SL₂(Z)-Equivalence Classes Overview

• SL₂(Z)-equivalence classes are defined by the group’s action on quadratic forms, lattices, and noncommutative tori, encapsulating deep arithmetic invariants.
• They establish bijections with ideal class groups and partition forms into genera, thereby uniting classical number theory with modern geometric insights.
• Explicit computational algorithms and geometric interpretations facilitate classification, with applications spanning Lefschetz fibrations, C*-algebras, and modular orbifold studies.

The classification of objects up to $\mathrm{SL}_2(\mathbb{Z})$-equivalence is central in the study of quadratic forms, noncommutative tori, monodromy representations, and their connections to algebraic and geometric structures. The group $\mathrm{SL}_2(\mathbb{Z})$ acts in a highly nontrivial manner on spaces such as binary quadratic forms, lattices, and noncommutative $C^\ast$-algebras, organizing them into equivalence classes that encode deep arithmetic, topological, and analytic invariants.

1. $\mathrm{SL}_2(\mathbb{Z})$-Action and Equivalence on Quadratic Forms

Let $f(x, y) = a x^2 + b x y + c y^2$ be a primitive binary quadratic form over $\mathbb{Z}$, with discriminant $\Delta = b^2 - 4ac$. $\mathrm{SL}_2(\mathbb{Z})$ acts on forms by linear change of variables: for $$\gamma = \begin{pmatrix}p & q \ r & s\end{pmatrix}$$, define $f^\gamma(x, y) = f(px + qy, rx + sy)$. Two forms are properly equivalent if their orbits coincide under this action (Bitan, 2019).

The quotient set of primitive forms of given discriminant by this group action is finite for each discriminant, and the resulting equivalence classes have a rich arithmetic structure encoded by the class group of the associated quadratic order.

2. Gauss–Dedekind Correspondence and the Class Group

A foundational result of Gauss, refined by Dedekind, is the bijection:

$${\{\text{SL}_2(\mathbb{Z})\text{-classes of primitive positive-definite quadratic forms of disc } \Delta < 0\}} \longleftrightarrow \operatorname{Cl}(\mathcal{O}_K),$$

where $K = \mathbb{Q}(\sqrt{\Delta})$ and $\mathcal{O}_K$ is its ring of integers. The association sends a form $[a, b, c]$ to the ideal $(a, (b+\sqrt{\Delta})/2)$ and inverts via the norm form induced by a basis of a fractional ideal. The group structure on $\operatorname{Cl}(\Delta)$ arises from Gauss's composition law, which can be described via congruences or, geometrically, by explicit parametrizations using Pell equations (Bitan, 2019, Simon, 2022).

3. Genus Theory and Fine Partition into Genera

Beyond the class group, quadratic forms of discriminant $\Delta$ are partitioned into genera, determined by their equivalence over local fields $\mathbb{Z}_p$ and $\mathbb{R}$. Two forms are in the same genus if they become locally equivalent everywhere. The genera correspond to cosets of the squares in the class group, yielding the exact sequence

$$1 \to \operatorname{Cl}(\Delta)^2 \to \operatorname{Cl}(\Delta) \to \prod_{p \mid \Delta} \{\pm1\} \to 1,$$

where the map is given by evaluating local Hilbert symbols $(\Delta, a_Q)_p$ on the first coefficients of reduced forms (Simon, 2022, Bitan, 2019). Gauss's principal genus theorem states that the square of any class falls into the principal genus.

4. Geometric and Representation-Theoretic Extensions

The theory of $\mathrm{SL}_2(\mathbb{Z})$-equivalence extends into geometric contexts:

• Noncommutative Tori: The crossed product $A_\theta \rtimes_A \mathbb{Z}$, for $A \in \mathrm{SL}_2(\mathbb{Z})$, encodes the action of a cyclic subgroup on the irrational rotation algebra. Bönicke–Chakraborty–He–Liao provide a full classification: two such crossed products are $*$-isomorphic if and only if $\theta \equiv \pm\theta'$ modulo $\mathbb{Z}$ and $I - A^{-1}$ is matrix-equivalent (via Smith normal form) to $I - B^{-1}$. Morita equivalence is governed by the $GL_2(\mathbb{Z})$-orbits of $\theta$ and the equivalence of these matrices. The K-theory invariants, traces, and bimodule constructions (e.g., Rieffel's Heisenberg module) are central to the proof (Bönicke et al., 2017).
• Lefschetz Fibrations and Monodromy Factorizations: The classification of Lefschetz elliptic fibrations over the disk is controlled by the conjugacy class in $\mathrm{SL}_2(\mathbb{Z})$ of the total monodromy. The Hurwitz equivalence problem for factorizations in the modular group traces the topological equivalence between fibrations and is algorithmically addressed via reduced words in the free product structure of $\mathrm{PSL}_2(\mathbb{Z})$ (Vélez et al., 2013).
• Lie Algebra Orbits: For a field $K$ of characteristic $\neq 2$, the adjoint orbits of $\mathrm{PSL}_2(K)$ on $\mathfrak{sl}_2(K)$ correspond bijectively to equivalence classes of nondegenerate quadratic forms, as parametrized geometrically by cross-ratio invariants. The Gauss composition is realized via Pell-type norm equations. Partitioning into genus is reconstructed via Hilbert symbols and explicit local invariants (Simon, 2022).

5. Modern Generalizations: Positive Characteristic and Étale Cohomology

In the geometric Gauss–Dedekind setting, the correspondence is extended to Dedekind domains over global function fields of positive characteristic. Here, the set of $\mathrm{SL}_2(\mathcal{O})$-equivalence classes of primitive forms with given discriminant $-a(\mathcal{O}^\times)^2$ is isomorphic as an abelian group to the Picard group $\operatorname{Pic}(\mathcal{O}_K)$, with the classification carried out through étale cohomology. Genera are orbits classified by the adelic norm group, again mirroring the classical genus theory but situated in the arithmetic geometry of global function fields. The principal genus remains defined via the fibre over the trivial class in the Nisnevich exact sequence (Bitan, 2019).

6. Explicit Constructions and Computational Algorithms

The effective computation and characterization of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes rely on explicit presentations:

• For quadratic forms, reduction algorithms and neighborhood searches are guided by the action on the hyperbolic plane or the structure of reduced forms.
• For monodromy factorizations, combinatorial algorithms generate Hurwitz-complete sets in the modular group via recursive enumeration in free product coordinates and controlled insertions of trivial Dehn-twist pairs (Vélez et al., 2013).

A summary table of key correspondences is given below:

Structure/Invariant	$\mathrm{SL}_2(\mathbb{Z})$-Class Set	Main Classification Principle
Binary quadratic forms	Primitive forms with fixed $\Delta$ modulo $\mathrm{SL}_2(\mathbb{Z})$	Bijection with $\operatorname{Cl}(\mathcal{O}_K)$
Crossed products $A_\theta \rtimes_A \mathbb{Z}$	C*-algebra up to $*$-iso/Morita eq.	$(\theta, I - A^{-1})$, Smith normal equivalence, $GL_2(\mathbb{Z})$-orbits
Lefschetz fibration monodromies	Factorizations modulo Hurwitz + conjugacy	Equivalence up to Hurwitz moves and overall conjugation
Genera of forms	Cosets of class group squares	Local equivalence via Hilbert symbols

7. Geometric and Topological Interpretation

The action of $\mathrm{SL}_2(\mathbb{Z})$ translates algebraic equivalence problems into geometric ones:

• On the modular orbifold $\mathrm{PSL}_2(\mathbb{Z}) \backslash \mathbb{H}$, quadratic forms of negative discriminant correspond to closed geodesics. The angle or length between associated geodesics encodes equivalence at the level of modular forms and knots.
• Intersection patterns and linking numbers of modular geodesics trace the arithmetic of genus splitting, connecting group-theoretic properties with topological invariants. This includes explicit computation of intersection angles and linking in the unit tangent bundle, revealing the arithmetic data in geometric form (Simon, 2022).

---

The theory of $\mathrm{SL}_2(\mathbb{Z})$-equivalence classes thus unifies algebraic, analytic, topological, and arithmetic perspectives, serving as an archetypal domain for the interplay of discrete group actions, class field theory, noncommutative geometry, and low-dimensional topology (Bitan, 2019, Simon, 2022, Bönicke et al., 2017, Vélez et al., 2013).