Rigid Meromorphic Cocycles

• Rigid meromorphic cocycles are cocycle-theoretic objects constructed from meromorphic functions and divisors, defined across multiple frameworks in arithmetic geometry.
• They utilize both multiplicative and additive cohomological methods, incorporating modular symbols, group cohomology, and deformation complexes to encode arithmetic invariants.
• Rigidity ensures that global structures are determined by localized residue or Stokes data, leading to applications in p-adic singular moduli, Borcherds-type lifts, and the classification of meromorphic connections.

Rigid meromorphic cocycles are cocycle-theoretic objects built from meromorphic data and constrained by strong local-to-global principles. In $p$-adic arithmetic geometry, the phrase most commonly refers either to multiplicative cohomology classes with values in rigid meromorphic functions on the Drinfeld upper half-plane or on more general $p$-adic symmetric spaces, or to additive modular-symbol cocycles of higher weight with rigid meromorphic values (Fust et al., 2023, Negrini, 2022, Darmon et al., 2023). In a distinct but related cohomological usage, rigid irreducible meromorphic connections on $\mathbb{P}^1$ are viewed as cocycles in a deformation complex, and rigidity means vanishing of the deformation space with fixed formal local data (Sabbah, 2022). Across these settings, meromorphicity supplies divisors, residues, or Stokes data, while rigidity means that global objects are determined by restricted local information, often up to finite ambiguity.

1. Terminology and principal meanings

In current usage, “rigid meromorphic cocycles” does not denote a single category-independent object. The literature employs the phrase in at least three nearby frameworks: multiplicative $p$-adic group cohomology, additive higher-weight modular-symbol theory, and cohomological interpretations of meromorphic connections on $\mathbb{P}^1$ (Fust et al., 2023, Negrini, 2022, Sabbah, 2022).

Framework	Basic object	Rigidity condition
$p$-adic multiplicative theory	Class in $H^1(\Gamma,\mathcal M^\times)$ or $H^s(\Gamma,\mathcal M_X)$	Divisor constrained by quadratic or Kudla–Millson data
Higher-weight additive theory	Element of $\mathrm{MS}_\Gamma(M_k)$	Period-function classification, residue control, Hecke constraints
Meromorphic connections on $\mathbb P^1$	Meromorphic connection viewed as a cocycle in a deformation complex	$p$0 or $p$1

A recurrent source of confusion is the coexistence of multiplicative and additive theories. In the Darmon–Vonk framework, the basic object is multiplicative: a cocycle with values in the multiplicative group of nonzero rigid meromorphic functions. In Negrini’s higher-weight theory, the coefficient module is additive, and cocycles are $p$2-invariant modular symbols with values in rigid meromorphic functions. The shared terminology reflects the common role of meromorphic rigid-analytic functions on $p$3-adic symmetric spaces and the common expectation that divisors, residues, and special values should encode arithmetic information (Fust et al., 2023, Negrini, 2022).

A second distinction concerns “rigid” itself. In the $p$4-adic arithmetic literature, rigidity is partly geometric, referring to rigid-analytic spaces and rigid meromorphic functions. In Sabbah’s reinterpretation of meromorphic connections, rigidity is deformation-theoretic: a cocycle is rigid when no nontrivial deformation exists with fixed formal types at the singular points (Sabbah, 2022).

2. Drinfeld upper half-plane, group cohomology, and modular symbols

Fix a prime $p$5. The Drinfeld $p$6-adic upper half-plane is

$p$7

and the Ihara group

$p$8

acts on $p$9 by $\mathbb{P}^1$0-adic Möbius transformations. Writing

$\mathbb{P}^1$1

the classical Darmon–Vonk notion of rigid meromorphic cocycle is a cohomology class in

$\mathbb{P}^1$2

represented by a $\mathbb{P}^1$3-cocycle $\mathbb{P}^1$4 satisfying

$\mathbb{P}^1$5

Two such cocycles are identified when they differ by a multiplicative coboundary $\mathbb{P}^1$6 (Fust et al., 2023).

The same framework contains theta cocycles, defined in

$\mathbb{P}^1$7

and distinguished subclasses. If $\mathbb{P}^1$8 is the stabilizer of $\mathbb{P}^1$9, a class is parabolic when its restriction to $p$0 is trivial, and quasi-parabolic when its restriction lies in the image of $p$1. A key fact is that every class in $p$2 has a unique representative $p$3 with $p$4; these are precisely the classes that can be evaluated at real quadratic points (Fust et al., 2023).

The higher-weight additive theory replaces multiplicative group cohomology by modular symbols. For each integer $p$5, let $p$6 be the group of rigid analytic functions on $p$7 and $p$8 the group of rigid meromorphic functions, both with the weight-$p$9 action

$\mathbb{P}^1$0

A rigid meromorphic cocycle of weight $\mathbb{P}^1$1 is then an element of

$\mathbb{P}^1$2

that is, a $\mathbb{P}^1$3-invariant modular symbol

$\mathbb{P}^1$4

satisfying antisymmetry, the modular-symbol relation, and $\mathbb{P}^1$5-invariance (Negrini, 2022). In this setting, the “period function” $\mathbb{P}^1$6 already carries substantial information, and the action of the involution

$\mathbb{P}^1$7

induces the $\mathbb{P}^1$8-even/$\mathbb{P}^1$9-odd dichotomy central to the classification of meromorphic period functions (Negrini, 2022).

These two formalisms are linked by residues. Although the 2024 Shintani paper works with rigid analytic cocycles, it explicitly identifies the original Darmon–Vonk theory as the theory of rigid meromorphic cocycles and emphasizes that annular residues along the Bruhat–Tits tree are the analytic relic of meromorphicity: residues are exactly the data that would detect principal parts once poles are allowed (Negrini, 2024).

3. Orthogonal and quaternionic generalizations

The orthogonal-group framework extends rigid meromorphic cocycles from $p$0 to arbitrary real signature. Let $p$1 be a non-degenerate quadratic space over $p$2, let $p$3, and choose a prime $p$4 such that $p$5 admits a self-dual lattice. The associated $p$6-adic symmetric space is

$p$7

where $p$8 is the quadric of isotropic lines. For $p$9, $H^1(\Gamma,\mathcal M^\times)$0 is the Drinfeld upper half-plane, and in split dimension $H^1(\Gamma,\mathcal M^\times)$1 one has

$H^1(\Gamma,\mathcal M^\times)$2

For anisotropic $H^1(\Gamma,\mathcal M^\times)$3, the rational quadratic divisor

$H^1(\Gamma,\mathcal M^\times)$4

defines an element of the $H^1(\Gamma,\mathcal M^\times)$5-module $H^1(\Gamma,\mathcal M^\times)$6 of locally finite combinations of rational quadratic divisors (Darmon et al., 2023).

Kudla–Millson geometry at infinity supplies distinguished divisor-valued cohomology classes $H^1(\Gamma,\mathcal M^\times)$7, where $H^1(\Gamma,\mathcal M^\times)$8 is the negative index in the real signature $H^1(\Gamma,\mathcal M^\times)$9. Let $H^s(\Gamma,\mathcal M_X)$0 be the multiplicative group of rigid meromorphic functions on $H^s(\Gamma,\mathcal M_X)$1 whose divisors lie in $H^s(\Gamma,\mathcal M_X)$2. Then rigid meromorphic cocycles of level $H^s(\Gamma,\mathcal M_X)$3 are defined as classes

$H^s(\Gamma,\mathcal M_X)$4

whose divisor lies in the subgroup generated by Kudla–Millson divisors: $H^s(\Gamma,\mathcal M_X)$5 The exact sequence

$H^s(\Gamma,\mathcal M_X)$6

and the surjectivity of the divisor map show that the obstruction to lifting a divisor-valued class to a meromorphic cocycle lies in $H^s(\Gamma,\mathcal M_X)$7; the paper then proves finiteness results for this obstruction group and constructs $H^s(\Gamma,\mathcal M_X)$8-adic Borcherds-type lifts in selected signatures (Darmon et al., 2023).

The explicit existence results are strongest in the definite case $H^s(\Gamma,\mathcal M_X)$9 and the hyperbolic case $\mathrm{MS}_\Gamma(M_k)$0. In the definite case, for suitable coefficient data $\mathrm{MS}_\Gamma(M_k)$1 orthogonal to vector-valued modular forms of weight $\mathrm{MS}_\Gamma(M_k)$2, there exists $\mathrm{MS}_\Gamma(M_k)$3 with

$\mathrm{MS}_\Gamma(M_k)$4

In signature $\mathrm{MS}_\Gamma(M_k)$5, under compactness and orthogonality hypotheses, one obtains a nonzero integer $\mathrm{MS}_\Gamma(M_k)$6 and $\mathrm{MS}_\Gamma(M_k)$7 such that

$\mathrm{MS}_\Gamma(M_k)$8

These are $\mathrm{MS}_\Gamma(M_k)$9-adic analogues of Borcherds’ singular theta lift, now valued in cohomology with rigid meromorphic coefficients rather than in automorphic products on complex orthogonal Shimura varieties (Darmon et al., 2023).

A complementary generalization appears in Gehrmann’s quaternionic theory. There, for a number field $\mathbb P^1$0, a quaternion algebra $\mathbb P^1$1 split at a finite place $\mathbb P^1$2, and a $\mathbb P^1$3-arithmetic congruence subgroup $\mathbb P^1$4, a rigid meromorphic cocycle is an element

$\mathbb P^1$5

where $\mathbb P^1$6 denotes rigid meromorphic functions on the $\mathbb P^1$7-adic upper half-plane attached to $\mathbb P^1$8. The divisor of such a cocycle is a class in cohomology with coefficients in locally finite divisors, and Bieri–Eckmann duality shows that in the critical degree $\mathbb P^1$9 its support is a finite union of $p$00-orbits of quadratic points of prescribed splitting type at infinity. Below that degree, the natural map

$p$01

is an isomorphism, so there are no genuinely meromorphic cocycles there (Gehrmann, 2020).

4. Higher-weight theory, residues, and lift constructions

Higher-weight additive rigid meromorphic cocycles were developed systematically for odd weights $p$02. The basic period functions are the values at $p$03,

$p$04

and satisfy explicit functional equations under the standard generators $p$05 of the Ihara group. A central classification theorem states that if $p$06 is odd and a rigid meromorphic period function of weight $p$07 has poles only in Galois-conjugate RM pairs with opposite residues, then it is a finite linear combination of explicit period functions $p$08 attached to RM points, plus a rigid analytic period function (Negrini, 2022).

The principal explicit family is

$p$09

defined for odd $p$10 and discriminant $p$11. This converges to a rigid meromorphic function on $p$12, and it is rigid analytic when $p$13. The resulting map

$p$14

is a rigid meromorphic cocycle of weight $p$15 (Negrini, 2022).

A second structural ingredient is the higher-weight Schneider–Teitelbaum lift. Harmonic cocycles on the Bruhat–Tits tree with values in polynomial duals are identified with rigid analytic cocycles via

$p$16

and there is a residue map

$p$17

satisfying

$p$18

For the special cocycles $p$19, one computes

$p$20

where $p$21 is a polynomial-valued modular symbol and, simultaneously, the period polynomial of a classical level-$p$22 Zagier form $p$23. This produces a cocycle-valued kernel

$p$24

hence a Shimura–Shintani style correspondence

$p$25

This realizes higher-weight rigid analytic cocycles as $p$26-adic targets of half-integral weight modular forms (Negrini, 2022).

Negrini’s Shintani lift deepens the residue formalism. For $p$27, a weight $p$28 rigid analytic cocycle is an element of $p$29, and the residue map

$p$30

is obtained by annular residues along a standard edge $p$31 of the Bruhat–Tits tree. On the cuspidal subspace, the Shintani-type map

$p$32

has Fourier coefficients given by pairings of residue polynomials with powers of binary quadratic forms, and satisfies

$p$33

Although the paper is restricted to rigid analytic cocycles, it explicitly states that the residue formalism is structurally prepared for a meromorphic setting and that one expects an extension to genuinely meromorphic cocycles with controlled poles (Negrini, 2024).

5. Special values, singular moduli, and class-field phenomena

In the original $p$34 theory, the arithmetic focus is evaluation at real quadratic points. If $p$35, then its stabilizer in $p$36 is isomorphic to $p$37, and for a quasi-parabolic cocycle $p$38 normalized by $p$39 one defines

$p$40

This value is independent of the chosen representative of the cohomology class and constant on $p$41-orbits. It is called a real quadratic singular modulus. The central conjecture states that for every $p$42, the value $p$43 is algebraic and lies in the compositum of the narrow ring class field $p$44 of the order attached to $p$45 and a field $p$46 determined by the poles of the period function of $p$47 (Fust et al., 2023).

The same survey records arithmetic evidence of a more precise kind. For the Dedekind–Rademacher theta cocycle $p$48, Darmon–Pozzi–Vonk show that for a real quadratic point $p$49 of discriminant prime to $p$50,

$p$51

for some $p$52, where $p$53 is the narrow Hilbert class field of $p$54. In particular, $p$55 is algebraic and lies in $p$56 (Fust et al., 2023).

The orthogonal theory generalizes special values from RM points to special points on $p$57 attached to maximal tori. For an oriented special point $p$58 and $p$59 regular at $p$60, one obtains a value

$p$61

The associated reciprocity conjecture predicts that these values lie in class fields $p$62 of the reflex field $p$63, up to a fixed finitely generated subgroup $p$64, and satisfy an adelic reciprocity law governed by the relative class group $p$65 (Darmon et al., 2023).

The Bianchi $p$66-case is the first setting in which genuinely new computations beyond the $p$67 theory have been carried out systematically. Here $p$68, $p$69, and $p$70. The cocycles arise as modular-symbol valued classes

$p$71

and can be evaluated at small RM points, small CM points, and big ATR points. The field of definition of the cocycle attached to $p$72 is $p$73, computed explicitly from local spinor norms (Gehrmann et al., 26 Nov 2025).

The 2025 Bianchi evaluation paper reports the first numerical verification of the conjectured algebraicity of special values at big ATR points. In $p$74 big ATR cases, the quantities $p$75 and $p$76 were recognized as algebraic integers. The same computations yielded $p$77 new small CM examples and $p$78 small RM examples. A further observation is a Bruinier–Yang-type divisibility pattern for the rational primes appearing in the norms of the recognized algebraic values, numerically paralleling the behaviour of CM values of Borcherds products on Hilbert modular surfaces (Gehrmann et al., 26 Nov 2025).

6. Meromorphic connections on $p$79 as rigid cocycles

A different but mathematically adjacent use of the phrase appears in Sabbah’s discussion of rigid irreducible meromorphic connections on $p$80. Let $p$81 be a nonempty Zariski open subset and $p$82 an algebraic vector bundle with integrable connection on $p$83. Such an object may be regarded as a meromorphic connection on $p$84, as a holonomic $p$85-module via middle extension, or as a Stokes-filtered local system under irregular Riemann–Hilbert. In the cocycle language adopted there, a meromorphic connection is a $p$86-cocycle in the deformation complex controlling connections with prescribed formal types, and rigidity means vanishing of the tangent space to deformations (Sabbah, 2022).

For an irreducible meromorphic bundle with connection, the rigidity index is

$p$87

Physical rigidity is equivalent to cohomological rigidity: $p$88 Equivalently, any other irreducible connection on the same $p$89 with the same formal type at each puncture is globally isomorphic to $p$90 (Sabbah, 2022).

The Arinkin–Deligne–Katz algorithm gives a structural classification. Starting from an irreducible holonomic $p$91-module with specified Levelt–Turrittin data, one applies rank-one twists, middle convolution $p$92, and Fourier–Laplace transform $p$93. Rigidity is preserved by these operations, and an irreducible $p$94 is rigid if and only if a finite sequence of such operations reduces it to the trivial rank-one connection $p$95. In this sense, all rigid irreducible meromorphic connections arise from the trivial one by iterating these functors (Sabbah, 2022).

Sabbah’s note then derives three global consequences for quasi-unipotent rigid irreducible connections on $p$96. First, they are of exponential-geometric origin: after passage to a suitable finite cover and cohomological direct image, they arise inside pushforwards of exponential connections $p$97. Second, their Stokes-filtered local systems admit integral structures over cyclotomic rings $p$98. Third, for fixed $p$99, bounded rank, bounded quasi-unipotence order, and a prescribed finite set $\mathbb{P}^1$00 of exponential factors, only finitely many such rigid irreducible connections exist. These results justify reading rigid meromorphic connections on $\mathbb{P}^1$01 as a precise geometric incarnation of “rigid meromorphic cocycles” (Sabbah, 2022).

A plausible implication is that the two large literatures considered above—$\mathbb{P}^1$02-adic rigid meromorphic cocycles and rigid meromorphic connections—share a common organizing principle: in both, cocycle classes are governed by sharply constrained divisor, residue, or Stokes data, and rigidity is expressed by the collapse of global ambiguity to finite or deformation-trivial choices.