Quadratic Functions in Abelian Groups

• Quadratic functions over abelian groups are maps defined by symmetric, bilinear polarization with vanishing third-order differences, establishing a degree-2 structure.
• They provide a unifying framework linking classical functional equations, categorical polynomial functors, and extended quadratic forms in arithmetic and geometry.
• Investigations highlight the critical role of 2-primary phenomena, bilinear cross-effects, and representation-theoretic enhancements such as Heisenberg actions in refining quadratic invariants.

Searching arXiv for the cited papers to ground the article in current metadata and ensure accurate citations. Quadratic functions over abelian groups are maps whose second-order behavior is bilinear and whose third-order variation vanishes. In the most classical additive setting, a map $q\colon X\to A$ on an abelian group is quadratic when its polarization $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$ is symmetric and biadditive, equivalently when $q$ satisfies identities such as $q(x+y)+q(x-y)=2q(x)+2q(y)$ and all third finite differences vanish (Feldman, 2021, Aichinger et al., 2019). This notion admits several technically distinct but closely related realizations: as degree-$\le 2$ polynomial maps detected by iterated differences, as reduced functors $F\colon \mathcal C\to Ab$ with vanishing third cross-effect, as solutions of functional equations on abelian groups and semigroups with involutions, as quadratic forms with coefficients in arithmetic groups, as quadratic $\ell$-adic sheaves enhancing finite-group quadratic forms, and as extended quadratic forms governed by quadratic form parameters over $\mathbb Z$ [(Hartl et al., 2008); (EL-Fassi, 2021); (Barańczuk, 2017); (Takeuchi, 2023); (Crowley et al., 2024)].

1. Classical definitions, polarization, and finite differences

For a map $q\colon X\to A$ between abelian groups, the classical quadratic condition is encoded by the polarization

$B(x,y)=q(x+y)-q(x)-q(y)+q(0).$

In the functional-equation framework, a quadratic function $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$0 is one for which there exists a symmetric biadditive function $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$1 such that $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$2, equivalently

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$3

for all $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$4; when $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$5, the associated polarization is

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$6

(Feldman, 2021). In this sense, “quadratic” is not defined by a coordinate expression but by second-order additivity.

A parallel characterization uses finite-difference operators. For abelian groups $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$7, the functional degree of a map $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$8 is defined through the augmentation ideal of $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$9, and degree $q$0 is equivalent to the vanishing of all $q$1-fold iterated differences. The quadratic specialization is exact: $q$2 is quadratic precisely when

$q$3

for all $q$4. In that case the second difference

$q$5

is independent of the base point $q$6, biadditive in each variable, and symmetric (Aichinger et al., 2019). The same paper records the canonical decomposition

$q$7

where $q$8 is constant, $q$9 has degree $q(x+y)+q(x-y)=2q(x)+2q(y)$0, and $q(x+y)+q(x-y)=2q(x)+2q(y)$1 is quadratic with $q(x+y)+q(x-y)=2q(x)+2q(y)$2 (Aichinger et al., 2019).

These two descriptions are compatible. The finite-difference criterion explains why polarization is bilinear: once third differences vanish, the second difference becomes translation-invariant and hence depends only on increments. A plausible implication is that the functional-equation and finite-difference approaches are best viewed as two normal forms of the same degree-$q(x+y)+q(x-y)=2q(x)+2q(y)$3 phenomenon: one emphasizes the Jensen-type identity, the other the annihilation of the third layer of difference operators.

2. Quadratic functors with values in $q(x+y)+q(x-y)=2q(x)+2q(y)$4

A categorical generalization replaces a single group by a pointed category $q(x+y)+q(x-y)=2q(x)+2q(y)$5 with finite coproducts and a small regular projective generator $q(x+y)+q(x-y)=2q(x)+2q(y)$6. In this framework, a reduced functor $q(x+y)+q(x-y)=2q(x)+2q(y)$7 is polynomial of degree $q(x+y)+q(x-y)=2q(x)+2q(y)$8 when its $q(x+y)+q(x-y)=2q(x)+2q(y)$9-st cross-effect vanishes; in particular, $\le 2$0 is linear iff $\le 2$1, and quadratic iff $\le 2$2 (Hartl et al., 2008). The second cross-effect is defined by

$\le 2$3

with a split short exact sequence

$\le 2$4

and for $\le 2$5-valued functors $\le 2$6 is additive in each variable (Hartl et al., 2008). When the target is $\le 2$7, a reduced functor admits a decomposition over finite coproducts in terms of its cross-effects; for quadratic functors only $\le 2$8 and $\le 2$9 remain nonzero (Hartl et al., 2008).

The paper identifies a precise analogue of classical polarization. For a normalized map $F\colon \mathcal C\to Ab$0 of morphism-sets, a second cross-effect $F\colon \mathcal C\to Ab$1 is defined using the reduced standard projective functor $F\colon \mathcal C\to Ab$2, and $F\colon \mathcal C\to Ab$3 is quadratic iff this cross-effect is bilinear, i.e. factors through the bilinearization $F\colon \mathcal C\to Ab$4 (Hartl et al., 2008). It also introduces the quadratization functor $F\colon \mathcal C\to Ab$5, the universal quadratic approximation left adjoint to the inclusion of degree-$F\colon \mathcal C\to Ab$6 functors, and proves that $F\colon \mathcal C\to Ab$7 (Hartl et al., 2008).

The main classification theorem states that reduced quadratic functors $F\colon \mathcal C\to Ab$8 preserving filtered colimits and suitable coequalizers are equivalent to quadratic $F\colon \mathcal C\to Ab$9-modules relative to $\ell$0. The minimal algebraic data are

$\ell$1

together with structure maps $\ell$2, an involution $\ell$3, and a map $\ell$4 subject to relations $\ell$5 and $\ell$6; conversely, a quadratic functor is reconstructed from such data by a quadratic tensor product $\ell$7 (Hartl et al., 2008). When $\ell$8 is a cogroup, the structure simplifies to maps $\ell$9 and $\mathbb Z$0, with $\mathbb Z$1, and relations $\mathbb Z$2–$\mathbb Z$3 make explicit the roles of the second Hopf invariant and the Whitehead product (Hartl et al., 2008). This yields the specializations to groups and to $\mathbb Z$4-modules: for the category of free groups, quadratic $\mathbb Z$5-modules become abelian square groups; for finitely generated free $\mathbb Z$6-modules, they become quadratic $\mathbb Z$7-modules (Hartl et al., 2008).

3. Functional equations and the emergence of a common quadratic core

Quadratic functions over abelian groups arise naturally as the rigid part of several functional equations. For the Kac–Bernstein equation

$\mathbb Z$8

on an arbitrary abelian group $\mathbb Z$9, passing to logarithms yields an inhomogeneous d’Alembert-type equation. The proof applies finite-difference operators to show that the logarithms $q\colon X\to A$0 and $q\colon X\to A$1 are degree-$q\colon X\to A$2 Djoković polynomials, up to a term constant on cosets of $q\colon X\to A$3 (Feldman, 2021). The resulting structure theorem is exact: if $q\colon X\to A$4, then there exist a real-valued quadratic function $q\colon X\to A$5, additive real functions $q\colon X\to A$6, and a real function $q\colon X\to A$7 constant on cosets of $q\colon X\to A$8 such that

$q\colon X\to A$9

for all $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$0, and conversely every such pair satisfies the Kac–Bernstein equation (Feldman, 2021). When $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$1, the cosetwise constant term reduces to a global constant (Feldman, 2021).

The same paper makes explicit that the quadratic parts of $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$2 and $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$3 coincide. This common $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$4 satisfies the quadratic identity

$B(x,y)=q(x+y)-q(x)-q(y)+q(0).$5

and on groups such as $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$6 or $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$7 it can be written as $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$8 for a real symmetric matrix $B(x,y)=q(x+y)-q(x)-q(y)+q(0).$9 (Feldman, 2021). In the Hermitian complex-valued case, unitary characters and sign functions constant on $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$00-cosets appear, but the same real quadratic core persists (Feldman, 2021).

A related semigroup-theoretic result treats the quadratic-type equation

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$01

on a commutative semigroup $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$02 with involutions $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$03, with values in an abelian group uniquely divisible by $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$04. The general solution has the form

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$05

where $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$06 is additive and invariant under $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$07, and $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$08 is symmetric and biadditive and anti-invariant in its first argument; moreover,

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$09

is obtained by polarization (EL-Fassi, 2021). For an abelian group with negation involution, the equation becomes

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$10

a parallelogram-type identity recovering the standard quadratic/bilinear split (EL-Fassi, 2021).

These results show that, in several nontrivial functional equations, the genuinely nonlinear part is not arbitrary: it is forced into a quadratic form whose polarization controls the entire second-order behavior. This suggests that quadraticity often appears as the maximal nonlinearity compatible with strong additive symmetries.

4. Explicit models on abelian groups and finite-group arithmetic

On finitely generated abelian groups, quadratic maps admit concrete representatives. For $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$11, a quadratic map is determined up to affine-linear terms by a symmetric biadditive form $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$12, and one convenient representative uses binomial coefficients: $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$13 where $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$14 (Aichinger et al., 2019). This avoids division by $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$15 and is therefore valid even when the codomain has $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$16-torsion (Aichinger et al., 2019). The cyclic example

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$17

has $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$18, while in two variables

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$19

yields

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$20

(Aichinger et al., 2019).

The functional degree formalism is also used to generalize Chevalley–Warning. If $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$21 and $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$22 are finite abelian $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$23-groups and $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$24 are functions, then under the inequality

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$25

the common zero set is not a singleton, and its cardinality is divisible by $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$26; when each $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$27 is quadratic, $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$28, giving the stated quadratic corollaries (Aichinger et al., 2019). In the square domain–codomain case $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$29, if $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$30, then the number of common zeros is divisible by $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$31 and the zero set is not a singleton (Aichinger et al., 2019).

A different arithmetic manifestation appears in diagonal quadratic forms with coefficients in an abelian group $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$32 equipped with reduction maps $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$33. The form

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$34

is studied under assumption $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$35 controlling $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$36-adic orders of reductions of independent points (Barańczuk, 2017). The main local–global theorem states that for binary and ternary diagonal forms, local representability modulo almost all $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$37 implies global representability in $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$38, up to torsion (Barańczuk, 2017). The result holds for Mordell–Weil groups of abelian varieties with $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$39, odd algebraic $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$40-theory groups $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$41, and finitely generated subgroups of $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$42 (Barańczuk, 2017). For rank $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$43, however, the Hasse–Minkowski principle fails: the explicit form

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$44

represents torsion modulo every $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$45 but has no nontrivial global torsion representation (Barańczuk, 2017).

5. Harmonic analysis, sheaf theory, and representation-theoretic enhancements

Quadratic structures on finite abelian groups admit analytic and geometric lifts. For a finite abelian group $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$46 of odd order $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$47, the convolution-square equation

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$48

defines the notion of a critical value $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$49: a complex number for which the equation has a nonzero solution (Benoist, 2022). The paper interprets the equation as quadratic both because it contains the pointwise square $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$50 and because characters satisfy $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$51 (Benoist, 2022). Its principal construction uses theta functions on abelian varieties with complex multiplication. Theorem 4.3 gives a critical value

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$52

attached to a principally polarized abelian variety and a unitary endomorphism $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$53, and Theorem 5.5 rewrites it in CM-field terms as

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$54

(Benoist, 2022). The same paper proves structural constraints on critical values: they are algebraic integers, all Galois conjugates are critical, $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$55 with equality only for $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$56, and only finitely many critical values exist on $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$57 (Benoist, 2022).

The function-sheaf analogue is developed for connected unipotent commutative groups over finite fields. A quadratic $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$58-adic sheaf $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$59 on a perfect unipotent commutative group $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$60 is an invertible étale sheaf such that

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$61

is bimultiplicative (Takeuchi, 2023). If the associated morphism $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$62 is finite étale, $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$63 is called isogeneous; if $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$64, it is non-degenerate (Takeuchi, 2023). Over $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$65, symmetric bimultiplicative sheaves are classified by additive endomorphisms $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$66 through sheaves of the form $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$67 (Takeuchi, 2023).

The associated cohomology realizes a finite Heisenberg representation. If $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$68 is isogeneous and $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$69, then $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$70 carries an action of a finite Heisenberg group $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$71, and Theorem 5.5 states that this cohomology is the unique irreducible $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$72-representation with the prescribed central character, of dimension $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$73, where $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$74 (Takeuchi, 2023). Cohomology is concentrated in degree $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$75, and Frobenius eigenvalues are of the form $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$76 with $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$77 a root of unity (Takeuchi, 2023). In the non-degenerate case, the trace function $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$78 on $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$79 is a non-degenerate quadratic form in the sense of finite abelian groups, and the Gauss sum

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$80

equals $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$81, where $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$82 is the Frobenius eigenvalue on middle cohomology (Takeuchi, 2023). A Hasse–Davenport-type relation over $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$83 is also proved (Takeuchi, 2023).

This sheaf-theoretic enhancement places the classical polarization identity inside a geometric package: the bicharacter becomes the bimultiplicative sheaf $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$84, non-degeneracy becomes the isogeny $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$85, and Gauss sums become Frobenius traces. A plausible implication is that the classical theory of quadratic forms on finite abelian groups is the trace-level shadow of a richer representation-theoretic mechanism.

6. Extended quadratic forms, Witt groups, and algebraic classification over $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$86

A further generalization replaces scalar-valued quadratic maps by quadratic refinements controlled by a form parameter $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$87 over $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$88, where

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$89

(Crowley et al., 2024). A $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$90-form on a finitely generated abelian group $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$91 is a triple $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$92 with $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$93 an $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$94-symmetric bilinear form and $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$95 satisfying

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$96

(Crowley et al., 2024). Here the “polar” of $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$97 is

$B(x,y)=q(x+y)-q(x)-q(y)+q(0)$98

so the classical pattern survives, but the target is no longer merely an abelian group: it is the coefficient object of an extended quadratic theory (Crowley et al., 2024).

The paper classifies all quadratic form parameters over $B(x,y)=q(x+y)-q(x)-q(y)+q(0)$99. The category $q$00 of symmetric parameters is equivalent to the slice category $q$01, while the anti-symmetric category $q$02 is equivalent to the coslice category $q$03 (Crowley et al., 2024). Indecomposable parameters include the symmetric $q$04, $q$05, $q$06, $q$07, and the anti-symmetric $q$08, $q$09, $q$10; every parameter admits a maximal splitting into one of these indecomposables plus an abelian-group summand (Crowley et al., 2024).

For nonsingular $q$11-forms, the corresponding Witt group $q$12 is computed explicitly. The principal examples are: $q$13 in the symmetric case, and

$q$14

in the anti-symmetric case (Crowley et al., 2024). The invariants are the signature $q$15, the Arf invariant $q$16, and the refined $q$17-primary invariants $q$18 built from characteristic elements (Crowley et al., 2024). For split parameters $q$19, the Witt group decomposes as

$q$20

where the quadratic tensor product is governed by exact sequences involving the symmetric square in the symmetric case and the anti-symmetric square quotient in the anti-symmetric case (Crowley et al., 2024).

This framework connects directly to Wall’s extended quadratic forms in manifold theory: the parameter $q$21 controls $q$22-connected $q$23-manifolds with connected boundary, and the resulting Witt groups govern boundary homotopy spheres and surgery-theoretic classifications (Crowley et al., 2024). The broader lesson is that quadraticity over abelian groups is not confined to scalar functions; it also underlies the structure of refinements of bilinear forms, their metabolic theory, and their Witt-theoretic invariants.

7. Scope, specializations, and recurring structural themes

Across these frameworks, several structural themes recur. First, the decisive invariant is always a second-order object: a symmetric biadditive form, a second cross-effect, a bimultiplicative sheaf, or a bilinear form $q$24 together with a quadratic refinement $q$25 [(Hartl et al., 2008); (Takeuchi, 2023); (Crowley et al., 2024)]. Second, the disappearance of third-order information is the operative degree-$q$26 condition, whether expressed as $q$27, $q$28, or the solvability of a quadratic-type functional equation [(Hartl et al., 2008); (Aichinger et al., 2019); (EL-Fassi, 2021)]. Third, $q$29-primary phenomena are exceptional and structurally important: they govern cosetwise constants on $q$30, the sharp $q$31-behavior of sign functions, the need for binomial-coefficient representatives when division by $q$32 is unavailable, and the torsion structure of Witt groups (Feldman, 2021, Aichinger et al., 2019, Crowley et al., 2024).

There are also important limitations and non-equivalences. Diagonalization is not generally available for quadratic forms with coefficients in an arbitrary abelian group equipped with reduction maps, and Hasse–Minkowski holds only in ranks $q$33 and $q$34 in the group-valued setting studied in (Barańczuk, 2017). In the categorical theory, equivalence with quadratic modules requires preservation of filtered colimits and suitable coequalizers, or reflexive coequalizers in the Mal’cev and Barr-exact case (Hartl et al., 2008). In the convolution-square theory, the odd-order hypothesis is intrinsic because the doubling map must be invertible (Benoist, 2022). In the sheaf-theoretic theory, isogeneity of $q$35 is the condition ensuring cohomological concentration and the Heisenberg action (Takeuchi, 2023).

Taken together, these results describe a coherent mathematical landscape. Quadratic functions over abelian groups are characterized by bilinear polarization, vanishings of third-order difference or cross-effect operators, and reconstruction from small algebraic data. Their manifestations range from explicit formulas on finite $q$36-groups and rigidity phenomena in functional equations to Heisenberg representations, Gauss sums, local–global principles, and Witt groups of extended quadratic forms (Aichinger et al., 2019, Feldman, 2021, Barańczuk, 2017, Takeuchi, 2023, Crowley et al., 2024). This suggests that “quadratic over an abelian group” is less a single definition than a stable degree-$q$37 paradigm whose algebraic core persists across category theory, arithmetic, harmonic analysis, and geometry.