Fungrim entry: 9bda2f

S_{j}\!\left(a, b, c, d\right) = \begin{cases} 1, & j = 1\\T\!\left(c, d\right), & j = 2\\T\!\left(a + c, b + d\right), & j = 3\\T\!\left(a, b\right), & j = 4\\ \end{cases}\; \text{ where } T\!\left(m, n\right) = \begin{cases} 1, & \left(m, n\right) \equiv \left(0, 0\right) \pmod {2}\\2, & \left(m, n\right) \equiv \left(0, 1\right) \pmod {2}\\4, & \left(m, n\right) \equiv \left(1, 0\right) \pmod {2}\\3, & \left(m, n\right) \equiv \left(1, 1\right) \pmod {2}\\ \end{cases}

Assumptions:

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

References:

Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Section 81.

TeX:

S_{j}\!\left(a, b, c, d\right) = \begin{cases} 1, & j = 1\\T\!\left(c, d\right), & j = 2\\T\!\left(a + c, b + d\right), & j = 3\\T\!\left(a, b\right), & j = 4\\ \end{cases}\; \text{ where } T\!\left(m, n\right) = \begin{cases} 1, & \left(m, n\right) \equiv \left(0, 0\right) \pmod {2}\\2, & \left(m, n\right) \equiv \left(0, 1\right) \pmod {2}\\4, & \left(m, n\right) \equiv \left(1, 0\right) \pmod {2}\\3, & \left(m, n\right) \equiv \left(1, 1\right) \pmod {2}\\ \end{cases}

j \in \left\{1, 2, 3, 4\right\} \;\mathbin{\operatorname{and}}\; \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \operatorname{PSL}_2(\mathbb{Z})

Definitions:

Fungrim symbol	Notation	Short description
`JacobiThetaPermutation`	$S_{j}\!\left(a, b, c, d\right)$	Index permutation in modular transformation of Jacobi theta functions
`Matrix2x2`	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$	Two by two matrix
`PSL2Z`	$\operatorname{PSL}_2(\mathbb{Z})$	Modular group (canonical representatives)

Source code for this entry:

Entry(ID("9bda2f"),
    Formula(Equal(JacobiThetaPermutation(j, a, b, c, d), Where(Cases(Tuple(1, Equal(j, 1)), Tuple(T(c, d), Equal(j, 2)), Tuple(T(Add(a, c), Add(b, d)), Equal(j, 3)), Tuple(T(a, b), Equal(j, 4))), Equal(T(m, n), Cases(Tuple(1, CongruentMod(Tuple(m, n), Tuple(0, 0), 2)), Tuple(2, CongruentMod(Tuple(m, n), Tuple(0, 1), 2)), Tuple(4, CongruentMod(Tuple(m, n), Tuple(1, 0), 2)), Tuple(3, CongruentMod(Tuple(m, n), Tuple(1, 1), 2))))))),
    Variables(j, a, b, c, d),
    Assumptions(And(Element(j, Set(1, 2, 3, 4)), Element(Matrix2x2(a, b, c, d), PSL2Z))),
    References("Hans Rademacher (1973), Topics in Analytic Number Theory, Springer. Section 81."))

Topics using this entry

Lattice transformations for Jacobi theta functions