Discriminant Quadratic Module

• Discriminant quadratic modules are finite quadratic structures derived from the dual quotient of an integral lattice associated with binary quadratic forms.
• They feature explicit quadratic and bilinear forms that enable unique orthogonal decompositions and link discriminants to class groups via Gauss composition.
• Research covers lattice realizations, arithmetic applications, and cryptographic methods that exploit module properties to bridge forms of different discriminants.

A discriminant quadratic module is the finite quadratic object attached to an integral quadratic form or lattice through its dual quotient. For a binary quadratic form

$q(x,y)=a x^2+b x y+c y^2,$

with discriminant

$\operatorname{disc}(q)=D:=b^2-4ac,$

one associates the lattice $L_q=\mathbb Z^2$ equipped with the bilinear form determined by $a,b,c$, its dual lattice $L_q^*$, and the finite abelian group

$$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$

This group carries a quadratic form and an associated bilinear form, yielding what the literature also calls a finite quadratic module. In the abstract setting, a finite quadratic module is a finite abelian group $M$ together with a nondegenerate quadratic form $Q:M\to \mathbb Q/\mathbb Z$. These constructions organize the passage between forms of discriminant $D$ and $Df^2$, the class groups of quadratic orders, and explicit realizations by even lattices (Arnault, 2014, Zhu, 2021).

1. Binary quadratic forms, discriminants, and the module $\operatorname{disc}(q)=D:=b^2-4ac,$0

A binary quadratic form is a homogeneous polynomial

$\operatorname{disc}(q)=D:=b^2-4ac,$1

with $\operatorname{disc}(q)=D:=b^2-4ac,$2. Its discriminant is

$\operatorname{disc}(q)=D:=b^2-4ac,$3

and the standing assumptions in the arithmetic setting are that $\operatorname{disc}(q)=D:=b^2-4ac,$4 is non-square and congruent to $\operatorname{disc}(q)=D:=b^2-4ac,$5 or $\operatorname{disc}(q)=D:=b^2-4ac,$6. The form is primitive if $\operatorname{disc}(q)=D:=b^2-4ac,$7 (Arnault, 2014).

From such a form one defines the integral lattice $\operatorname{disc}(q)=D:=b^2-4ac,$8 with bilinear form

$\operatorname{disc}(q)=D:=b^2-4ac,$9

Its dual lattice is

$L_q=\mathbb Z^2$0

and the discriminant module is the finite abelian group

$L_q=\mathbb Z^2$1

The module carries a quadratic form

$L_q=\mathbb Z^2$2

and an associated bilinear form

$L_q=\mathbb Z^2$3

This realizes the discriminant as a finite quadratic datum rather than only an integer invariant. When $L_q=\mathbb Z^2$4 varies through proper equivalence classes with fixed discriminant $L_q=\mathbb Z^2$5, the resulting arithmetic structure is compatible with Gauss composition; in the same framework, proper equivalence classes of primitive forms of discriminant $L_q=\mathbb Z^2$6 correspond to the ideal class group $L_q=\mathbb Z^2$7 of the quadratic order $L_q=\mathbb Z^2$8 (Arnault, 2014).

2. Finite quadratic modules in the abstract sense

Let $L_q=\mathbb Z^2$9 be a finite abelian group written additively. A quadratic form on $a,b,c$0 is a map

$a,b,c$1

satisfying two conditions:

1. $a,b,c$2 in $a,b,c$3 for all integers $a,b,c$4 and all $a,b,c$5.
2. The associated map

$a,b,c$6

is $a,b,c$7-bilinear and symmetric.

The pair $a,b,c$8 is a finite quadratic module if, in addition, $a,b,c$9 is nondegenerate in the strong sense that the map

$L_q^*$0

is an isomorphism of finite $L_q^*$1-modules (Zhu, 2021).

Several basic consequences are stated explicitly. Nondegeneracy implies in particular that $L_q^*$2 and that $L_q^*$3. The literature also uses the bilinear notation $L_q^*$4, but the quadratic datum $L_q^*$5 is finer and is forced by even lattices. This is relevant when comparing the lattice-theoretic convention $L_q^*$6 with the abstract convention $L_q^*$7: both encode a finite quadratic structure, but the precise target depends on the normalization adopted in the source (Zhu, 2021).

3. Orthogonal decomposition and indecomposable building blocks

The main structure theorem states that every finite quadratic module $L_q^*$8 decomposes uniquely, up to isometry, as an orthogonal sum of indecomposable $L_q^*$9-primary modules. Moreover, each $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$0-primary summand decomposes uniquely into orthogonal indecomposables of standard types (Zhu, 2021).

For odd primes $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$1, the indecomposable modules are cyclic:

$$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$2

where $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$3 and $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$4. These have order $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$5, and varying $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$6 modulo squares in $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$7 gives two isometry classes for each $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$8.

For $$A_q=L_q^*/L_q,\qquad |A_q|=|\operatorname{disc}(q)|.$$9, four families are required. The cyclic case is

$M$0

with $M$1 odd. In addition there are two non-cyclic indecomposable families:

$M$2

and

$M$3

The decomposition proof proceeds through primary decomposition, splitting off cyclic summands for $M$4, and a more delicate cyclic-or-binary splitting for $M$5 via explicit congruence arguments.

A notable nuance is that the exposition also states that uniqueness of decomposition fails in general, citing Wall’s relations and giving $M$6 with $M$7 as an example of two non-equivalent splittings into copies of $M$8. This indicates that canonical orthogonal decompositions and arbitrary splittings should not be conflated (Zhu, 2021).

4. Embedded discriminants, semi-equivalence, and the map $M$9

Fix an integer $Q:M\to \mathbb Q/\mathbb Z$0. Given a primitive form

$Q:M\to \mathbb Q/\mathbb Z$1

of discriminant $Q:M\to \mathbb Q/\mathbb Z$2 with $Q:M\to \mathbb Q/\mathbb Z$3, define

$Q:M\to \mathbb Q/\mathbb Z$4

In matrix notation, with

$Q:M\to \mathbb Q/\mathbb Z$5

this is written $Q:M\to \mathbb Q/\mathbb Z$6, and

$Q:M\to \mathbb Q/\mathbb Z$7

More generally, if $Q:M\to \mathbb Q/\mathbb Z$8, then

$Q:M\to \mathbb Q/\mathbb Z$9

Thus the discriminant changes by a square under integral change of variables with determinant different from $D$0 (Arnault, 2014).

For $D$1, define

$D$2

and

$D$3

If $D$4, then

$D$5

with

$D$6

Exactly

$D$7

distinct primitive lifts occur, where $D$8 is the Kronecker symbol.

The paper introduces semi-equivalence for primitive forms $D$9 of discriminant $Df^2$0: they are semi-equivalent if there exist primitive forms $Df^2$1 of discriminant $Df^2$2 with $Df^2$3 under $Df^2$4-equivalence, integers $Df^2$5, and matrices $Df^2$6 such that

$Df^2$7

A central result is Theorem 2.11 of Arnault: if $Df^2$8 is a discriminant and $Df^2$9 an odd prime, then the map

$\operatorname{disc}(q)=D:=b^2-4ac,$00

sending the class of $\operatorname{disc}(q)=D:=b^2-4ac,$01 to the class of any $\operatorname{disc}(q)=D:=b^2-4ac,$02 with $\operatorname{disc}(q)=D:=b^2-4ac,$03, is well-defined and surjective. Under the standard bijection between proper equivalence classes of primitive forms and ideal classes, this recovers the natural surjection

$\operatorname{disc}(q)=D:=b^2-4ac,$04

At the discriminant-module level, the same lift produces a chain of lattices

$\operatorname{disc}(q)=D:=b^2-4ac,$05

and an exact sequence

$\operatorname{disc}(q)=D:=b^2-4ac,$06

compatible with the quadratic forms. This exhibits $\operatorname{disc}(q)=D:=b^2-4ac,$07 as a quotient of $\operatorname{disc}(q)=D:=b^2-4ac,$08 (Arnault, 2014).

5. Realization by even lattices of minimal rank

For any indecomposable finite quadratic module $\operatorname{disc}(q)=D:=b^2-4ac,$09, one can construct an even lattice $\operatorname{disc}(q)=D:=b^2-4ac,$10, and also a positive-definite lattice, of the least rank whose discriminant module is $\operatorname{disc}(q)=D:=b^2-4ac,$11. The constructions are explicit: the lattices are given by Gram matrices over $\operatorname{disc}(q)=D:=b^2-4ac,$12 (Zhu, 2021).

For the odd-primary cyclic modules $\operatorname{disc}(q)=D:=b^2-4ac,$13, the minimal rank depends on $\operatorname{disc}(q)=D:=b^2-4ac,$14, on the parity of $\operatorname{disc}(q)=D:=b^2-4ac,$15, and on the congruence class of $\operatorname{disc}(q)=D:=b^2-4ac,$16. If $\operatorname{disc}(q)=D:=b^2-4ac,$17, rank $\operatorname{disc}(q)=D:=b^2-4ac,$18 suffices. If $\operatorname{disc}(q)=D:=b^2-4ac,$19 and $\operatorname{disc}(q)=D:=b^2-4ac,$20 is even, rank $\operatorname{disc}(q)=D:=b^2-4ac,$21 still suffices. If $\operatorname{disc}(q)=D:=b^2-4ac,$22, $\operatorname{disc}(q)=D:=b^2-4ac,$23 is odd, and $\operatorname{disc}(q)=D:=b^2-4ac,$24, rank $\operatorname{disc}(q)=D:=b^2-4ac,$25 again suffices. If $\operatorname{disc}(q)=D:=b^2-4ac,$26, $\operatorname{disc}(q)=D:=b^2-4ac,$27 is odd, and $\operatorname{disc}(q)=D:=b^2-4ac,$28, no rank $\operatorname{disc}(q)=D:=b^2-4ac,$29 lattice can realize the module, and rank $\operatorname{disc}(q)=D:=b^2-4ac,$30 is used.

For the $\operatorname{disc}(q)=D:=b^2-4ac,$31-primary cyclic modules $\operatorname{disc}(q)=D:=b^2-4ac,$32, the rank depends on $\operatorname{disc}(q)=D:=b^2-4ac,$33. If $\operatorname{disc}(q)=D:=b^2-4ac,$34, rank $\operatorname{disc}(q)=D:=b^2-4ac,$35 suffices, with Gram matrix $\operatorname{disc}(q)=D:=b^2-4ac,$36. If $\operatorname{disc}(q)=D:=b^2-4ac,$37 and $\operatorname{disc}(q)=D:=b^2-4ac,$38 is odd, a rank $\operatorname{disc}(q)=D:=b^2-4ac,$39 model is constructed. If $\operatorname{disc}(q)=D:=b^2-4ac,$40 and $\operatorname{disc}(q)=D:=b^2-4ac,$41 is even, a rank $\operatorname{disc}(q)=D:=b^2-4ac,$42 model is constructed. For the non-cyclic modules $\operatorname{disc}(q)=D:=b^2-4ac,$43 and $\operatorname{disc}(q)=D:=b^2-4ac,$44, rank $\operatorname{disc}(q)=D:=b^2-4ac,$45 lattices are exhibited explicitly.

To pass to positive-definite models, the construction uses Milgram’s formula

$\operatorname{disc}(q)=D:=b^2-4ac,$46

From the resulting signature congruence, the minimal positive dimension is determined, and explicit positive-definite Gram matrices are then written down for the indecomposables. The stated ranks include $\operatorname{disc}(q)=D:=b^2-4ac,$47-dimensional models for certain $\operatorname{disc}(q)=D:=b^2-4ac,$48, rank $\operatorname{disc}(q)=D:=b^2-4ac,$49 or $\operatorname{disc}(q)=D:=b^2-4ac,$50 in odd-$\operatorname{disc}(q)=D:=b^2-4ac,$51 cases, and rank $\operatorname{disc}(q)=D:=b^2-4ac,$52, or $\operatorname{disc}(q)=D:=b^2-4ac,$53 for $\operatorname{disc}(q)=D:=b^2-4ac,$54, $\operatorname{disc}(q)=D:=b^2-4ac,$55, and $\operatorname{disc}(q)=D:=b^2-4ac,$56.

Small examples illustrate the general theory. The module $\operatorname{disc}(q)=D:=b^2-4ac,$57 is realized by the rank $\operatorname{disc}(q)=D:=b^2-4ac,$58 lattice with Gram matrix $\operatorname{disc}(q)=D:=b^2-4ac,$59. The module $\operatorname{disc}(q)=D:=b^2-4ac,$60, with $\operatorname{disc}(q)=D:=b^2-4ac,$61, is realized by the positive-definite rank $\operatorname{disc}(q)=D:=b^2-4ac,$62 lattice

$\operatorname{disc}(q)=D:=b^2-4ac,$63

The binary $\operatorname{disc}(q)=D:=b^2-4ac,$64-groups $\operatorname{disc}(q)=D:=b^2-4ac,$65 and $\operatorname{disc}(q)=D:=b^2-4ac,$66 are likewise realized by explicit root-lattice-type constructions (Zhu, 2021).

6. Arithmetic applications, worked example, and terminological distinction

A concrete embedded-discriminant example takes

$\operatorname{disc}(q)=D:=b^2-4ac,$67

for which

$\operatorname{disc}(q)=D:=b^2-4ac,$68

With $\operatorname{disc}(q)=D:=b^2-4ac,$69,

$\operatorname{disc}(q)=D:=b^2-4ac,$70

and

$\operatorname{disc}(q)=D:=b^2-4ac,$71

Since $\operatorname{disc}(q)=D:=b^2-4ac,$72, the lifted form $\operatorname{disc}(q)=D:=b^2-4ac,$73 is primitive, and under Gauss composition the class of $\operatorname{disc}(q)=D:=b^2-4ac,$74 maps to that of $\operatorname{disc}(q)=D:=b^2-4ac,$75 (Arnault, 2014).

The same framework is used in cryptography and computational number theory. Systems such as NICE use forms of discriminant $\operatorname{disc}(q)=D:=b^2-4ac,$76 and $\operatorname{disc}(q)=D:=b^2-4ac,$77. The lift from $\operatorname{disc}(q)=D:=b^2-4ac,$78 to $\operatorname{disc}(q)=D:=b^2-4ac,$79 with $\operatorname{disc}(q)=D:=b^2-4ac,$80 allows key-generation and encryption in the larger group $\operatorname{disc}(q)=D:=b^2-4ac,$81, while decryption recovers classes in $\operatorname{disc}(q)=D:=b^2-4ac,$82. The cited attacks exploit semi-equivalence classes and the explicit description of the kernel of

$\operatorname{disc}(q)=D:=b^2-4ac,$83

Several factorization algorithms, including those of Shanks, Schnorr–Lenstra, and Buchmann–Jakobi, are described as moving between forms of smaller and larger discriminant through the $\operatorname{disc}(q)=D:=b^2-4ac,$84-lifts and tracking the kernel size $\operatorname{disc}(q)=D:=b^2-4ac,$85. In lattice theory, discriminant modules encode the glue between primitive sublattices in the classification of even integral lattices, and the explicit constructions are used to produce concrete examples of Jacobi forms of lattice index via their Weil representations (Zhu, 2021).

A recurring terminological confusion is with Quadratic Discriminant Analysis. In statistical learning, Quadratic Discriminant Analysis assumes class-conditional Gaussian distributions and uses discriminant functions of the form

$\operatorname{disc}(q)=D:=b^2-4ac,$86

with quadratic decision boundaries when the class covariances differ (Ghojogh et al., 2019). This terminology should be distinguished from the arithmetic and lattice-theoretic discriminant quadratic module: the former concerns probabilistic classification, whereas the latter concerns finite quadratic structures attached to lattices, quadratic forms, and discriminants.