# Elliptic and Modular Functions More information on elliptic functions can be found for example in Chandrasekharan [[Chandrasekharan, 1985](../../references.md#cite-chand85)], and for modular functions and their use see Koblitz [[Koblitz, 1984](../../references.md#cite-koblitz84)]. ## 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` ```magma 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 $E_{4}(z)$. ```magma > C := ComplexField(); > R := PowerSeriesRing(C); > E4 := 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) ``` 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 $z1$. ```magma > q1 := Exp(2*Pi(RealField())*i*(2.5 +i)); > Evaluate(E4, q1); 0.559302852856190773766762411942 + 3.67329046709782088758389413820E-31*i ``` If we are interested only in the value of $E_{4}$ at a single point, then we can compute it directly: ```magma > Eisenstein(4, 2.5 + i); 0.559302852856190773766762411942 + 3.67329046709782088758389413820E-31*i ``` ## Weierstrass Series ### `WeierstrassSeries(z, q): RngSerElt, RngSerElt -> RngSerElt` ```magma 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](../../references.md#cite-lang87)]. ### `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 $f_{2}(z)$. ```magma > C := ComplexField(); > R := PowerSeriesRing(C); > f2 := 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 + ... ```