Elliptic and Modular Functions

More information on elliptic functions can be found for example in Chandrasekharan [Chandrasekharan, 1985], and for modular functions and their use see Koblitz [Koblitz, 1984].

Eisenstein Series

Let \(f(z)\) be a modular function. Then \(f(z)\) may be written as a Fourier series \(f(z) = \sum_{n \in {\mathbb{Z}}} a_{n} q^{n},\) where \(q = e^{2 \pi i z}\), which has at most finitely many nonzero coefficients \(a_{n}\) with \(n<0\). Such a Fourier expansion of a modular function is called its \(q\)-expansion. In this and the next section we present intrinsics for \(q\)-expansions of the Eisenstein series and the Weierstrass \(\wp\)-function.

Let \(z\) be a point in the upper half-plane and let \(L\) be a lattice in \({\mathbb{C}}\). The Eisenstein series are defined as the coefficients of the Laurent Series expansion of the Weierstrass \(\wp\)-function:

\[\wp(z, L) = {{1}\over{z^2}} + \sum_{2\le k} G_k(L)(2k-1)z^{2k-2}\]

where \(G_k(L)\) are the Eisenstein series. The normalization \(E_{2n}(z) = {{1}\over{2 \zeta(2n)}} G_{2n}(z)\) ensures that \(E_{2n}(z)\) has a rational \(q\)-expansion.

Eisenstein(k, z): RngIntElt, RngSerElt -> RngSerElt

Precision: RngIntElt                    Default: 

Given a positive even integer \(k = 2n\) and a complex power series \(z\) with positive valuation, return the \(q\)-expansion of the normalized Eisenstein series \(E_{2n}(z)\). If \(z\) has finite precision this is the default for Precision otherwise the default precision of the parent of \(z\) is used.

Eisenstein(k, t): RngIntElt, FldComElt -> FldComElt

Given a positive even integer \(k = 2n\) and a point \(t\) in the upper half plane, return the value of \(E_{2n}(z)\) at \(t\).

Eisenstein(k, L): RngIntElt, SeqEnum -> FldComElt

Given a positive even integer \(k = 2n\) and a lattice \(L = [a, b]\) in the complex plane, return the value of the Eisenstein series \(E_{2n}(z)\) relative to the lattice \(L\).

Eisenstein(k, F): RngIntElt, QuadBinElt -> RngSerElt

Given a positive even integer \(k = 2n\) and a binary quadratic form \(F = ax^2+bxy+cy^2\), return the value of the Eisenstein series \(E_{2n}(z)\) at the point \(\tau = \left(-b+\sqrt{b^2-4ac} \right)/(2a)\) where \(z\) is \(e^{2\pi*{\mathrm{i}}*\tau}\).

Example: Eisenstein (ex-db3e10)

We compute the \(q\)-expansion for the normalized Eisenstein series \(E_{4}(z)\).

> C<i> := ComplexField();
> R<z> := PowerSeriesRing(C);
> E4<q> := Eisenstein(4, z);
> E4;
1.00000000000000000000000000000 +
    240.000000000000000000000000000*q +
    2160.00000000000000000000000000*q^2 +
    6720.00000000000000000000000000*q^3 +
    17520.0000000000000000000000000*q^4 +
    30240.0000000000000000000000000*q^5 +
    60480.0000000000000000000000000*q^6 +
    82560.0000000000000000000000000*q^7 +
    140400.000000000000000000000000*q^8 +
    181680.000000000000000000000000*q^9 +
    272160.000000000000000000000000*q^10 +
    319680.000000000000000000000000*q^11 +
    490560.000000000000000000000000*q^12 +
    527520.000000000000000000000000*q^13 +
    743040.000000000000000000000000*q^14 +
    846720.000000000000000000000000*q^15 +
    1123440.00000000000000000000000*q^16 +
    1179360.00000000000000000000000*q^17 +
    1635120.00000000000000000000000*q^18 +
    1646400.00000000000000000000000*q^19 + O(q^20)

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We now evaluate this series at the point \(z1 = 2.5 + i\). Since the expansion is in terms of \(q\) rather than \(z\) we first must calculate the point \(q1\) that corresponds to \(z1\).

> q1 := Exp(2*Pi(RealField())*i*(2.5 +i));
> Evaluate(E4, q1);
0.559302852856190773766762411942 +
3.67329046709782088758389413820E-31*i

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If we are interested only in the value of \(E_{4}\) at a single point, then we can compute it directly:

> Eisenstein(4, 2.5 + i);
0.559302852856190773766762411942 +
3.67329046709782088758389413820E-31*i

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Weierstrass Series

WeierstrassSeries(z, q): RngSerElt, RngSerElt -> RngSerElt

Precision: RngIntElt                    Default: 

Return a normalized \(q\)-expansion of the Weierstrass \(\wp\)-function:

\[\wp(z, L) = {{1}\over{z^2}} + \sum_{2\le k} G_k(L)(2k-1)z^{2k-2}\]

where \(G_k(L)\) are the Eisenstein series and

\[\hbox{\tt WeierstrassSeries}(z,q) = (2 \pi i)^{-2}\wp(q, z/(2 \pi i))\]

Each term is an Eisenstein series, calculated to precision Precision, which is by default the precision of \(q\).

WeierstrassSeries(z, t): RngSerElt, FldComElt -> RngSerElt

Given a complex power series \(z\) with positive valuation and a point \(t =\tau\) in the upper-half complex plane, return the normalized \(q\)-expansion of the Weierstrass \(\wp\)-function. This is equivalent to evaluating the \(q\)-series expansion at \(q = e^{2\pi i\tau}\).

WeierstrassSeries(z, L): RngSerElt, SeqEnum -> RngSerElt

Given a complex power series \(z\) with positive valuation and a lattice \(L = [a, b]\) in the complex plane, returns the normalized \(q\)-expansion of the Weierstrass \(\wp\)-function relative to the lattice \(L\).

WeierstrassSeries(z, F): RngSerElt, QuadBinElt -> RngSerElt

Given a complex power series \(z\) with positive valuation and a binary quadratic form \(F = ax^2+bxy+cy^2\), this function returns the \(q\)-expansion of the Weierstrass \(\wp\)-function at \(\tau = \left(-b+\sqrt{b^2-4ac} \right)/(2a)\).

The Jacobi \(\theta\) and Dedekind \(\eta\)-functions

The first Jacobi \(\theta\)-function, \(\theta(q, z)\), is defined by

\[\theta(q, z)={1\over{\mathrm{i}}}\sum_{n=-\infty}^\infty(-1)^nq^{(n+{1\over2})^2}e^{(2n+1){\mathrm{i}}z}=2\sum_{n=0}^\infty(-1)^nq^{(n+{1\over2})^2}\sin(2n+1)z.\]

Defined this way, \(\theta\) satisfies \(\theta(q, -z)=-\theta(q, z)\), it is periodic with period \(2\pi\) in the second variable: \(\theta(q, z+2\pi)= \theta(q, z)\), and its zeroes are of the form \(m_1\pi+m_2{\log x\over{\mathrm{i}}}\) for any integers \(m_1, m_2\).

JacobiTheta(q, z): FldReElt, RngSerElt[FldRe] -> RngSerElt

JacobiTheta(q, z): FldComElt, RngSerElt[FldCom] -> RngSerElt

For a real or complex number \(q\) satisfying \(\vert q\vert<1\), return the first of Jacobi’s theta functions \(\theta(q,z)\) as a power series expansion in \(z\), a series over the complex numbers. Pari is used here.

JacobiTheta(q, z): FldReElt, FldReElt -> FldReElt

JacobiTheta(q, z): FldComElt, FldComElt -> FldComElt

For real or complex numbers \(q, z\) satisfying \(\vert q\vert<1\), return the value of \(\theta(q, z)\), the first of Jacobi’s theta functions. Pari is used here.

JacobiThetaNullK(q, k): FldReElt, RngIntElt -> FldReElt

For integer \(k\geq 0\), return the \(k\)-th derivative \(\theta^{(k)}(q, 0)\) of \(\theta(q, z)\) at \(z=0\). Pari is used here.

DedekindEta(z): RngSerElt -> RngSerElt

Given a complex power series \(z\) with positive valuation, return the \(q\)-expansion of Dedekind’s \(\eta\)-function. Note that the unnormalized series is returned, that is, the factor \(q^{1/24}\) is not removed. See [Lang, 1987].

DedekindEta(s): FldComElt -> FldComElt

For complex argument \(s\) with positive imaginary part, this returns the actual value of Dedekind’s \(\eta\)-function which is defined by \(\eta(s)=e^{2\pi{\mathrm{i}}s\over 24} \left(1 + \sum_{n=1}^\infty(-1)^n (q^{n(3n-1)/2} + q^{n(3n+1)/2}) \right)\) where \(q = e^{2\pi{\mathrm{i}}s}\).

The \(j\)-Invariant and the Discriminant

The discriminant of the elliptic curve corresponding to the complex lattice \(L_{\tau}\), spanned by \(1\) and \(\tau\) is given by

\[\Delta(\tau) = q \left(1 + \sum_{n=1}^\infty(-1)^n (q^{n(3n-1)/2} + q^{n(3n+1)/2}) \right)\]

where \(q = e^{2\pi{\mathrm{i}}\tau}\).

jInvariant(q): RngSerElt -> RngSerElt

Given a power series \(q\) over a real or complex field with positive valuation, return the \(q\)-expansion of the elliptic \(j\)-invariant. The expansion begins with

\[j(q) = q^{-1} + 744 + 196884 q + \cdots.\]

Note that:

\[j(q) = {{{E_4(q)}^3}\over{\Delta(q)}}\]

where \(E_4(q)\) = Eisenstein(4, q) and \(\Delta(q)\) = Delta(q).

jInvariant(s): FldComElt -> FldComElt

For complex argument \(s\) with positive imaginary part, this returns the value of the elliptic \(j\)-invariant at \(s\). This is a modular function of weight \(0\) whose Fourier expansion starts with

\[j(s)=e^{-2\pi{\mathrm{i}}s}+744+196884e^{2\pi{\mathrm{i}}s}+\cdots.\]

jInvariant(L): SeqEnum -> FldComElt

Given a lattice \(L = [a, b]\) in the complex plane, this function returns the value of the elliptic \(j\)-invariant of \(L\). This is the \(j\)-invariant of \(\tau\) where \(\tau = a/b\) or \(\tau = b / a\), whichever is in the upper half complex plane.

jInvariant(F): QuadBinElt -> FldComElt

For a binary quadratic form \(F = ax^2+bxy+cy^2\) with negative discriminant, this returns the elliptic \(j\)-invariant of \(F\). This is the \(j\)-invariant of \(\tau\) where \(\tau = \left(-b + \sqrt{b^2-4ac}\right) / (2a)\).

Delta(z): RngSerElt -> RngSerElt

Given a complex power series \(z\), this function returns a \(q\)-series expansion of the discriminant \(\Delta(z)\).

Delta(t): FldComElt -> FldComElt

Given a point \(t\) in the upper half plane, return the \(q\)-series expansion of the discriminant \(\Delta(q)\) evaluated at \(q = e^{2\pi{\mathrm{i}}t}\).

Delta(L): SeqEnum -> FldComElt

Given a pair \(L\) = [\(a\),\(b\)] of complex numbers generating a lattice in \({\mathbb{C}}\), return the \(q\)-series expansion of the discriminant \(\Delta(q)\) evaluated at \(q = e^{2\pi{\mathrm{i}}\tau}\) where \(\tau = a/b\) or \(\tau = b / a\), whichever is in the upper half complex plane.

Weber’s Functions

WeberF(s): FldComElt -> FldComElt

For complex argument \(s\) in the upper half-plane, this returns the value of Weber’s function \(f\), defined in such a way that

\[j(s)={(f(s)^{24}-16)^3\over f(s)^{24}}.\]

WeberF2(g): RngSerElt -> RngSerElt

For a complex power series \(g\) having positive valuation, this function returns the \(q\)-expansion of Weber’s \(f_2\) function

\[f_2(x)={\eta(2x)\sqrt{2}\over\eta(x)}\]

defined in such a way that

\[j(s)={(f_2(s)^{24}+16)^3\over f_2(s)^{24}}.\]

WeberF1(s): FldComElt -> FldComElt

WeberF2(s): FldComElt -> FldComElt

For complex number \(s\) lying in the upper half-plane, these return the value of Weber’s functions \(f_1\) and \(f_2\), defined in such a way that

\[j(s)={(f_{1/2}(s)^{24}+16)^3\over f_{1/2}(s)^{24}}.\]

In fact, \(f_2\) is as defined above and

\[f_1(x)=f_2(-1/x)={\eta(x/2)/\eta(x)}.\]

Example: Eisenstein (ex-3718fd)

We compute the \(q\)-expansion for the Weber function \(f_{2}(z)\).

> C<i> := ComplexField();
> R<x> := PowerSeriesRing(C);
> f2<q> := WeberF2(x);
> f2;
1.41421356237309504880168872421 +
    (1.41421356237309504880168872421 +
    0.370240244846530520584656749172*i)*q +
    (1.36574922765338060759226121771 +
    0.370240244846530520584656749172*i)*q^2 +
    (2.77996279002647565639394994192 +
    0.366010933793292419482272977081*i)*q^3 +
    (2.78023959778761313408864734217 +
    0.736251178639822940066929726253*i)*q^4 +
    (4.14598882544099374168090855987 +
    0.736265672260303709036837819528*i)*q^5 +
    (5.56020175541059398072755542234 +
    1.10227660605359612851911079661*i)*q^6 +
    ...

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