In mathematics, the modular group is the projective special linear group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2 {\displaystyle 2\times 2} matrices with integer coefficients and determinant 1 {\displaystyle 1} , such that the matrices A {\displaystyle A} and − A {\displaystyle -A} are identified. The modular group acts on the upper-half of the complex plane by linear fractional transformations. The name "modular group" comes from the relation to moduli spaces, and not from modular arithmetic.
Definition The modular group Γ is the group of fractional linear transformations of the complex upper half-plane, which have the form
z ↦ a z + b c z + d , {\displaystyle z\mapsto {\frac {az+b}{cz+d}},}
where a , b , c , d {\displaystyle a,b,c,d} are integers, and a d − b c = 1 {\displaystyle ad-bc=1} . The group operation is function composition. This group of transformations is isomorphic to the projective special linear group PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} , which is the quotient of the 2-dimensional special linear group SL ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )} by its center { I , − I } {\displaystyle \{I,-I\}} . In other words, PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} consists of all matrices
( a b c d ) {\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}
where a , b , c , d {\displaystyle a,b,c,d} are integers, a d − b c = 1 {\displaystyle ad-bc=1} , and pairs of matrices A {\displaystyle A} and − A {\displaystyle -A} are considered to be identical. The group operation is usual matrix multiplication. Some authors define the modular group to be PSL ( 2 , Z ) {\displaystyle \operatorname {PSL} (2,\mathbb {Z} )} , and still others define the modular group to be the larger group SL ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )} . Some mathematical relations require the consideration of the group GL ( 2 , Z ) {\displaystyle \operatorname {GL} (2,\mathbb {Z} )} of matrices with determinant plus or minus one. ( SL ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )} is a subgroup of this group.) Similarly, PGL ( 2 , Z ) {\displaystyle \operatorname {PGL} (2,\mathbb {Z} )} is the quotient group GL ( 2 , Z ) / { I , − I } {\displaystyle \operatorname {GL} (2,\mathbb {Z} )/\{I,-I\}} . Since all 2 × 2 {\displaystyle 2\times 2} matrices with determinant 1 are symplectic matrices, then SL ( 2 , Z ) = Sp ( 2 , Z ) {\displaystyle \operatorname {SL} (2,\mathbb {Z} )=\operatorname {Sp} (2,\mathbb {Z} )} , the symplectic group of 2 × 2 {\displaystyle 2\times 2} matrices.
Finding elements To find an explicit matrix
( a x b y ) {\displaystyle {\begin{pmatrix}a&x\\b&y\end{pmatrix}}}
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