Symbol: SL2Z — SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) — Modular group
Whether
SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z )
or
PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z )
should be called "the modular group" is an arbitrary convention. Here we allow any element of
SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z )
to represent an element of the modular group, but we use
PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z )
when uniqueness is desired.
Definitions:
Fungrim symbol Notation Short description SL2Z SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) Modular group PSL2Z PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z ) Modular group (canonical representatives)
Source code for this entry:
Entry(ID("094772"),
SymbolDefinition(SL2Z, SL2Z, "Modular group"),
Description("Whether", SL2Z, "or", PSL2Z, "should be called \"the modular group\" is an arbitrary convention. Here we allow any element of", SL2Z, "to represent an element of the modular group, but we use", PSL2Z, "when uniqueness is desired.")) Symbol: PSL2Z — PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z ) — Modular group (canonical representatives)
Definitions:
Fungrim symbol Notation Short description PSL2Z PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z ) Modular group (canonical representatives)
Source code for this entry:
Entry(ID("1e211d"),
SymbolDefinition(PSL2Z, PSL2Z, "Modular group (canonical representatives)")) Domain Codomain γ ∈ SL 2 ( Z ) and τ ∈ H \gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} γ ∈ S L 2 ( Z ) a n d τ ∈ H γ ∘ τ ∈ H \gamma \circ \tau \in \mathbb{H} γ ∘ τ ∈ H
Table data:
( P , Q ) \left(P, Q\right) ( P , Q )
such that
( P ) ⟹ ( Q ) \left(P\right) \;\implies\; \left(Q\right) ( P ) ⟹ ( Q ) Definitions:
Fungrim symbol Notation Short description ModularGroupAction γ ∘ τ \gamma \circ \tau γ ∘ τ Action of modular group SL2Z SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) Modular group HH H \mathbb{H} H Upper complex half-plane
Source code for this entry:
Entry(ID("76de9d"),
SymbolDefinition(ModularGroupAction, ModularGroupAction(gamma, tau), "Action of modular group"),
Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(Tuple(And(Element(gamma, SL2Z), Element(tau, HH)), Element(ModularGroupAction(gamma, tau), HH))))) Definitions:
Source code for this entry:
Entry(ID("dc2c26"),
SymbolDefinition(ModularGroupFundamentalDomain, ModularGroupFundamentalDomain, "Fundamental domain for action of the modular group")) H = { τ : τ ∈ C and Im ( τ ) > 0 } \mathbb{H} = \left\{ \tau : \tau \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}(\tau) > 0 \right\} H = { τ : τ ∈ C a n d I m ( τ ) > 0 } TeX:
\mathbb{H} = \left\{ \tau : \tau \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}(\tau) > 0 \right\} Definitions:
Fungrim symbol Notation Short description HH H \mathbb{H} H Upper complex half-plane CC C \mathbb{C} C Complex numbers Im Im ( z ) \operatorname{Im}(z) I m ( z ) Imaginary part
Source code for this entry:
Entry(ID("d7962e"),
Formula(Equal(HH, Set(tau, ForElement(tau, CC), Greater(Im(tau), 0))))) SL 2 ( Z ) = { ( a b c d ) : a ∈ Z and b ∈ Z and c ∈ Z and d ∈ Z and a d − b c = 1 } \operatorname{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; c \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; d \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; a d - b c = 1 \right\} S L 2 ( Z ) = { ( a c b d ) : a ∈ Z a n d b ∈ Z a n d c ∈ Z a n d d ∈ Z a n d a d − b c = 1 } TeX:
\operatorname{SL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; c \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; d \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; a d - b c = 1 \right\} Definitions:
Fungrim symbol Notation Short description SL2Z SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) Modular group Matrix2x2 ( a b c d ) \begin{pmatrix} a & b \\ c & d \end{pmatrix} ( a c b d ) Two by two matrix ZZ Z \mathbb{Z} Z Integers
Source code for this entry:
Entry(ID("c84f3f"),
Formula(Equal(SL2Z, Set(Matrix2x2(a, b, c, d), For(Tuple(a, b, c, d)), And(Element(a, ZZ), Element(b, ZZ), Element(c, ZZ), Element(d, ZZ), Equal(Sub(Mul(a, d), Mul(b, c)), 1)))))) PSL 2 ( Z ) = { ( a b c d ) : ( a b c d ) ∈ SL 2 ( Z ) and ( c > 0 or ( c = 0 and d > 0 ) ) } \operatorname{PSL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \left(c > 0 \;\mathbin{\operatorname{or}}\; \left(c = 0 \;\mathbin{\operatorname{and}}\; d > 0\right)\right) \right\} P S L 2 ( Z ) = { ( a c b d ) : ( a c b d ) ∈ S L 2 ( Z ) a n d ( c > 0 o r ( c = 0 a n d d > 0 ) ) } TeX:
\operatorname{PSL}_2(\mathbb{Z}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \left(c > 0 \;\mathbin{\operatorname{or}}\; \left(c = 0 \;\mathbin{\operatorname{and}}\; d > 0\right)\right) \right\} Definitions:
Fungrim symbol Notation Short description PSL2Z PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z ) Modular group (canonical representatives) Matrix2x2 ( a b c d ) \begin{pmatrix} a & b \\ c & d \end{pmatrix} ( a c b d ) Two by two matrix SL2Z SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) Modular group
Source code for this entry:
Entry(ID("80279d"),
Formula(Equal(PSL2Z, Set(Matrix2x2(a, b, c, d), For(Tuple(a, b, c, d)), And(Element(Matrix2x2(a, b, c, d), SL2Z), Or(Greater(c, 0), And(Equal(c, 0), Greater(d, 0)))))))) ( a b c d ) ∘ τ = a τ + b c τ + d \begin{pmatrix} a & b \\ c & d \end{pmatrix} \circ \tau = \frac{a \tau + b}{c \tau + d} ( a c b d ) ∘ τ = c τ + d a τ + b Assumptions: ( a b c d ) ∈ SL 2 ( Z ) and τ ∈ H \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} ( a c b d ) ∈ S L 2 ( Z ) a n d τ ∈ H TeX:
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \circ \tau = \frac{a \tau + b}{c \tau + d}
\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} Definitions:
Fungrim symbol Notation Short description ModularGroupAction γ ∘ τ \gamma \circ \tau γ ∘ τ Action of modular group Matrix2x2 ( a b c d ) \begin{pmatrix} a & b \\ c & d \end{pmatrix} ( a c b d ) Two by two matrix SL2Z SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) Modular group HH H \mathbb{H} H Upper complex half-plane
Source code for this entry:
Entry(ID("127a52"),
Formula(Equal(ModularGroupAction(Matrix2x2(a, b, c, d), tau), Div(Add(Mul(a, tau), b), Add(Mul(c, tau), d)))),
Assumptions(And(Element(Matrix2x2(a, b, c, d), SL2Z), Element(tau, HH)))) ( γ η ) ∘ τ = γ ∘ ( η ∘ τ ) \left(\gamma \eta\right) \circ \tau = \gamma \circ \left(\eta \circ \tau\right) ( γ η ) ∘ τ = γ ∘ ( η ∘ τ ) Assumptions: γ ∈ SL 2 ( Z ) and η ∈ SL 2 ( Z ) and τ ∈ H \gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \eta \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} γ ∈ S L 2 ( Z ) a n d η ∈ S L 2 ( Z ) a n d τ ∈ H TeX:
\left(\gamma \eta\right) \circ \tau = \gamma \circ \left(\eta \circ \tau\right)
\gamma \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \eta \in \operatorname{SL}_2(\mathbb{Z}) \;\mathbin{\operatorname{and}}\; \tau \in \mathbb{H} Definitions:
Fungrim symbol Notation Short description ModularGroupAction γ ∘ τ \gamma \circ \tau γ ∘ τ Action of modular group SL2Z SL 2 ( Z ) \operatorname{SL}_2(\mathbb{Z}) S L 2 ( Z ) Modular group HH H \mathbb{H} H Upper complex half-plane
Source code for this entry:
Entry(ID("5636db"),
Formula(Equal(ModularGroupAction(Parentheses(Mul(gamma, eta)), tau), ModularGroupAction(gamma, Parentheses(ModularGroupAction(eta, tau))))),
Variables(gamma, eta, tau),
Assumptions(And(Element(gamma, SL2Z), Element(eta, SL2Z), Element(tau, HH)))) F = { τ : τ ∈ H and Re ( τ ) ∈ [ − 1 2 , 1 2 ) and ( ∣ τ ∣ > 1 or ( ∣ τ ∣ = 1 and Re ( τ ) ≤ 0 ) ) } \mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \;\mathbin{\operatorname{and}}\; \left(\left|\tau\right| > 1 \;\mathbin{\operatorname{or}}\; \left(\left|\tau\right| = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \le 0\right)\right) \right\} F = { τ : τ ∈ H a n d R e ( τ ) ∈ [ − 2 1 , 2 1 ) a n d ( ∣ τ ∣ > 1 o r ( ∣ τ ∣ = 1 a n d R e ( τ ) ≤ 0 ) ) } The choice to include the left or right boundary is arbitrary; the present definition follows Cohen and simplifies the treatment of reduced binary quadratic forms.
References:
H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993 TeX:
\mathcal{F} = \left\{ \tau : \tau \in \mathbb{H} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \in \left[-\frac{1}{2}, \frac{1}{2}\right) \;\mathbin{\operatorname{and}}\; \left(\left|\tau\right| > 1 \;\mathbin{\operatorname{or}}\; \left(\left|\tau\right| = 1 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(\tau) \le 0\right)\right) \right\} Definitions:
Fungrim symbol Notation Short description ModularGroupFundamentalDomain F \mathcal{F} F Fundamental domain for action of the modular group HH H \mathbb{H} H Upper complex half-plane Re Re ( z ) \operatorname{Re}(z) R e ( z ) Real part ClosedOpenInterval [ a , b ) \left[a, b\right) [ a , b ) Closed-open interval Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value
Source code for this entry:
Entry(ID("a637cd"),
Formula(Equal(ModularGroupFundamentalDomain, Set(tau, For(tau), And(Element(tau, HH), Element(Re(tau), ClosedOpenInterval(Neg(Div(1, 2)), Div(1, 2))), Or(Greater(Abs(tau), 1), And(Equal(Abs(tau), 1), LessEqual(Re(tau), 0))))))),
Description("The choice to include the left or right boundary is arbitrary; the present definition follows Cohen and simplifies the treatment of reduced binary quadratic forms."),
References("H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993")) TeX:
i \in \mathcal{F} Definitions:
Source code for this entry:
Entry(ID("1d1028"),
Formula(Element(ConstI, ModularGroupFundamentalDomain))) e 2 π i / 3 = − 1 + 3 i 2 ∈ F {e}^{2 \pi i / 3} = \frac{-1 + \sqrt{3} i}{2} \in \mathcal{F} e 2 π i / 3 = 2 − 1 + 3 i ∈ F Corner of the fundamental domain.
TeX:
{e}^{2 \pi i / 3} = \frac{-1 + \sqrt{3} i}{2} \in \mathcal{F} Definitions:
Fungrim symbol Notation Short description Exp e z {e}^{z} e z Exponential function Pi π \pi π The constant pi (3.14...) ConstI i i i Imaginary unit Sqrt z \sqrt{z} z Principal square root ModularGroupFundamentalDomain F \mathcal{F} F Fundamental domain for action of the modular group
Source code for this entry:
Entry(ID("21b67f"),
Formula(EqualAndElement(Exp(Div(Mul(Mul(2, Pi), ConstI), 3)), Div(Add(-1, Mul(Sqrt(3), ConstI)), 2), ModularGroupFundamentalDomain)),
Description("Corner of the fundamental domain.")) ∣ e 2 π i τ ∣ < 0.004334 \left|{e}^{2 \pi i \tau}\right| < 0.004334 ∣ ∣ e 2 π i τ ∣ ∣ < 0 . 0 0 4 3 3 4 Assumptions: τ ∈ F \tau \in \mathcal{F} τ ∈ F TeX:
\left|{e}^{2 \pi i \tau}\right| < 0.004334
\tau \in \mathcal{F} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value Exp e z {e}^{z} e z Exponential function Pi π \pi π The constant pi (3.14...) ConstI i i i Imaginary unit ModularGroupFundamentalDomain F \mathcal{F} F Fundamental domain for action of the modular group
Source code for this entry:
Entry(ID("e28209"),
Formula(Less(Abs(Exp(Mul(Mul(Mul(2, Pi), ConstI), tau))), Decimal("0.004334"))),
Variables(tau),
Assumptions(Element(tau, ModularGroupFundamentalDomain))) { γ ∘ τ : τ ∈ F and γ ∈ PSL 2 ( Z ) } = H \left\{ \gamma \circ \tau : \tau \in \mathcal{F} \;\mathbin{\operatorname{and}}\; \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\} = \mathbb{H} { γ ∘ τ : τ ∈ F a n d γ ∈ P S L 2 ( Z ) } = H TeX:
\left\{ \gamma \circ \tau : \tau \in \mathcal{F} \;\mathbin{\operatorname{and}}\; \gamma \in \operatorname{PSL}_2(\mathbb{Z}) \right\} = \mathbb{H} Definitions:
Fungrim symbol Notation Short description ModularGroupAction γ ∘ τ \gamma \circ \tau γ ∘ τ Action of modular group ModularGroupFundamentalDomain F \mathcal{F} F Fundamental domain for action of the modular group PSL2Z PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) P S L 2 ( Z ) Modular group (canonical representatives) HH H \mathbb{H} H Upper complex half-plane
Source code for this entry:
Entry(ID("fd53ab"),
Formula(Equal(Set(ModularGroupAction(gamma, tau), For(Tuple(gamma, tau)), And(Element(tau, ModularGroupFundamentalDomain), Element(gamma, PSL2Z))), HH))) π 4 ∑ q = 1 ∞ φ ( q ) q 4 = π 4 ζ ( 3 ) ζ ( 4 ) = 45 ζ ( 3 ) 2 π 3 \frac{\pi}{4} \sum_{q=1}^{\infty} \frac{\varphi(q)}{{q}^{4}} = \frac{\pi}{4} \frac{\zeta\!\left(3\right)}{\zeta\!\left(4\right)} = \frac{45 \zeta\!\left(3\right)}{2 {\pi}^{3}} 4 π q = 1 ∑ ∞ q 4 φ ( q ) = 4 π ζ ( 4 ) ζ ( 3 ) = 2 π 3 4 5 ζ ( 3 ) TeX:
\frac{\pi}{4} \sum_{q=1}^{\infty} \frac{\varphi(q)}{{q}^{4}} = \frac{\pi}{4} \frac{\zeta\!\left(3\right)}{\zeta\!\left(4\right)} = \frac{45 \zeta\!\left(3\right)}{2 {\pi}^{3}} Definitions:
Fungrim symbol Notation Short description Pi π \pi π The constant pi (3.14...) Sum ∑ n f ( n ) \sum_{n} f(n) ∑ n f ( n ) Sum Totient φ ( n ) \varphi(n) φ ( n ) Euler totient function Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity RiemannZeta ζ ( s ) \zeta\!\left(s\right) ζ ( s ) Riemann zeta function
Source code for this entry:
Entry(ID("8a9884"),
Formula(Equal(Mul(Div(Pi, 4), Sum(Div(Totient(q), Pow(q, 4)), For(q, 1, Infinity))), Mul(Div(Pi, 4), Div(RiemannZeta(3), RiemannZeta(4))), Div(Mul(45, RiemannZeta(3)), Mul(2, Pow(Pi, 3))))))