j-invariant
In mathematics, Felix Klein's Template:Mvar-invariant or Template:Mvar 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 numbers. It is the unique such function that is holomorphic away from a simple pole at the cusp such that
<math display="block">j\big(e^{2\pi i/3}\big) = 0, \quad j(i) = 1728 = 12^3.</math>
Rational functions 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 curves 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
The Template:Mvar-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 display="block">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, also since 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 display="block">g_2(\tau) = 60G_4(\tau) = 60\sum_{(m,n) \neq (0,0)} \left(m + n\tau\right)^{-4}</math> <math display="block">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 Fourier series,
<math display="block">\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,
<math display="block">\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). The Template:Mvar-invariant can then be directly expressed in terms of the Eisenstein series as,
<math display="block">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>Template:Cite arXiv The paper uses a non-equivalent definition of <math>\Delta</math>, but this has been accounted for in this article.</ref>
<math display="block">\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 exact value,
<math display="block">\eta(2i) = \frac{\Gamma \left(\frac14\right)}{2^{11/8} \pi^{3/4}} </math>
implying the transcendental numbers,
<math display="block">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 display="block">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 Template:Math as representing an isomorphism class of elliptic curves. Every elliptic curve Template:Mvar over Template:Math is a complex torus, and thus can be identified with a rank 2 lattice; that is, a two-dimensional lattice of Template:Math. This lattice can be rotated and scaled (operations that preserve the isomorphism class), so that it is generated by Template:Math and Template:MvarTemplate:Math. This lattice corresponds to the elliptic curve <math>y^2=4x^3-g_2(\tau)x-g_3(\tau)</math> (see Weierstrass elliptic functions).
Note that Template:Mvar is defined everywhere in Template:Math as the modular discriminant is non-zero. This is due to the corresponding cubic polynomial having distinct roots.
The fundamental region
It can be shown that Template:Math is a modular form of weight twelve, and Template:Math one of weight four, so that its third power is also of weight twelve. Thus their quotient, and therefore Template:Mvar, is a modular function of weight zero, in particular a holomorphic function Template:Math invariant under the action of Template:Math. Quotienting out by its centre Template:Math yields the modular group, which we may identify with the projective special linear group Template:Math.
By a suitable choice of transformation belonging to this group,
<math display="block"> \tau \mapsto \frac{a\tau + b}{c\tau +d}, \qquad ad-bc =1,</math>
we may reduce Template:Mvar to a value giving the same value for Template:Mvar, and lying in the fundamental region for Template:Mvar, which consists of values for Template:Mvar satisfying the conditions
<math display="block">\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 Template:Math when restricted to this region still takes on every value in the complex numbers Template:Math exactly once. In other words, for every Template:Mvar in Template:Math, there is a unique τ in the fundamental region such that Template:Math. Thus, Template:Mvar has the property of mapping the fundamental region to the entire complex plane.
Additionally two values Template:Math produce the same elliptic curve iff Template:Math for some Template:Math. This means Template:Math provides a bijection from the set of elliptic curves over Template:Math to the complex plane.<ref>Gareth A. Jones and David Singerman. (1987) Complex functions: an algebraic and geometric viewpoint. Cambridge UP. [1]</ref>
As a Riemann surface, the fundamental region has genus Template:Math, and every (level one) modular function is a rational function in Template:Mvar; and, conversely, every rational function in Template:Mvar is a modular function. In other words, the field of modular functions is Template:Math.
Class field theory and Template:Mvar
Template:Further The Template:Mvar-invariant has many remarkable properties:
- If Template:Mvar is any point of the upper half plane whose corresponding elliptic curve has complex multiplication (that is, if Template:Mvar is any element of an imaginary quadratic field with positive imaginary part, so that Template:Mvar is defined), then Template:Math is an algebraic integer.<ref>Template:Cite book</ref> These special values are called singular moduli.
- The field extension Template:Math is abelian, that is, it has an abelian Galois group.
- Let Template:Math be the lattice in Template:Math generated by Template:Math It is easy to see that all of the elements of Template:Math which fix Template:Math under multiplication form a ring with units, called an order. The other lattices with generators Template:Math associated in like manner to the same order define the algebraic conjugates Template:Math of Template:Math over Template:Math. Ordered by inclusion, the unique maximal order in Template:Math is the ring of algebraic integers of Template:Math, and values of Template:Mvar having it as its associated order lead to unramified extensions of Template:Math.
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 Template:Mvar is a quadratic irrational number in the upper half plane then Template:Math is an algebraic integer. In addition he proved that if Template:Mvar is an algebraic number but not imaginary quadratic then Template:Math is transcendental.
The Template:Mvar 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 Template:Mvar is in the upper half plane, then Template:Math and Template:Math are never both simultaneously algebraic. Stronger results are now known, for example if Template:Math is algebraic then the following three numbers are algebraically independent, and thus at least two of them transcendental:
<math display="block">j(\tau), \frac{j^\prime(\tau)}{\pi}, \frac{j^{\prime\prime}(\tau)}{\pi^2}</math>
The Template:Mvar-expansion and moonshine
Several remarkable properties of Template:Mvar have to do with its [[q-expansion|Template:Mvar-expansion]] (Fourier series expansion), written as a Laurent series in terms of Template:Math, which begins:
<math display="block">j(\tau) = q^{-1} + 744 + 196884 q + 21493760 q^2 + 864299970 q^3 + 20245856256 q^4 + \cdots</math>
Note that Template:Mvar has a simple pole at the cusp, so its Template:Mvar-expansion has no terms below Template:Math.
All the Fourier coefficients are integers, which results in several almost integers, notably Ramanujan's constant:
<math display="block">e^{\pi \sqrt{163}} \approx 640320^3 + 744.</math>
The asymptotic formula for the coefficient of Template:Math is given by
<math display="block">\frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}},</math>
as can be proved by the Hardy–Littlewood circle method.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
Moonshine
More remarkably, the Fourier coefficients for the positive exponents of Template:Mvar 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 Template:Math is the dimension of grade-Template:Mvar part of the moonshine module, the first example being the Griess algebra, which has dimension 196,884, corresponding to the term Template:Math. This startling observation, first made by 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 display="block">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>Template:Cite journal</ref>
Alternate expressions
We have
<math display="block">j(\tau) = \frac{256\left(1-x\right)^3}{x^2} </math>
where Template:Math and Template:Mvar is the modular lambda function
<math display="block"> \lambda(\tau) = \frac{\theta_2^4(e^{\pi i\tau})}{\theta_3^4(e^{\pi i\tau})} = k^2(\tau)</math>
a ratio of Jacobi theta functions Template:Math, and is the square of the elliptic modulus Template:Math.<ref name=C108>Chandrasekharan (1985) p.108</ref> The value of Template:Mvar is unchanged when Template:Mvar is replaced by any of the six values of the cross-ratio:<ref name=C110>Template:Citation</ref>
<math display="block">\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 Template:Mvar are at Template:Math, so that Template:Mvar is a Belyi function.<ref>Template:Citation</ref>
Expressions in terms of theta functions
Define the nome Template:Math and the Jacobi theta function,
<math display="block">\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 here. Let,
<math display="block">\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 Template:Math and Template:Math are alternative notations, and Template:Math. Then we have the for modular invariants Template:Math, Template:Math,
<math display="block">\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 display="block">\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 Template:Math. The Template:Math can then be rapidly computed,
<math display="block">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 Template:Mvar as a function of a complex variable. However, as an invariant for isomorphism classes of elliptic curves, it can be defined purely algebraically.<ref>Template:Cite book</ref> Let
<math display="block">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 Template:Math (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 display="block">\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 Template:Math and Template:Math. We also have the discriminant
<math display="block">\Delta = -b_2^2b_8 + 9b_2b_4b_6 - 8b_4^3 - 27b_6^2.</math>
The Template:Mvar-invariant for the elliptic curve may now be defined as
<math display="block">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 display="block">j = 1728\frac{c_4^3}{c_4^3-c_6^2}.</math>
Inverse function
The inverse function of the Template:Mvar-invariant can be expressed in terms of the hypergeometric function Template:Math (see also the article Picard–Fuchs equation). Explicitly, given a number Template:Mvar, to solve the equation Template:Math for Template:Mvar can be done in at least four ways.
Method 1: Solving the sextic in Template:Mvar,
<math display="block">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 Template:Math, and Template:Mvar is the modular lambda function so the sextic can be solved as a cubic in Template:Mvar. Then,
<math display="block">\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 Template:Mvar, where Template:Math is the arithmetic–geometric mean.<ref group="note">The equality holds if the arithmetic–geometric mean <math>\operatorname{M}(a,b)</math> of 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 The Arithmetic-Geometric Mean of Gauss by David A. Cox.</ref>
Method 2: Solving the quartic in Template:Mvar,
<math display="block">j(\tau) = \frac{27\left(1 + 8\gamma\right)^3}{\gamma\left(1 - \gamma\right)^3} </math>
then for any of the four roots,
<math display="block">\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 in Template:Mvar,
<math display="block">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 display="block">\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 in Template:Mvar,
<math display="block">j(\tau)=\frac{1728}{4\alpha(1-\alpha)}</math>
then,
<math display="block">\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 Template:Mvar, and the other gives Template:Math, but since Template:Math, it makes no difference which Template:Mvar 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.Template:Cn A related result is the expressibility via quadratic radicals of the values of Template:Mvar 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 Template:Math of order 2 is cubic.<ref>Template:Cite book Theorem 4.8</ref>
Pi formulas
The Chudnovsky brothers found in 1987,<ref>Template:Citation.</ref>
<math display="block">\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 display="block">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
<math display="block">\begin{align} E_1: &\text{ } y^2 = x^3 - 25x \\ E_2: &\text{ } y^2 = x^3 - 4x, \end{align}</math>
both having <math>j</math>-invariant <math>1728</math>. Then, the rational points of <math>E_2</math> can be computed as
<math display="block">E_2(\mathbb{Q}) = \{\infty, (2,0), (-2,0), (0,0) \}</math>
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
<math display="block">\{ n(-4,6) : n \in \mathbb{Z} \}</math>
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:
<math display="block">n = \frac{
-9 \pm \sqrt{81 - 4\cdot 16\cdot(-25)}
}{2\cdot 16} = \frac{-9 \pm 41}{32}.</math>
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
<math display="block">(x,y)\mapsto (\mu^2x,\mu^3y) \ \text{ where }\ \mu = \frac{\sqrt{10}}{2}.</math>
References
Notes
Other
- Template:Citation. Provides a very readable introduction and various interesting identities.
- Template:Citation. Provides a variety of interesting algebraic identities, including the inverse as a hypergeometric series.
- Template:Citation Introduces the j-invariant and discusses the related class field theory.
- Template:Citation. Includes a list of the 175 genus-zero modular functions.
- Template:Citation. Provides a short review in the context of modular forms.
- Template:Citation.