Curious Properties of j-Function Formulas By Titus Piezas III Contents: I. Introduction II. Other Formulas for the j-function: prime order n = 2,3,5,7,13 III. Some Conjectures for prime order n IV. Square orders n = 22 , 32 , 52 V. Continued Fractions and the j-function VI. Conclusion

I. Introduction The most famous value of the j-function j(τ) is at the argument τ = (1+√-163)/2 and which explains the amazing approximation involving eπ√163 also known as Ramanujan’s constant. Not so well known though is when τ = (1+√-427)/2 giving the even closer approximation, eπ√427 = 52803 (236674+30303√61)3 + 743.99999999999999999999… which goes for 22 nines before other numbers appear. These two are the largest negative discriminants D (as absolute values) with class number 1,2, respectively. The j-function can be calculated in Mathematica using the command N[1728KleinInvariantJ[τ], n] for arbitrary precision n. Note that the number 1728 = 123 (or more fundamentally, 12 and 24) interestingly will appear a lot when dealing with certain elliptic functions. More formally, the j-function is the modular function defined by, j(τ) = 1728J(τ) where J(τ) is Klein’s absolute invariant, J (τ ) =

4( y 2 − y + 1) 3 27 ( y 2 − y ) 2

with y = λ(τ) the elliptic lambda function, and τ as the half-period ratio. While this is one wellknown definition, other formulas of similar simplicity can be given using Eisenstein series as well as the Dedekind eta function. For the latter, it turns out that given a genus zero group Γ and a genus zero subgroup G = Γ0 (n), then a modular relation can be found between the j-function and a Hauptmodul associated to G ⊂ Γ. Such relations can be found explicitly for subgroup G with n1 dividing 24 (Chen, Yui, p.260). There are then exactly eight possible orders, which we can divide into prime 1 n = 2,3,5,9,13 and square n = 4,9,25. II. Other Formulas for the j-function: prime order n = 2,3,5,7,13

For the sake of brevity, j(τ) will be given simply as j. The primary formula given above is then, j=

4 4 ( y 2 − y + 1) 3 ( y 2 − y) 2

As a sextic equation in y, this has the polynomial discriminant D as, D = 2 −50 ( j − 1728 ) 3 j 4

a form which should become familiar later. But there are other formulas given by, ( x − 16 ) 3 x (x + 3) 3 ( x + 27 ) j= x 2 (x + 10 x + 5) 3 j= x 2 ( x + 5 x + 1)3 ( x 2 + 13 x + 49) j= x ( x 4 + 7 x 3 + 20 x 2 + 19 x + 1) 3 ( x 2 + 5 x + 13 ) j= x j=

(eq.1) (eq.2) (eq.3) (eq.4) (eq.5)

for orders n = 2,3,5,7,13, respectively (Yui, Chen, p. 260, with small modifications). Note that as equations in x, these have degree n+1. They also have the beautifully consistent and simple polynomial discriminants Dn . D2 = 2 2 ( j − 1728 ) j 2 D3 = −3 3 ( j − 1728 ) 2 j 2 D5 = 5 5 ( j − 1728 ) 2 j 4 D7 = −7 7 ( j − 1728 ) 4 j 4 D13 = 1313 ( j − 1728 ) 6 j 8

As formulas, they have the obvious general form j = Numerato r/Denominato r and the surprising thing is there seems to be a relationship between these two. Defining the expression S n as, S n = – Numerator + 1728Denominator then S n is either a square or a near-square! Explicitly, these are, S 2 = ( x − 64)( x + 8) 2 S 3 = ( x 2 + 18 x − 27 ) 2 S 5 = ( x 2 + 22 x + 125 )(x 2 + 4 x − 1) 2

S 7 = ( x 4 + 14 x 3 + 63 x 2 + 70 x − 7) 2 S13 = ( x 2 + 6 x + 13)( x 6 + 10 x 5 + 46 x 4 + 108 x 3 + 122 x 2 + 38 x − 1) 2

Note that when n = 4m-1, S n is a perfect square. Some of these polynomials surprisingly can be found in contexts other than as a j-function formula. S 5 appears naturally in 5th degree equations when transforming the Brioschi into the De Moivre quintic (see “Solvable Brioschi quintics, other one-parameter forms, and the j -function” by this author) while S 7 , after scaling, appears in Klein’s approach to solving the general 7th degree equation which uses, as well as other things, the group theoretic properties of the heptagon (see his “On the order-seven transformation of elliptic functions”, p. 22, given in the references). Perhaps S 13 may yet turn up in order-thirteen identities of elliptic functions! What then are the x? While the formula in the Introduction used the elliptic lambda function, the x here are expressed in terms of the Dedekind eta function, η(τ). This is usually defined in terms of the infin ite product, η (τ ) = q 1 / 24

∞

∏ (1 − q

n

)

n

where q = exp(2πiτ) and n = 1 to ∞. It was first introduced in 1877 by W.R. Dedekind (18311916) and can be interpreted as the 24th root of the modular discriminant of an elliptic curve. It is then perhaps not surprising the integers 12 and 24 are so ubiquitous. Defining the following functions f i : For n = 2: f 2 (τ ) = 21 / 2

η (2τ ) , η (τ )

f 1− k (τ ) = exp(−πik / 24 )

η (τ k ) η (τ )

where τ k = (τ + k ) / 2 for k = 0 to 1, then three roots x of eq.1 are x = f i 24 for i = 0 to 2. The case n = 2 involve the Weber functions, f, f1 , f 2 (Yui, Zagier, p.1646) named after Heinrich Weber (1842-1913) and the higher cases are just analogous versions with some differences since they are now odd primes. For n = 3: f 33 (τ ) = 31 / 2

η (3τ ) , η (τ )

f 30+ k (τ ) = exp(πik / 12 )

η (τ k ) η (τ )

where τ k = (τ + k ) / 3 for k = 0 to 2, then four roots x of eq.2 are x = f i 12 for i = 30 to 33. This case in turn was discussed by Ramanujan for particular τ. (See “Extending Ramanujan’s Dedekind eta quotients”.) For n = 5: f 55 (τ ) = 51 / 2

η (5τ ) , η(τ )

f 50+ k (τ ) = exp(πik / 6)

η (τ k ) η (τ )

where τ k = (τ + 2k ) / 5 for k = 0 to 4, then six roots x of eq.3 are x = f i 6 for i = 50 to 55. For n = 7: f 77 (τ ) = 7 1/ 2

η (7τ ) , η (τ )

f 70+ k (τ ) = exp(πik / 4 )

η (τ k ) η (τ )

where τ k = (τ + 3 k ) / 7 for k = 0 to 6, then eight roots x of eq.4 are x = f i 4 for i = 70 to 77. For n = 13: f 143 (τ ) = 131 / 2

η (13τ ) , η (τ )

f 130+ k (τ ) = exp(πik / 2)

η (τ k ) η (τ )

where τ k = (τ + 6 k ) / 13 for k = 0 to 12, then fourteen roots x of eq.5 are x = f i 2 for i = 130 to 143. For the odd primes n = 3,5,7,13, by looking at how the functions are defined, one can see that they have a general form determined by n. III. Some Conjectures for prime order n The Weber functions obey the known product and sum identities, f f 1 f 2 = √2;

f 18 + f 2 8 = f 8

and the others for higher n seem to follow analogous versions. I have no rigorous proof for the following seven observations, just empirical data. If you can prove (or disprove) them, I’d very much appreciate to know. These are: (f 30 f 31 f 32 f 33)3 = -3√3 f 50 f 51 f 52 f 53 f 54 f 55 = √5 f 70 f 71 f 72 f 73 f 74 f 75f 76 f 77 = √7 f 130 f131 f132… f 143 = √13 and as sums, f 30 6 + f 31 6 + f 326 = f 33 6 f 50 m + f 51 m + f52 m + f 53 m + f 54 m + f55 m = 0, for m = 2,4,8 f 70 2 + f 71 2 + f 722 + f 732 + f 742 + f75 2 + f76 2 = f 77 2 IV. Square orders n = 22 , 3 2, 5 2

For the square orders n = m2 , two formulas per n will be given. Technically, there should be three, involving the three possible eta quotients, η (τ / m 2 ) η (τ / m) η (τ ) , , η (τ ) η (m τ ) η (m 2 τ )

though only the first two will be given. For n = 22 : j=

( y 2 − 48) 3 y − 64 2

j=

,

( z 2 + 192 ) 3 ( z 2 − 64) 2

8

8

 η (τ / 4)   η(τ / 2)   + 8 , and z =   + 8 . with y =   η (τ )   η (2τ ) 

For n = 32 : j=

y 3 ( y 3 − 24 ) 3 y 3 − 27

,

j=

z 3 ( z 2 + 216 ) 3 ( z 3 − 27 ) 3

3

3

 η (τ / 9 )   η(τ / 3)   + 3 , and z =   + 3 . with y =   η (τ )   η (3τ ) 

For n = 52 : j =−

(s 20 + 12 s 15 + 14 s 10 − 12 s 5 + 1) 3 s 25 ( s10 + 11s 5 − 1)

with s −1 − s =

,

η (τ / 25 ) η(τ / 5) + 1 , and r −1 − r = +1. η(τ ) η (5τ )

The discriminants, again, are beautifully simple, D4a = −216 ( j − 1728 ) 3 j 4 D4b = 2 70 ( j − 1728 ) 3 j 4 D9a = −3 27 ( j − 1728 ) 6 j 8 D9b = −3147 ( j − 1728 ) 6 j 8 D25a = 5125 ( j − 1728 ) 30 j 40 D25b = 5 785 ( j − 1728 ) 30 j 40

and S n as,

j =−

(r 20 − 228 r 15 + 494 r 10 + 228 r 5 + 1)3 r 5 (r 10 + 11r 5 − 1) 5

S 4 a = − ( y 3 − 72 y ) 2 S 4b = −( z 3 − 576 z) 2 S 9a = −( y 6 − 36 y 3 + 216 ) 2 S 9b = − ( z 6 − 540 z 3 − 5832 ) 2 S 25a = ( s 30 + 18 s 25 + 75 s 20 + 75 s10 − 18 s 5 + 1) 2 S 25b = (r 30 + 522 r 25 − 10005 r 20 − 10005 r 10 − 522 r 5 + 1) 2

which are all squares! The last, S 25b , is in fact a polynomial invariant of the icosahedral group H120 (Dickson, p. 235). Note that these polynomials are also highly factorable. V. Continued Fractions and the j-function In Duke’s “Continued Fractions and Modular Functions” he discusses the following continued fractions, other than the second one. The first, the Rogers-Ramanujan R(q) is given as, R( q) =

q1 / 5 q 1+ q2 1+ q3 1+ 1 + ...

while the Ramanujan cubic continued fraction C(q) is, C (q ) =

q1 / 3 q + q2 1+ q2 + q4 1+ q3 + q6 1+ 1 + ...

and the Ramanujan-Gordon-Gollnitz V(q), q1 / 2

V ( q) =

q + q2

1+

q4

1+ 1+

q3 + q6 1+

q8 1 + ...

finally the octic continued fraction U(q),

U (q ) =

2 q 1/ 8 q

1+

q2

1+q + 1 + q2 +

q3 1 + q 3 + ...

the last of which also studied by Ramanujan. Surprisingly, for the form q = e2πiτ = exp(2πiτ) these particular subsets, call it r(τ), c(τ), v(τ), u(τ), or simply r, c, v, u for short, an explicit modular relation can be found between it and the j-function j(τ)! Simply stated, one can find more formulas for j in terms of r, c, v, u. These are given by, j =−

j=

j=

j=

(r 20 − 228 r 15 + 494 r 10 + 228 r 5 + 1)3 r 5 (r 10 + 11r 5 − 1) 5

(256 c 12 + 256 c 9 + 960 c 6 + 232 c 3 + 1) 3 (c 4 + c) 3 (8c 3 − 1) 6 4( x 4 + 60 x 3 + 134 x 2 + 60 x + 1) 3 x ( x + 1) 2 ( x − 1)8

, where 4 x + 2 = v 2 + v −2

16 (u 16 + 14u 8 + 1) 3 u 8 (u 8 − 1) 4

the first we’ve come across before, the second was derived by this author using a formula of Ramanujan’s based on (ultimately) the Weber functions, while the third and fourth ones can be found in Duke’s paper (p.21). These have the familiar discriminants, Dr = 5 785 ( j − 1728 ) 30 j 40 Dc = −2 −11043 495 ( j − 1728 )18 j 24 Dx = 2 188 ( j − 1728 ) 6 j 8 Du = 2 272 ( j − 1728 ) 24 j 32

S n which are all squares, S r = (r 30 + 522 r 25 − 10005 r 20 − 10005 r10 − 522 r 5 + 1) 2 S c = −( 4096 c 18 + 6144 c 15 − 30720 c 12 − 24640 c 9 − 12072 c 6 − 516 c 3 + 1) 2 S x = −2 2 ( x 6 − 126 x 5 − 1041 x 4 − 1764 x 3 − 1041 x 2 − 126 x + 1) 2 S u = −4 2 (u 24 − 33u 16 − 33u 8 + 1) 2

where the first and last (up to scaling) are polynomial invariants of the icosahedral group H120 and the octahedral group H48 (Dickson, p. 230). The variables can also be expressed in terms of η(τ) where as we’ve seen r is a quadratic relation,

r −1 − r =

η(τ / 5) +1 η (5τ )

but c is now a cubic relation, 3

4c 3 + 1  η (τ / 3)   + 3 =  c  η (3τ ) 

and the third and fourth (Duke, p.21-22) are, v 2 + v −2 =

η 2 (4τ )η 4 (τ ) 4η 2 (2τ )η 4 (8τ )

+ 6,

u=

2η (τ )η 2 (4τ ) η 3 (2τ )

VI. Conclusion The purpose of this paper was simply to present certain observations about some formulas for the j-function which may lead to further investigation on aspects of it. Some obvious questions to ask would be: 1. Given a formula for j a rational polynomial in some function x. If seen as an equation in x, is the discriminant always as simple as just a constant multiplied by powers of j and (j 1728)? 2. Why is the expression Sn a near-square or even sometimes a square? Another would be if there were other continued fractions that could be used as a formula for j. There is in fact one associated with the case n = 2, the Weber functions (Duke, p.23) which leads to the question, 3. Are there continued fractions for each prime order n = 2,3,5,7,13? The Rogers-Ramanujan r can be connected to Γ(5), this author has speculated in a previous paper that Ramanujan’s cubic c is for n = 3 (or n = 32 ), so that just leaves n = 7,13 and it would be interesting to know if there are continued fractions for them. Finally, there are approximations of a different form, two of which are, eπ√58 = 3964 – 104.000000177… eπ√190 = (12√19)4 (481+340√2)4 – 104.000000000000… which goes for 15 zeros before other numbers appear. But those use still another formula… --End-Footnotes:

1. In Ken Ono’s “P -adic properties of values of the modular j-function”, in Theorem 2 he discusses p-adic numbers for p = 2,3,5,7. I’ve always been meaning to email him that there might be a p=13 as well.)

© Titus Piezas III June 15, 2006 [email protected] (Pls. remove “III”) www.geocities.com/titus_piezas/ramanujan.html References: 1. Dickson, L., “Algebraic Theories”, Dover Publications, Inc., NY, 1959. 2. Duke, W., “Continued Fractions and Modular Functions”, Bulletin in the AMS, www.math.ucla.edu/~duke/preprints/bams4.pdf. 3. Klein, F., “On the order-seven transformation of elliptic functions” in “The Eightfold Way: The Beauty of Klein’s Quartic Curve”. 4. Ono, K., and Papanikolas, M., “P-adic properties of values of the modular j-function”, http://www.math.wisc.edu/~ono/reprints/079.pdf 5. Piezas, T., “Solvable Brioschi quintics, other one-parameter forms, and the j-function”, http://www.geocities.com/titus_piezas/Brioschi_page.html 6. Piezas, T., “Extending Ramanujan’s Dedekind eta quotients (Part I)”, http://www.geocities.com/titus_piezas/ramanujan_page5.html 7. Weisstein, E., “Klein’s Absolute Invariant”, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, 1999, or at http://mathworld.wolfram.com 8. Yui, N., and Chen., I, “Singular values of Thompson Series”, Mathematical Research Institute of Ohio State Uni., Monster Bash, 1993. 9. Yui, N., and Zagier, N., “On the singular values of Weber modular functions”, Math. of Computation, Vol. 66, No 220, Oct 1997.