Editing J-invariant

{{Short description|Modular function in mathematics}}
{{DISPLAYTITLE:''j''-invariant}}
[[File:KleinInvariantJ.jpg|right|thumb|200px|Klein's {{mvar|j}}-invariant in the complex plane]]

In [[mathematics]], [[Felix Klein]]'s '''{{mvar|j}}-invariant''' or '''{{mvar|j}} function''' is a [[modular function]] of weight zero for the [[special linear group]] <math>\operatorname{SL}(2,\Z)</math> defined on the [[upper half-plane]] of [[complex number]]s. It is the unique such function that is [[Holomorphic function|holomorphic]] away from a simple pole at the [[Cusp (singularity)|cusp]] such that

:<math>j\big(e^{2\pi i/3}\big) = 0, \quad j(i) = 1728 = 12^3.</math>

[[Rational function]]s of <math>j</math> are modular, and in fact give all modular functions of weight 0. Classically, the <math>j</math>-invariant was studied as a parameterization of [[elliptic curve]]s over <math>\mathbb{C}</math>, but it also has surprising connections to the symmetries of the [[Monster group]] (this connection is referred to as [[monstrous moonshine]]).

==Definition==
[[File:J-inv-real.jpeg|right|thumb|200px|Real part of the {{mvar|j}}-invariant as a function of the square of the [[Nome (mathematics)|nome]] on the unit disk]]
[[File:J-inv-phase.jpeg|thumb|200px|Phase of the {{mvar|j}}-invariant as a function of the square of the nome on the unit disk]]
{{Further|Elliptic curve#Elliptic curves over the complex numbers|Modular forms}}

The {{mvar|j}}-invariant can be defined as a function on the [[upper half-plane]] <math>\mathcal{H}=\{\tau\in\C \mid \operatorname{Im}(\tau)>0\}</math>, by

:<math>j(\tau) = 1728 \frac{g_2(\tau)^3}{\Delta(\tau)} = 1728 \frac{g_2(\tau)^3}{g_2(\tau)^3 - 27g_3(\tau)^2} = 1728 \frac{g_2(\tau)^3}{(2\pi)^{12}\,\eta^{24}(\tau)}</math>

with the third definition implying <math>j(\tau)</math> can be expressed as a [[Cube (algebra)|cube]], also since [[1728 (number)|1728]]<math>{} = 12^3</math>. The function cannot be continued analytically beyond the upper half-plane due to the [[natural boundary]] at the real line.

The given functions are the [[modular discriminant]] <math>\Delta(\tau) = g_2(\tau)^3 - 27g_3(\tau)^2 = (2\pi)^{12}\,\eta^{24}(\tau)</math>, [[Dedekind eta function]] <math>\eta(\tau)</math>, and modular invariants,

:<math>g_2(\tau) = 60G_4(\tau) = 60\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-4}</math>
:<math>g_3(\tau) = 140G_6(\tau) = 140\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-6}</math>

where <math>G_4(\tau)</math>, <math>G_6(\tau)</math> are [[Eisenstein_series#Fourier_series|Fourier series]],

:<math>\begin{align}
G_4(\tau)&=\frac{\pi^4}{45}\, E_4(\tau) \\[4pt]
G_6(\tau)&=\frac{2\pi^6}{945}\, E_6(\tau)
\end{align}</math>

and <math>E_4(\tau)</math>, <math>E_6(\tau)</math> are [[Eisenstein_series#Identities_involving_Eisenstein_series|Eisenstein series]],

:<math>\begin{align}
E_4(\tau)&= 1+ 240\sum_{n=1}^\infty \frac{n^3 q^n}{1-q^n} \\[4pt]
E_6(\tau)&= 1- 504\sum_{n=1}^\infty \frac{n^5 q^n}{1-q^n} 
\end{align}</math>

and <math>q=e^{2\pi i \tau}</math> (the square of the [[nome (mathematics)|nome]]). The {{mvar|j}}-invariant can then be directly expressed in terms of the Eisenstein series as,

:<math>j(\tau) = 1728 \frac{E_4(\tau)^3}{E_4(\tau)^3 - E_6(\tau)^2} </math>

with no numerical factor other than 1728. This implies a third way to define the modular discriminant,<ref>{{cite arXiv|last=Milne|first=Steven C.|year=2000|eprint=math/0009130v3|title=Hankel Determinants of Eisenstein Series}} The paper uses a non-equivalent definition of <math>\Delta</math>, but this has been accounted for in this article.</ref>

:<math>\Delta(\tau) = (2\pi)^{12}\,\frac{E_4(\tau)^3 - E_6(\tau)^2}{1728}</math>

For example, using the definitions above and <math>\tau = 2i</math>, then the Dedekind eta function <math>\eta(2i)</math> has the [[Dedekind_eta_function#Special_values|exact value]],

:<math>\eta(2i) = \frac{\Gamma \left(\frac14\right)}{2^{11/8} \pi^{3/4}} </math>

implying the [[transcendental numbers]],

:<math>g_2(2i) = \frac{11\,\Gamma \left(\frac14\right)^8}{2^{8} \pi^2},\qquad g_3(2i) = \frac{7\,\Gamma \left(\frac14\right)^{12}}{2^{12} \pi^3}</math>

but yielding the [[algebraic number]] (in fact, an [[integer]]),

:<math>j(2i) = 1728 \frac{g_2(2i)^3}{g_2(2i)^3 - 27g_3(2i)^2} = 66^3.</math>

In general, this can be motivated by viewing each {{math|τ}} as representing an isomorphism class of elliptic curves. Every elliptic curve {{mvar|E}} over {{math|'''C'''}} is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of {{math|'''C'''}}. This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by {{math|1}} and {{mvar|τ}}{{math| ∈ '''H'''}}. This lattice corresponds to the elliptic curve <math>y^2=4x^3-g_2(\tau)x-g_3(\tau)</math> (see [[Weierstrass's elliptic functions#General theory|Weierstrass elliptic functions]]).

Note that {{mvar|j}} is defined everywhere in {{math|'''H'''}} as the modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots.

==The fundamental region==
[[File:ModularGroup-FundamentalDomain.svg|thumb|400px|The usual choice of a fundamental domain (gray) for the modular group acting on the upper half plane.]]

It can be shown that {{math|Δ}} is a [[modular form]] of weight twelve, and {{math|''g''<sub>2</sub>}} one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore {{mvar|j}}, is a modular function of weight zero, in particular a holomorphic function {{math|'''H''' → '''C'''}} invariant under the action of {{math|SL(2, '''Z''')}}. Quotienting out by its centre {{math|{ ±I <nowiki>}</nowiki>}} yields the [[modular group]], which we may identify with the [[projective special linear group]] {{math|PSL(2, '''Z''')}}.

By a suitable choice of transformation belonging to this group,

:<math> \tau \mapsto \frac{a\tau + b}{c\tau +d}, \qquad ad-bc =1,</math>

we may reduce {{mvar|τ}} to a value giving the same value for {{mvar|j}}, and lying in the [[fundamental region]] for {{mvar|j}}, which consists of values for {{mvar|τ}} satisfying the conditions

:<math>\begin{align}
        |\tau| &\ge 1 \\[5pt]
  -\tfrac{1}{2} &< \mathfrak{R}(\tau) \le \tfrac{1}{2} \\[5pt]
  -\tfrac{1}{2} &< \mathfrak{R}(\tau) < 0 \Rightarrow |\tau| > 1
\end{align}</math>

The function {{math|''j''(''τ'')}} when restricted to this region still takes on every value in the [[complex number]]s {{math|'''C'''}} exactly once.  In other words, for every {{mvar|c}} in {{math|'''C'''}}, there is a unique τ in the fundamental region such that {{math|''c'' {{=}} ''j''(''τ'')}}. Thus, {{mvar|j}} has the property of mapping the fundamental region to the entire complex plane.

Additionally two values {{math|τ,τ' ∈'''H'''}}  produce the same elliptic curve iff {{math|τ {{=}} T(τ')}} for some {{math|T ∈ PSL(2, '''Z''')}}. This means {{math|''j''}} provides a bijection from the set of elliptic curves over {{math|'''C'''}} to the complex plane.<ref>Gareth A. Jones and David Singerman. (1987)  Complex functions: an algebraic and geometric viewpoint.  Cambridge UP.  [https://books.google.com/books?id=jJhWM4vAyVMC]</ref>

As a [[Riemann surface]], the fundamental region has genus {{math|0}}, and every ([[Modular form#Definition|level one]]) modular function is a [[rational function]] in {{mvar|j}}; and, conversely, every rational function in {{mvar|j}} is a modular function. In other words, the field of modular functions is {{math|'''C'''(''j'')}}.

==Class field theory and {{mvar|''j''}}==
{{further|Complex multiplication|Hilbert's twelfth problem}}
The {{mvar|j}}-invariant has many remarkable properties:

*If {{mvar|τ}} is any point of the upper half plane whose corresponding elliptic curve has [[complex multiplication]] (that is, if {{mvar|τ}} is any element of an imaginary [[quadratic field]] with positive imaginary part, so that {{mvar|j}} is defined), then {{math|''j''(''τ'')}} is an [[algebraic integer]].<ref>{{cite book | first=Joseph H. | last=Silverman | author-link=Joseph H. Silverman | title=The Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=106 | year=1986 | isbn=978-0-387-96203-0 | zbl=0585.14026 | page=339 }}</ref> These special values are called [[singular moduli]].
* The field extension {{math|'''Q'''[''j''(''τ''), ''τ'']/'''Q'''(''τ'')}} is abelian, that is, it has an abelian [[Galois group]].
* Let {{math|Λ}} be the lattice in {{math|'''C'''}} generated by {{math|{1, ''τ''}.}} It is easy to see that all of the elements of {{math|'''Q'''(''τ'')}} which fix {{math|Λ}} under multiplication form a ring with units, called an [[order (ring theory)|order]]. The other lattices with generators {{math|{1, ''τ{{prime}}''},}} associated in like manner to the same order define the [[conjugate element (field theory)|algebraic conjugate]]s {{math|''j''(''τ{{prime}}'')}} of {{math|''j''(''τ'')}} over {{math|'''Q'''(''τ'')}}. Ordered by inclusion, the unique maximal order in {{math|'''Q'''(''τ'')}} is the ring of algebraic integers of {{math|'''Q'''(''τ'')}}, and values of {{mvar|τ}} having it as its associated order lead to [[unramified extension]]s of {{math|'''Q'''(''τ'')}}.

These classical results are the starting point for the theory of complex multiplication.

==Transcendence properties==
In 1937 [[Theodor Schneider]] proved the aforementioned result that if {{mvar|τ}} is a quadratic irrational number in the upper half plane then {{math|''j''(''τ'')}} is an algebraic integer.  In addition he proved that if {{mvar|τ}} is an [[algebraic number]] but not imaginary quadratic then {{math|''j''(''τ'')}} is transcendental.

The {{mvar|j}} function has numerous other transcendental properties.  [[Kurt Mahler]] conjectured a particular transcendence result that is often referred to as Mahler's conjecture, though it was proved as a corollary of results by Yu. V. Nesterenko and Patrice Phillipon in the 1990s.  Mahler's conjecture (now proven) is that, if {{mvar|τ}} is in the upper half plane, then {{math|''e''<sup>2π''iτ''</sup>}} and {{math|''j''(''τ'')}} are never both simultaneously algebraic. Stronger results are now known, for example if {{math|''e''<sup>2π''iτ''</sup>}} is algebraic then the following three numbers are algebraically independent, and thus at least two of them transcendental:

:<math>j(\tau), \frac{j^\prime(\tau)}{\pi}, \frac{j^{\prime\prime}(\tau)}{\pi^2}</math>

==The {{mvar|q}}-expansion and moonshine==
Several remarkable properties of {{mvar|j}} have to do with its [[q-expansion|{{mvar|q}}-expansion]] ([[Fourier series]] expansion), written as a [[Laurent series]] in terms of {{math|''q'' {{=}} ''e''<sup>2π''iτ''</sup>}}, which begins:

:<math>j(\tau) = q^{-1} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \cdots</math>

Note that {{mvar|j}} has a [[simple pole]] at the cusp, so its {{mvar|q}}-expansion  has no terms below {{math|''q''<sup>−1</sup>}}.

All the Fourier coefficients are integers, which results in several [[almost integer]]s, notably [[Ramanujan's constant]]:

:<math>e^{\pi \sqrt{163}} \approx 640320^3 + 744</math>.

The [[asymptotic formula]] for the coefficient of {{math|''q<sup>n</sup>''}} is given by

:<math>\frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}</math>,

as can be proved by the [[Hardy–Littlewood circle method]].<ref>{{cite journal|first=Hans|last=Petersson|author-link=Hans Petersson|title=Über die Entwicklungskoeffizienten der automorphen Formen|journal=Acta Mathematica|volume=58|issue=1|year=1932|pages=169–215|mr=1555346|doi=10.1007/BF02547776|doi-access=free}}</ref><ref>{{cite journal|first=Hans|last=Rademacher|author-link=Hans Rademacher|title=The Fourier coefficients of the modular invariant j(τ)|journal=American Journal of Mathematics|volume=60|year=1938|pages=501–512|mr=1507331|issue=2|doi=10.2307/2371313|jstor=2371313}}</ref>

===Moonshine===
More remarkably, the Fourier coefficients for the positive exponents of {{mvar|q}} are the dimensions of the graded part of an infinite-dimensional [[graded algebra]] representation of the [[monster group]]  called the ''[[moonshine module]]'' – specifically, the coefficient of {{math|''q<sup>n</sup>''}} is the dimension of grade-{{mvar|n}} part of the moonshine module, the first example being the [[Griess algebra]], which has dimension 196,884, corresponding to the term {{math|196884''q''}}. This startling observation, first made by [[John McKay (mathematician)|John McKay]], was the starting point for [[moonshine theory]].

The study of the Moonshine conjecture led [[John Horton Conway]] and [[Simon P. Norton]] to look at the genus-zero modular functions. If they are normalized to have the form

:<math>q^{-1} + {O}(q)</math>

then [[John G. Thompson]] showed that there are only a finite number of such functions (of some finite level), and Chris J. Cummins later showed that there are exactly 6486 of them, 616 of which have integral coefficients.<ref name=Cum04>{{cite journal | last=Cummins | first=Chris J. | title=Congruence subgroups of groups commensurable with ''PSL''(2,'''Z''')$ of genus 0 and 1 | journal=Experimental Mathematics | volume=13 | number=3 | pages=361–382 | year=2004 | issn=1058-6458 | zbl=1099.11022 | doi=10.1080/10586458.2004.10504547| s2cid=10319627 | url=http://projecteuclid.org/euclid.em/1103749843 }}</ref>

==Alternate expressions==
We have

:<math>j(\tau) = \frac{256\left(1-x\right)^3}{x^2} </math>

where {{math|''x'' {{=}} ''λ''(1 − ''λ'')}} and {{mvar|λ}} is the [[modular lambda function]]

:<math> \lambda(\tau) = \frac{\theta_2^4(e^{\pi i\tau})}{\theta_3^4(e^{\pi i\tau})} = k^2(\tau)</math>

a ratio of [[theta function|Jacobi theta functions]] {{math|''θ<sub>m</sub>''}}, and is the square of the elliptic modulus {{math|''k''(''τ'')}}.<ref name=C108>Chandrasekharan (1985) p.108</ref> The value of {{mvar|j}} is unchanged when {{mvar|λ}} is replaced by any of the six values of the [[cross-ratio]]:<ref name=C110>{{citation | last=Chandrasekharan | first=K. | author-link=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=978-3-540-15295-8 | zbl=0575.33001 | page=110 }}</ref>

:<math>\left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace</math>

The branch points of {{mvar|j}} are at {{math|{0, 1, ∞<nowiki>}</nowiki>}}, so that {{mvar|j}} is a [[Belyi function]].<ref>{{citation | last1=Girondo | first1=Ernesto | last2=González-Diez | first2=Gabino | title=Introduction to compact Riemann surfaces and dessins d'enfants | series=London Mathematical Society Student Texts | volume=79 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2012 | isbn=978-0-521-74022-7 | zbl=1253.30001 | page=267 }}</ref>

==Expressions in terms of theta functions==

Define the [[Nome (mathematics)|nome]] {{math|''q'' {{=}} ''e''<sup>π''iτ''</sup>}} and the [[theta function|Jacobi theta function]],

:<math>\vartheta(0; \tau) = \vartheta_{00}(0; \tau) = 1 + 2 \sum_{n=1}^\infty \left(e^{\pi i\tau}\right)^{n^2} = \sum_{n=-\infty}^\infty q^{n^2}</math>

from which one can derive the auxiliary theta functions, defined [[Theta function#Auxiliary functions|here]]. Let,

:<math>\begin{align}
  a &= \theta_{2}(q) = \vartheta_{10}(0; \tau) \\
  b &= \theta_{3}(q) = \vartheta_{00}(0; \tau) \\
  c &= \theta_{4}(q) = \vartheta_{01}(0; \tau)
\end{align}</math>

where {{math|''ϑ<sub>ij</sub>''}} and {{math|''θ<sub>n</sub>''}} are alternative notations, and {{math|''a''<sup>4</sup> − ''b''<sup>4</sup> + ''c''<sup>4</sup> {{=}} 0}}.  Then we have the for [[Weierstrass's elliptic functions#The constants e1.2C e2 and e3|modular invariants]] {{math|''g''<sub>2</sub>}}, {{math|''g''<sub>3</sub>}}, 

:<math>\begin{align}
  g_2(\tau) &= \tfrac{2}{3}\pi^4 \left(a^8 + b^8 + c^8\right) \\
  g_3(\tau) &= \tfrac{4}{27}\pi^6 \sqrt{\frac{\left(a^8+b^8+c^8\right)^3-54\left(abc\right)^8}{2}} \\
\end{align}</math>

and modular discriminant,

:<math>\Delta = g_2^3-27g_3^2 = (2\pi)^{12} \left(\tfrac{1}{2}a b c\right)^8 = (2\pi)^{12}\eta(\tau)^{24}</math>

with [[Dedekind eta function]] {{math|''η''(''τ'')}}. The {{math|''j''(''τ'')}} can then be rapidly computed,

:<math>j(\tau) = 1728\frac{g_2^3}{g_2^3-27g_3^2} = 32 \frac{\left(a^8 + b^8 + c^8\right)^3 }{ \left(a b c\right)^8}</math>

==Algebraic definition==
So far we have been considering {{mvar|j}} as a function of a complex variable. However, as an invariant for isomorphism classes of elliptic curves, it can be defined purely algebraically.<ref>{{Cite book|title=Elliptic functions|last=Lang|first=Serge|author-link=Serge Lang|publisher=Springer-Verlag|year=1987|isbn=978-1-4612-9142-8|series=Graduate Texts in Mathematics|volume=112|location=New-York ect|pages=299–300|zbl=0615.14018}}</ref> Let

:<math>y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>

be a plane elliptic curve ''over any field''. Then we may perform successive transformations to get the above equation into the standard form {{math|''y''<sup>2</sup> {{=}} 4''x''<sup>3</sup> − ''g''<sub>2</sub>''x'' − ''g''<sub>3</sub>}} (note that this transformation can only be made when the characteristic of the field is not equal to 2 or 3). The resulting coefficients are:

:<math>\begin{align}
b_2 &= a_1^2 + 4a_2,\quad  &b_4 &= a_1a_3 + 2a_4,\\
b_6 &= a_3^2 + 4a_6,\quad  &b_8 &= a_1^2a_6 - a_1a_3a_4 + a_2a_3^2 + 4a_2a_6 - a_4^2,\\
c_4 &= b_2^2 - 24b_4,\quad &c_6 &= -b_2^3 + 36b_2b_4 - 216b_6,
\end{align}</math>

where {{math|''g''<sub>2</sub> {{=}} ''c''<sub>4</sub>}} and {{math|''g''<sub>3</sub> {{=}} ''c''<sub>6</sub>}}.  We also have the [[discriminant#low degrees|discriminant]]

:<math>\Delta = -b_2^2b_8 + 9b_2b_4b_6 - 8b_4^3 - 27b_6^2.</math>

The {{mvar|j}}-invariant for the elliptic curve may now be defined as

:<math>j = \frac{c_4^3}{\Delta}</math>

In the case that the field over which the curve is defined has characteristic different from 2 or 3, this is equal to

:<math>j = 1728\frac{c_4^3}{c_4^3-c_6^2}.</math>

==Inverse function==
The [[inverse function]] of the {{mvar|j}}-invariant can be expressed in terms of the [[hypergeometric function]] {{math|<sub>2</sub>''F''<sub>1</sub>}} (see also the article [[Picard–Fuchs equation]]). Explicitly, given a number {{mvar|N}}, to solve the equation {{math|''j''(''τ'') {{=}} ''N''}} for {{mvar|τ}} can be done in at least four ways.

'''Method 1''': Solving the [[sextic]] in {{mvar|λ}},

:<math>j(\tau) = \frac{256\bigl(1-\lambda(1-\lambda)\bigr)^3}{\bigl(\lambda(1-\lambda)\bigr)^2} = \frac{256\left(1-x\right)^3}{x^2} </math>

where {{math|''x'' {{=}} ''λ''(1 − ''λ'')}}, and {{mvar|λ}} is the [[modular lambda function]] so the sextic can be solved as a cubic in {{mvar|x}}. Then,

:<math>\tau = i \ \frac{{}_2F_1 \left (\tfrac{1}{2},\tfrac{1}{2},1;1 - \lambda \right )}{{}_2F_1 \left (\tfrac{1}{2},\tfrac{1}{2},1;\lambda \right)}=i\frac{\operatorname{M}(1,\sqrt{1-\lambda})}{\operatorname{M}(1,\sqrt{\lambda})}</math>

for any of the six values of {{mvar|λ}}, where {{math|M}} is the [[arithmetic–geometric mean]].<ref group="note">The equality holds if the arithmetic–geometric mean <math>\operatorname{M}(a,b)</math> of [[complex number|complex numbers]] <math>a,b</math> (such that <math>a,b\ne 0;a\ne \pm b</math>) is defined as follows: Let <math>a_0=a</math>, <math>b_0=b</math>, <math>a_{n+1}=(a_n+b_n)/2</math>, <math>b_{n+1}=\pm\sqrt{a_nb_n}</math> where the signs are chosen such that <math>|a_n-b_n|\le|a_n+b_n|</math> for all <math>n\in\mathbb{N}</math>. If <math>|a_n-b_n|=|a_n+b_n|</math>, the sign is chosen such that <math>\Im (b_n/a_n)>0</math>. Then <math>\operatorname{M}(a,b)=\lim_{n\to\infty}a_n=\lim_{n\to\infty}b_n</math>. When <math>a,b</math> are positive real (with <math>a\ne b</math>), this definition coincides with the usual definition of the arithmetic–geometric mean for positive real numbers. See [https://www.researchgate.net/publication/248675540_The_Arithmetic-Geometric_Mean_of_Gauss The Arithmetic-Geometric Mean of Gauss] by [[David A. Cox]].</ref>

'''Method 2''': Solving the [[quartic function|quartic]] in {{mvar|γ}},

:<math>j(\tau) = \frac{27\left(1 + 8\gamma\right)^3}{\gamma\left(1 - \gamma\right)^3} </math>

then for any of the four [[Root of a function|roots]],

:<math>\tau = \frac{i}{\sqrt{3}} \frac{{}_2F_1 \left (\tfrac{1}{3},\tfrac{2}{3},1;1-\gamma \right)}{{}_2F_1 \left(\tfrac{1}{3},\tfrac{2}{3},1;\gamma \right )}</math>

'''Method 3''': Solving the [[cubic function|cubic]] in {{mvar|β}},

:<math>j(\tau) = \frac{64\left(1+3\beta\right)^3}{\beta\left(1-\beta\right)^2} </math>

then for any of the three roots,

:<math>\tau = \frac{i}{\sqrt{2}} \frac{{}_2F_1 \left (\tfrac{1}{4},\tfrac{3}{4},1;1-\beta \right)}{{}_2F_1 \left(\tfrac{1}{4},\tfrac{3}{4},1;\beta \right )}</math>

'''Method 4''': Solving the [[quadratic equation|quadratic]] in {{mvar|α}},

:<math>j(\tau)=\frac{1728}{4\alpha(1-\alpha)}</math>

then,

:<math>\tau = i \ \frac{{}_2F_1 \left (\tfrac{1}{6},\tfrac{5}{6},1;1-\alpha \right)}{{}_2F_1 \left(\tfrac{1}{6},\tfrac{5}{6},1;\alpha \right )}</math>

One root gives {{mvar|τ}}, and the other gives {{math|−{{sfrac|1|''τ''}}}}, but since {{math|''j''(''τ'') {{=}} ''j''(−{{sfrac|1|''τ''}})}}, it makes no difference which {{mvar|α}} is chosen. The latter three methods can be found in [[Ramanujan]]'s theory of [[elliptic functions]] to alternative bases.

The inversion is applied in high-precision calculations of elliptic function periods even as their ratios become unbounded.{{cn|date=December 2021}} A related result is the expressibility via quadratic radicals of the values of {{mvar|j}} at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting [[compass and straightedge constructions]]). The latter result is hardly evident since the [[modular equation]] for {{math|''j''}} of order 2 is cubic.<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} Theorem 4.8</ref>

==Pi formulas==

The [[Chudnovsky algorithm|Chudnovsky brothers]] found in 1987,<ref>{{Citation|last1=Chudnovsky|first1=David V.|author1-link=Chudnovsky brothers|last2=Chudnovsky|first2=Gregory V.|author2-link=Chudnovsky brothers|title=The Computation of Classical Constants|year=1989|journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]]|issn=0027-8424|volume=86|issue=21|pages=8178–8182|doi=10.1073/pnas.86.21.8178|pmid=16594075|pmc=298242|jstor=34831|bibcode=1989PNAS...86.8178C|doi-access=free}}.</ref>

:<math>\frac{1}{\pi} = \frac{12}{640320^{3/2}} \sum_{k=0}^\infty \frac{(6k)! (163 \cdot 3344418k + 13591409)}{(3k)!\left(k!\right)^3 \left(-640320\right)^{3k}}</math>

a proof of which uses the fact that

:<math>j\left(\frac{1+\sqrt{-163}}{2}\right) = -640320^3.</math>

For similar formulas, see the [[Ramanujan–Sato series]].

== Failure to classify elliptic curves over other fields ==
The <math>j</math>-invariant is only sensitive to isomorphism classes of elliptic curves over the complex numbers, or more generally, an [[algebraically closed field]]. Over other fields there exist examples of elliptic curves whose <math>j</math>-invariant is the same, but are non-isomorphic. For example, let <math>E_1,E_2</math> be the elliptic curves associated to the polynomials<blockquote><math>\begin{align}
E_1: &\text{ } y^2 = x^3 - 25x \\
E_2: &\text{ } y^2 = x^3 - 4x,
\end{align}</math></blockquote>both having <math>j</math>-invariant <math>1728</math>. Then, the rational points of <math>E_2</math> can be computed as:<blockquote><math>E_2(\mathbb{Q}) = \{\infty, (2,0), (-2,0), (0,0) \}</math></blockquote>since <math>x^3 - 4x = x(x^2 - 4) = x(x-2)(x+2).
</math> There are no rational solutions with <math>y = a \neq 0</math>. This can be shown using [[Cardano's formula]] to show that in that case the solutions to <math>x^3 - 4x - a^2</math> are all irrational. 

On the other hand, on the set of points<blockquote><math>\{ n(-4,6) : n \in \mathbb{Z} \}</math></blockquote> the equation for <math>E_1</math> becomes <math>36n^2 = -64n^3 + 100n </math>. Dividing by <math>4n</math> to eliminate the <math>(0,0)</math> solution, the quadratic formula gives the rational solutions: <blockquote><math>n =  \frac{
    -9 \pm \sqrt{81 - 4\cdot 16\cdot(-25)}
}{2\cdot 16} = 
\frac{-9 \pm 41}{32}.</math></blockquote>If these curves are considered over <math>\mathbb{Q}(\sqrt{10})</math>, there is an isomorphism <math>E_1(\mathbb{Q}(\sqrt{10})) \cong E_2(\mathbb{Q}(\sqrt{10}))</math> sending<blockquote><math>(x,y)\mapsto (\mu^2x,\mu^3y) \ \text{ where }\ \mu = \frac{\sqrt{10}}{2}.</math></blockquote>

==References==
===Notes===
{{reflist|group=note}}
===Other===
{{Reflist}}
*{{citation|first=Tom M.|last=Apostol|author-link=Tom M. Apostol|title=Modular functions and Dirichlet Series in Number Theory|mr=0422157|year=1976|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=41|location=New York}}. Provides a very readable introduction and various interesting identities.
**{{citation|first=Tom M.|last=Apostol|author-link=Tom M. Apostol|title=Modular functions and Dirichlet Series in Number Theory|volume=41|edition=2nd|year=1990|isbn=978-0-387-97127-8|mr=1027834|doi=10.1007/978-1-4612-0999-7|series=Graduate Texts in Mathematics|url-access=registration|url=https://archive.org/details/modularfunctions0000apos}}
*{{citation|doi=10.4153/CMB-1999-050-1|doi-access=free|first1=Bruce C.|last1=Berndt|author1-link=Bruce C. Berndt|first2=Heng Huat|last2=Chan|title=Ramanujan and the modular j-invariant|journal=[[Canadian Mathematical Bulletin]]|volume=42|issue=4|year=1999|pages=427–440|mr=1727340}}. Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.
*{{citation|first=David A.|last=Cox|author-link=David A. Cox|title=Primes of the Form x^2 + ny^2: Fermat, Class Field Theory, and Complex Multiplication|mr=1028322|year=1989|publisher= Wiley-Interscience Publication, John Wiley & Sons Inc.|location=New York}} Introduces the j-invariant and discusses the related class field theory.
*{{citation|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Simon|last2=Norton|author2-link=Simon P. Norton|title=Monstrous moonshine|journal=Bulletin of the London Mathematical Society|volume=11|issue=3|year=1979|pages=308–339|mr=0554399|doi=10.1112/blms/11.3.308}}. Includes a list of the 175 genus-zero modular functions.
*{{citation|first=Robert A.|last=Rankin|author-link=Robert Alexander Rankin|title=Modular forms and functions|year=1977|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-21212-0|mr=0498390}}. Provides a short review in the context of modular forms.
*{{citation|first=Theodor|last=Schneider|author-link=Theodor Schneider|title=Arithmetische Untersuchungen elliptischer Integrale|journal=Math. Annalen|volume=113|year=1937|pages=1&ndash;13|mr=1513075|doi=10.1007/BF01571618|s2cid=121073687}}.

[[Category:Modular forms]]
[[Category:Elliptic functions]]
[[Category:Moonshine theory]]