Dedekind–Rademacher Cocycle: Cohomology & Computations

• Dedekind–Rademacher cocycle is a cohomological refinement of the transformation law of log Δ(z), integrating Dedekind’s function and Rademacher’s modifications.
• The explicit cocycle representative J_DR combines modular unit constructions, Bernoulli distributions, and correction terms to yield computable invariants linked to Eisenstein series.
• Generalizations of the cocycle extend to distribution-valued and rigid-analytic frameworks, connecting results to Gross–Stark units, p-adic measures, and higher-dimensional analogues.

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 (Sim, 2023, Darmon et al., 2021).

1. Classical cocycle on $\Delta$0

For $\Delta$1 and

$\Delta$2

Dedekind’s transformation law takes the form

$\Delta$3

for $\Delta$4, with $\Delta$5 arising from the multivaluedness of $\Delta$6 (Matsusaka, 2020). Dedekind showed that

$\Delta$7

where $\Delta$8 is the classical Dedekind sum (Matsusaka, 2020).

Rademacher modified $\Delta$9 by

$12$0

and showed that $12$1 satisfies

$12$2

for all $12$3, together with the $12$4-cocycle relation

$12$5

In particular, for hyperbolic $12$6 one has $12$7 (Matsusaka, 2020).

The classical symbol also admits integral and continued-fraction descriptions. Writing

$12$8

one has for hyperbolic $12$9

$\mathrm{SL}_2(\mathbb Z)$0

for any $\mathrm{SL}_2(\mathbb Z)$1 on the axis $\mathrm{SL}_2(\mathbb Z)$2; and if $\mathrm{SL}_2(\mathbb Z)$3 is conjugate to a purely periodic continued fraction, then

$\mathrm{SL}_2(\mathbb Z)$4

in the notation of the continued-fraction expansion (Matsusaka, 2020).

This classical construction provides the prototype for later distribution-valued and rigid-analytic formulations. It also underlies the level-$\mathrm{SL}_2(\mathbb Z)$5 homomorphism $\mathrm{SL}_2(\mathbb Z)$6, defined by

$\mathrm{SL}_2(\mathbb Z)$7

which appears as the period of the eta-quotient $\mathrm{SL}_2(\mathbb Z)$8 (Klinger-Logan et al., 22 Jul 2025).

2. Distribution-valued cohomology class $\mathrm{SL}_2(\mathbb Z)$9

A global formulation begins with $\Phi$0, viewed as row vectors with the topology generated by its $\Phi$1-lattices. A test function on $\Phi$2 is a compactly supported locally constant $\Phi$3-valued function,

$\Phi$4

and $\Phi$5 consists of those supported away from $\Phi$6 (Sim, 2023).

Let $\Phi$7 act on $\Phi$8 by

$\Phi$9

and on test functions by

$\Psi$0

For a right $\Psi$1-module $\Psi$2, one sets

$\Psi$3

with left $\Psi$4-action

$\Psi$5

This gives the ambient module for the Dedekind–Rademacher cohomology class (Sim, 2023).

Fix an integer $\Psi$6 with $\Psi$7, and let $\Psi$8. Scholl’s construction of Kato–Siegel units on the universal elliptic curve yields a $\Psi$9-invariant distribution

$1$0

where $1$1 denotes the $1$2-valued holomorphic functions on $1$3 with no zeros, equipped with the usual $1$4-action by slash operators (Sim, 2023).

Applying the long exact sequence induced by

$1$5

one obtains the Dedekind–Rademacher cohomology class

$1$6

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 (Sim, 2023).

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 $1$7. That shift makes possible both explicit cocycle formulas and comparison with $1$8-adic measure constructions.

3. Explicit cocycle representative $1$9

An explicit representative is obtained from a section

$$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$0

satisfying

$$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$1

Writing out the three factors in the $$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$2-expansion of the Kato–Siegel unit gives

$$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$3

where $$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$4 is the $$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$5-th Bernoulli distribution on $$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$6, $$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$7 and its log-variants come from cyclotomic distributions on $$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$8, and

$$\mu_{DR}\in H^1\bigl(\mathrm{GL}_2(\mathbb Q),\mathrm{Dist}(V,\mathbb Z)\bigr)$$9

is the theta-unit distribution; all are $J_{DR}$0-smoothed by the group-algebra element $J_{DR}$1 (Sim, 2023).

The section is then defined by

$J_{DR}$2

and the explicit Dedekind–Rademacher cocycle is

$J_{DR}$3

Sim describes this as the fully explicit cocycle of Dedekind–Rademacher (Sim, 2023).

Because $J_{DR}$4 is generated by the diagonal scalars $J_{DR}$5 together with

$J_{DR}$6

the cocycle is determined by its values on these generators. For every $J_{DR}$7,

$J_{DR}$8

For $J_{DR}$9,

$p$0

and

$p$1

where $p$2 is an explicit $p$3-valued distribution. All other values are determined by cocycle and invariance relations (Sim, 2023).

The derivation rests on a period comparison:

$p$4

for $p$5 with $p$6. One rewrites $p$7 as a sum of Eisenstein series of weight $p$8, applies Mellin-transform formulas of Stevens, and tracks Bernoulli-sum identities. The terms $p$9 yield the $\Delta$00 summands, and the correction by $\Delta$01 yields the $\Delta$02 terms (Sim, 2023).

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 $\Delta$03 be a group and

$\Delta$04

For a left $\Delta$05-module $\Delta$06, the $\Delta$07-smoothing operator is

$\Delta$08

For any subgroup $\Delta$09, this induces

$\Delta$10

In particular, if

$\Delta$11

then one may form the $\Delta$12-smoothed class

$\Delta$13

(Sim, 2023).

Fix a prime $\Delta$14. Darmon and Dasgupta constructed a family of period-of-Eisenstein measures

$\Delta$15

for $\Delta$16, which glue to a $\Delta$17-cocycle

$\Delta$18

characterized by the moment condition

$\Delta$19

(Sim, 2023).

After pulling $\Delta$20 down to $\Delta$21-adics and twisting by $\Delta$22, Sim proves that in cohomology

$\Delta$23

Equivalently, if $\Delta$24, $\Delta$25, $\Delta$26 is coprime to $\Delta$27, $\Delta$28, and

$\Delta$29

then

$\Delta$30

in $\Delta$31 (Sim, 2023).

This comparison identifies the global Dedekind–Rademacher class with the $\Delta$32-adic measures of Darmon–Dasgupta up to the explicit scalar $\Delta$33. 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 $\Delta$34-cocycle

$\Delta$35

valued in the multiplicative group of Drinfeld’s rigid meromorphic functions on the $\Delta$36-adic upper half-plane (Darmon et al., 2021). Their construction begins with Siegel units of $\Delta$37-power level, packaged into a $\Delta$38-invariant distribution

$\Delta$39

on the space $\Delta$40 of primitive vectors. After taking logarithms one obtains a measure-valued $\Delta$41-cocycle

$\Delta$42

whose restriction to $\Delta$43 recovers the classical Dedekind–Rademacher homomorphism $\Delta$44 given by periods of the weight-two Eisenstein series

$\Delta$45

The multiplicative Poisson transform

$\Delta$46

produces the cocycle $\Delta$47 (Darmon et al., 2021).

A point $\Delta$48 is called a real-multiplication point of discriminant $\Delta$49 if it is fixed by an order in a real quadratic field $\Delta$50 of discriminant $\Delta$51. If $\Delta$52 generates the infinite cyclic stabilizer, the RM-value of a rigid-analytic theta-cocycle $\Delta$53 at $\Delta$54 is

$\Delta$55

For fundamental discriminant $\Delta$56 with $\Delta$57, Darmon, Pozzi, and Vonk prove that

$\Delta$58

where $\Delta$59 is the Gross–Stark $\Delta$60-unit in the narrow Hilbert class field of the associated real quadratic field. In particular,

$\Delta$61

(Darmon et al., 2021).

In the formulation of Sim, the work of Darmon, Pozzi, and Vonk shows that the RM-values of the Dedekind–Rademacher cocycle $\Delta$62 are Gross–Stark units up to a controlled torsion, and the explicit formulas for $\Delta$63 make these values computable (Sim, 2023). The comparison theorem with Darmon–Dasgupta measures shows how these complex-analytic units are transformed into the $\Delta$64-adic Gross–Stark units built by Dasgupta (Sim, 2023).

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 $\Delta$65, one defines a generating series

$\Delta$66

whose modular defect is a weight-$\Delta$67 rational cocycle

$\Delta$68

Integrating in $\Delta$69 gives a weight-$\Delta$70 cocycle $\Delta$71, unique up to constants, with $\Delta$72, and one defines

$\Delta$73

Geometrically, $\Delta$74 recovers the signed intersection number of the closed geodesics $\Delta$75 and $\Delta$76 in $\Delta$77, equivalently the linking number of the corresponding modular knots (Matsusaka, 2020). 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 (Matsusaka, 2020).

A higher-dimensional analogue was developed for generalized Dedekind sums via Todd series of lattice cones. For a simplicial lattice cone $\Delta$78, Pommersheim’s construction yields an $\Delta$79-cocycle

$\Delta$80

with values in meromorphic functions. In the special cone $\Delta$81, the coefficients $\Delta$82 identify with generalized Dedekind sums $\Delta$83, and reciprocity follows from the cocycle property of the Todd series (Chae et al., 2014). For fixed even weight $\Delta$84 and dimension $\Delta$85, Chae, Jun, and Lee define a Laurent polynomial $\Delta$86 so that

$\Delta$87

and the corrected quantity

$\Delta$88

defines an $\Delta$89-cocycle on $\Delta$90; in the classical $\Delta$91 case one recovers Rademacher’s $\Delta$92-function (Chae et al., 2014).

A further extension to Bianchi groups constructs a Dedekind–Rademacher–Bianchi cocycle

$\Delta$93

where $\Delta$94 is the space of functions $\Delta$95. For

$\Delta$96

it is defined by

$\Delta$97

and satisfies

$\Delta$98

so $\Delta$99 (Klinger-Logan et al., 22 Jul 2025). The cocycle is related to a function $12$00 on hyperbolic $12$01-space through

$12$02

is Hecke-equivariant, parametrizes twisted $12$03-values, and yields integrality of these values up to the factor $12$04 (Klinger-Logan et al., 22 Jul 2025).

Within the subject itself, one recurring point of clarification concerns the optional “degree zero” condition in the smoothing divisor $12$05. In the original Darmon–Dasgupta setting, both

$12$06

were imposed. Sim shows that relaxing $12$07 merely introduces a global coboundary $12$08, whereas $12$09 is the essential condition needed to kill the unwanted $12$10-order. In particular, checking integrality forces exactly $12$11, but never requires $12$12 (Sim, 2023). 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 $12$13. It organizes transformation laws, Eisenstein periods, special values of $12$14-functions, and arithmetic units across classical, $12$15-adic, hyperbolic, higher-dimensional, and Bianchi-group settings (Sim, 2023, Klinger-Logan et al., 22 Jul 2025).