## 5. Real multiplication values and Gross–Stark units

Darmon, Pozzi, and Vonk construct a rigid-analytic $1$-cocycle
$$
J_{DR}\in H^1(T,\mathscr A^\times/p^{\mathbb Z}),
\qquad
T=\mathrm{SL}_2(\mathbb Z[1/p]),
$$
valued in the multiplicative group of Drinfeld’s rigid meromorphic functions on the $p$-adic upper half-plane [2103.02490]. Their construction begins with Siegel units of $p$-power level, packaged into a $\mathrm{GL}_2(\mathbb Z[1/p])$-invariant distribution
$$
\mu_{\mathrm{Siegel}}\in D(X,\mathcal O^{an}(\mathcal H)),
$$
on the space $X=(\mathbb Q_p^2)'$ of primitive vectors. After taking logarithms one obtains a measure-valued $1$-cocycle
$$
c_\mu\in Z^1\bigl(T,D(X,\mathbb Z)\bigr),
$$
whose restriction to $\mathrm{SL}_2(\mathbb Z)$ recovers the classical Dedekind–Rademacher homomorphism $Y_{DR}:\Gamma_0(p)\to\mathbb Z$ given by periods of the weight-two Eisenstein series
$$
E_P(z)=\frac d{dz}\log\bigl(A(pz)/A(z)\bigr)
=-1+24\sum_{n\ge1}L_0^{(p)}(n)\,q^n.
$$
The multiplicative Poisson transform
$$
\mathcal P:D^0(X,\mathbb Z)\longrightarrow\mathscr A^\times,\qquad
\mathcal P(\nu)(T)=\exp\!\Bigl(\int_{X_0}\log(xT+y)\,d\nu(x,y)\Bigr)
$$
produces the cocycle $J_{DR}$ [2103.02490].

A point $\tau\in\mathcal H_p$ is called a real-multiplication point of discriminant $D$ if it is fixed by an order in a real quadratic field $F=\mathbb Q(\sqrt D)$ of discriminant $D$. If $y_\tau$ generates the infinite cyclic stabilizer, the RM-value of a rigid-analytic theta-cocycle $J$ at $\tau$ is
$$
J[\tau]=J(y_\tau)(\tau)\in\mathbb C_p^\times.
$$
For fundamental discriminant $D>0$ with $(D,p)=1$, Darmon, Pozzi, and Vonk prove that
$$
J_{DR}[\tau]=u_\tau^2
\quad
\bigl(\mathrm{mod}\;(p^{\mathbb Z})\text{-torsion and powers of }p\bigr),
$$
where $u_\tau$ is the Gross–Stark $p$-unit in the narrow Hilbert class field of the associated real quadratic field. In particular,
$$
J_{DR}[\tau]\in \bigl(\mathcal O_H[1/p]^\times\bigr)\otimes_{\mathbb Z}\mathbb Z[1/2]
$$
[2103.02490].

In the formulation of Sim, the work of Darmon, Pozzi, and Vonk shows that the RM-values of the Dedekind–Rademacher cocycle $J_{DR}$ are Gross–Stark units up to a controlled torsion, and the explicit formulas for $J_{DR}$ make these values computable [2307.00425]. The comparison theorem with Darmon–Dasgupta measures shows how these complex-analytic units are transformed into the $p$-adic Gross–Stark units built by Dasgupta [2307.00425].

## 6. Variants, analogues, and generalizations

A hyperbolic analogue of the Rademacher symbol was constructed by Duke–Imamoğlu–Tóth and studied through explicit formulas by Matsusaka. Fixing a primitive hyperbolic $\gamma\in\Gamma$, one defines a generating series
$$
F_\gamma(z)=\sum_{m\ge0}\mathrm{Val}_m(\gamma)q^m,
$$
whose modular defect is a weight-$2$ rational cocycle
$$
r_\gamma(\sigma,z)=j(\sigma,z)^{-2}F_\gamma(\sigma z)-F_\gamma(z).
$$
Integrating in $z$ gives a weight-$0$ cocycle $R_\gamma(\sigma,z)$, unique up to constants, with $R_\gamma(T,z)=0$, and one defines
$$
\Phi_\gamma(\sigma)=\tfrac2\pi\lim_{y\to\infty}R_\gamma(\sigma,iy),
\qquad
\Psi_\gamma(\sigma)=\lim_{n\to\infty}\Phi_\gamma(\sigma^n)/n.
$$
Geometrically, $\Psi_\gamma(\sigma)$ recovers the signed intersection number of the closed geodesics $S_\gamma$ and $S_\sigma$ in $\Gamma\backslash\mathcal H$, equivalently the linking number of the corresponding modular knots [2003.12354]. This construction contrasts the parabolic axis underlying the classical symbol with a hyperbolic axis and is accompanied by Kronecker limit type formulas for parabolic, elliptic, and hyperbolic Eisenstein series [2003.12354].

A higher-dimensional analogue was developed for generalized Dedekind sums via Todd series of lattice cones. For a simplicial lattice cone $C$, Pommersheim’s construction yields an $n$-cocycle
$$
C\mapsto S(C):=\frac{\mathrm{Todd}_C(x)}{(\det M_C)\,x_1\cdots x_n}
$$
with values in meromorphic functions. In the special cone $C(q;p)$, the coefficients $t_r(q;p)$ identify with generalized Dedekind sums $S_r(q;p)$, and reciprocity follows from the cocycle property of the Todd series [1406.3563]. For fixed even weight $N$ and dimension $n$, Chae, Jun, and Lee define a Laurent polynomial $f_r(p)$ so that
$$
R_r(q;p)\equiv f_r(p)\pmod q,
$$
and the corrected quantity
$$
\widehat S_r(q;p):=S_r(q;p)-\frac{r!}{d_{N,n}\,q^{N-n+1}}\,f_r(p)
$$
defines an $n$-cocycle on $\mathrm{SL}_n(\mathbb Z)$; in the classical $n=2$ case one recovers Rademacher’s $\phi$-function [1406.3563].

A further extension to Bianchi groups constructs a Dedekind–Rademacher–Bianchi cocycle
$$
\Phi_N:\Gamma_0(\mathfrak n)\longrightarrow\mathscr F,
\qquad
\Gamma_0(\mathfrak n)\subset \mathrm{SL}_2(\mathcal O_L)\subset \mathrm{SL}_2(\mathbb C),
$$
where $\mathscr F$ is the space of functions $(\mathbb C/L)^2\to\mathbb C$. For
$$
A=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in \Gamma_0(\mathfrak n),
$$
it is defined by
$$
\Phi_N(A)(p,q)
=
\Phi\!\begin{pmatrix}a&bN\\ c/N&d\end{pmatrix}(p,q)
-
\Phi\!\begin{pmatrix}a&b\\ c&d\end{pmatrix}(p,q),
$$
and satisfies
$$
\Phi_N(AB)(p,q)=\Phi_N(A)(p,q)+\Phi_N(B)\bigl((p,q)A\bigr),
$$
so $\Phi_N\in Z^1\bigl(\Gamma_0(\mathfrak n),\mathscr F\bigr)$ [2507.16671]. The cocycle is related to a function $H_N(u)=H(N\cdot u)-H(u)$ on hyperbolic $3$-space through
$$
\Phi_N(A)=H_N(A\cdot u)-H_N(u),
$$
is Hecke-equivariant, parametrizes twisted $L$-values, and yields integrality of these values up to the factor $(\alpha-\alpha')$ [2507.16671].

Within the subject itself, one recurring point of clarification concerns the optional “degree zero” condition in the smoothing divisor $\delta$. In the original Darmon–Dasgupta setting, both
$$
\sum_{D\mid N}n_D=0
\qquad\text{and}\qquad
\sum_{D\mid N}n_DD=0
$$
were imposed. Sim shows that relaxing $\sum n_D=0$ merely introduces a global coboundary $\Psi$, whereas $\sum n_DD=0$ is the essential condition needed to kill the unwanted $q$-order. In particular, checking integrality forces exactly $\sum n_DD=0$, but never requires $\sum n_D=0$ [2307.00425]. This resolves a technical issue already remarked upon by Dasgupta–Kakde and Fleischer–Liu in the sources summarized by Sim.

Taken together, these variants show that the Dedekind–Rademacher cocycle is not an isolated construction attached only to $\mathrm{SL}_2(\mathbb Z)$. It organizes transformation laws, Eisenstein periods, special values of $L$-functions, and arithmetic units across classical, $p$-adic, hyperbolic, higher-dimensional, and Bianchi-group settings [2307.00425] [2507.16671].

Source: https://www.emergentmind.com/topics/dedekind-rademacher-cocycle