Local Realizability at a Prime

• Local realizability at a prime is defined as the condition where a global structure exhibits specific p-dependent local properties, enabling local-to-global analysis.
• It plays a crucial role across algebraic dynamics, homotopy theory, and number theory by identifying prime-specific obstructions and congruence phenomena.
• Applications include matching fixed point counts in dynamical systems, realizing cohomological invariants in topology, and decomposing ideals in commutative algebra.

Local realizability at a prime is a mathematical property or condition asserting that a given algebraic, topological, or dynamical object exhibits a specific kind of “realizability” when analyzed with respect to a fixed prime number. This notion is used across diverse areas—including algebraic topology, algebraic geometry, number theory, and dynamics—to formalize the passage from global to local behavior and to identify obstructions, local-to-global principles, and congruence phenomena sensitive to the arithmetic of primes.

1. Formal Definition and Local-Global Principle

Local realizability at a prime is generally described as the property that a global structure (e.g., an integer sequence, a Galois representation, an unstable module, a chain complex) has a partial realization or manifestation that is controlled by the arithmetic, algebraic, or cohomological data associated to a fixed prime $p$. In typical settings, "local" means one of the following:

• The $p$-primary (or $p$-component) part of a structure (e.g., $p$-torsion, $p$-adic, $p$-localization, $p$-primary group, or the $p$-part of a sequence).
• A structure considered over a local object such as $\mathbb{Z}_p$, a local ring, $p$-adic fields, or the Sylow $p$-subgroups of a group.
• A variant of a realization problem restricted to the context of a fixed prime (e.g., fusion systems over a discrete $p$-toral group, or local deformation rings at $p$).

The local-global principle in this context asserts that a global realization problem can be analyzed by first understanding its realizability at every prime and then patching the local information to answer the original question. For example, the principle “a sequence is nilpotently realizable globally if and only if its localized pieces are realizable at every prime” appears in the context of algebraic dynamics for integer sequences (Jaidee et al., 14 Aug 2025).

2. Local Realizability in Arithmetic Dynamics and Integer Sequences

In algebraic dynamics, a sequence $a = (a_n)$ is said to be algebraically realizable if there exists a group endomorphism $T: G \to G$ so that for all $n$, $\#\mathrm{Fix}(T^n) = a_n$, i.e., the number of fixed points of $T^n$ coincides with $a_n$. The notion of local realizability at a prime $p$ is then defined as the property that the sequence of $p$-parts, i.e., $(\lfloor a_n\rfloor_p) = (p^{\nu_p(a_n)})$, is itself algebraically realizable, typically by an endomorphism of a $p$-group (Jaidee et al., 14 Aug 2025).

A fundamental theorem in this context is:

• $a$ is nilpotently realizable globally if and only if for every rational prime $p$ the sequence $(p^{\nu_p(a_n)})$ is (nilpotently) algebraically realizable.

Examples illustrate the scope of local realizability:

• The Bernoulli denominators: For $B_n$ the sequence of Bernoulli numbers, the denominator sequence $b_n$ given by

$b_n = 2 \prod_{p-1 \mid 2n} p^{1+\nu_p(n)}$

is locally algebraically realizable at every prime. That is, for each $p$ there exists a group endomorphism whose periodic-point count matches $\lfloor b_n\rfloor_p$.

• The Euler numbers: The sequence $e_n = (-1)^n E_{2n}$ (where $E_k$ are the Euler numbers) is globally realizable but fails to be locally nilpotently realizable at some primes (e.g., at $p=61$, $\lfloor e \rfloor_{61} = (1,1,61,1,1,\dots)$ violates necessary congruence properties). Therefore, no nilpotent group endomorphism can realize this sequence globally, explaining the relative scarcity of congruences for Euler numbers (Jaidee et al., 14 Aug 2025).

This ties local realizability directly to congruence phenomena and provides a dynamical interpretation of number-theoretic regularity notions, as in the case of Bernoulli regular primes: a prime $q$ is regular if and only if the $q$-part of the Bernoulli numerator sequence is locally algebraically realizable; irregularity manifests as a local obstruction (Jaidee et al., 14 Aug 2025).

3. Local Realizability in Homotopy Theory and Algebraic Topology

In algebraic topology, local realizability at a prime arises in the context of realizing algebraic invariants (e.g., unstable modules over the Steenrod algebra) as the actual cohomology of topological spaces, or of spectra, at a given prime.

• In the Arone–Goodwillie spectral sequence for iterated loop space functors (Büscher et al., 2011), local realizability at an odd prime $p$ is addressed by analyzing when an unstable module over the Steenrod algebra $A_p$ with specific module-theoretic and numerical properties can be realized as the cohomology of a space. A key realization condition is the numerical inequality

$2p^k \le (p^2 - 1)m + p(m - l)$

where $k$ is the desuspension index, and $[l, m]$ is the range of degrees. The spectral sequence carries Dyer–Lashof and Browder operations, and the differentials encode obstructions to realizability; failure of these conditions implies nonrealizability at the given prime (Büscher et al., 2011).

• In chromatic homotopy theory, local realizability is central to the paper of the exotic $K(n)$-local Picard group at a prime (Beaudry et al., 2022); at $n = 2$, $p = 2$, the subgroup $\kappa_2$ measures the failure of all algebraically defined invertible objects (rank-one Morava modules) to be realized by actual spectra, highlighting intricate phenomena unique to small primes.
• In resolutions at primes in stable homotopy theory, e.g., at $p = 2$, towers of fixed-point spectra allow for explicit homotopical resolutions of the $K(2)$-local sphere, with local behavior governed by the algebraic and group-theoretic structure at the prime (Bobkova et al., 2016, Beaudry, 2017).

4. Local Realizability in Algebraic Geometry and Commutative Algebra

• In the theory of ideals, "localization at a prime" refers to focusing on behaviors or structures near a fixed prime ideal $P$. Algorithms such as the Local Primary Algorithm exploit double ideal quotient techniques to compute the $P$-primary component of an ideal $I$, emphasizing the determination of "local" properties (such as multiplicity, dimension, singularities) at a prime (Ishihara et al., 2020).
• In tropical geometry, the local independence data (the algebraic matroid structure) associated with a prime ideal $P$ in the coordinate ring is preserved under tropicalization. A tropical variety can be locally realized at a prime only if its independence complex coincides with that of some algebraic matroid (i.e., arises from a prime ideal) (Yu, 2015). Failure of this condition (as in the Bergman fans for non-algebraic matroids) signals genuine local non-realizability.
• In local ring theory, the gluing of minimal prime ideals can be carried out inside a local ring $B$ by constructing a subring $S$ with prescribed identification of minimal primes. The resulting local ring $S$ realizes a modified spectrum with the same formal local structure as $B$, offering a controlled form of local realizability of prime structures (Colbert et al., 2021).

5. Local Realizability in Number Theory and the Inverse Galois Problem

• In the inverse Galois problem with inertia conditions, a triple $(G, I, p)$ (finite group, subgroup, prime) is said to be $\mathbb{Q}$-realizable if $G$ arises as the Galois group of a number field $L/\mathbb{Q}$ and $I$ is the inertia subgroup at $p$ in the corresponding decomposition group. Neukirch's theorem and local embedding problem techniques show, for odd order $G$, that a global solution exists if and only if such inertia behavior is realizable locally over $\mathbb{Q}_p$ (Liu, 2017). The possibility of local realizability without global realizability (Grunwald–Wang-type counterexamples) underscores the subtlety of local-global reconciliation in this context.

6. Local Realizability in Analytic Number Theory and Diophantine Equations

• The notion arises in circle method treatments of Diophantine equations in the primes: for a given system of polynomial equations $F(x) = 0$, prime solutions are said to exist in abundance (with an asymptotic formula) if, for every prime $p$, the system admits a nonsingular $p$-adic solution; this is the "local realizability at a prime." Local solubility at all primes and at infinity ensures the existence of a positive main term in the prime solution count, thus lifting local realizability to a global density result (Cook et al., 2013, Yamagishi, 2017, Liu et al., 10 May 2024). Typically, if the local conditions (i.e., at each prime $p$ and at $\mathbb{R}$) are met, the set of prime solutions is Zariski dense in the variety defined by $F$ (Liu et al., 10 May 2024).

7. Other Contexts: Fusion Systems and Quantum Information

• In the theory of fusion systems, realizability at a prime is reflected in the property that a fusion system over a discrete $p$-toral group can be realized as the fusion system of an actual (possibly infinite) group, built from compatible colimits of finite groups. The finer notion of sequential realizability captures whether the fusion system can be approached via an ascending chain of realizable subsystems at $p$ (Broto et al., 15 Sep 2024).
• In quantum information theory, local realizability at a prime dimension may refer to the distinguishability of generalized Bell states via LOCC protocols. A simple, prime-field-based criterion (F-equivalence) determines when a set of $d$ (with $d$ prime) generalized Bell states is locally realizable as distinguishable states (Hashimoto et al., 2021).

Summary Table: Local Realizability at a Prime in Key Contexts

Area	What is Locally Realized at $p$?	Realizability Condition Example
Arithmetic Dynamics	Periodic point count sequence $p$-part	Existence of group endomorphism on $p$-group producing sequence $(p^{\nu_p(a_n)})$ (Jaidee et al., 14 Aug 2025)
Homotopy Theory	Unstable modules over $A_p$, $K(n)$-local spectra	Numerical constraints from spectral sequence and cohomology operations (Büscher et al., 2011, Beaudry et al., 2022)
Commutative Algebra	$P$-primary component of ideal; minimal primes	Double ideal quotient $(I : (I : P))$ criteria (Ishihara et al., 2020); gluing partitions (Colbert et al., 2021)
Number Theory	Galois group/inertia at $p$	Embedding problems over $\mathbb{Q}_p$ (Liu, 2017)
Analytic Number Theory	Existence of $p$-adic nonsingular solution	Local solubility and positive singular series in circle method (Cook et al., 2013, Liu et al., 10 May 2024)
Knot Floer Homology	Liftability of local chain complex structure	Algorithmic check of differential obstructions (Popović, 2023)
Fusion Systems	Realizability by (possibly infinite) group at $p$	Sequential/LT-realizability criteria (Broto et al., 15 Sep 2024)

Conclusion

Local realizability at a prime is a pervasive and unifying concept linking local structure at a prime (in the sense of arithmetic, algebraic, homotopical, or dynamical decomposition) to global realizability phenomena. It functions both as a diagnostic for local obstructions to global realization and as a mechanism by which local-to-global principles are formulated and investigated. Across disciplines, it serves to formalize when and how global algebraic, topological, or dynamical problems decompose into or are controlled by their prime-wise components, and to explain the depth of congruence phenomena, exceptional behavior at certain primes, and the precise mechanisms by which local properties influence or dictate global realizability.