Timeline for Computing the q-series of the j-invariant
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 1, 2011 at 19:16 | comment | added | Dan Kneezel | @N. Elkies: Thank you for your detailed comments. They have been quite enlightening! | |
| Jul 31, 2011 at 12:48 | comment | added | Noam D. Elkies | @D.Kneezel on $j = f(h_5)$: Yes, modulo a typo -- the next-to-last coefficient is $6\cdot 5^{13}$, not $63\cdot 5^{13}$, making it $j = (h_5^2+250h_5+5^5)^3/h_5^5$. In general, because each $\Gamma_0(e)$ is contained with finite index in $\Gamma(1)$, the $\Gamma(1)$ invariant $j$ is a rational function on $X_0(e)$, of degree $[\Gamma(1):\Gamma_0(e)]=:d_e$, say. When $X_0(e)$ is also rational, with coordinate $h_e$, this means $j$ is a rational function of $h_e$ with degree $d_e$. These rational functions show up in several contexts, but that's a bigger topic for a different time... | |
| Jul 31, 2011 at 7:52 | comment | added | Dan Kneezel | @N. Elkies: Oh! The multiplicator in the Zuckerman reference is a relation between $h_5$ and $j$. In particular, $$j(\tau)=h_5+6\cdot 5^3+\frac{63\cdot 5^5}{h_5}+\frac{52\cdot 5^8}{h_5^2}+\frac{63\cdot 5^{10}}{h_5^3}+\frac{63\cdot 5^{13}}{h_5^4}+\frac{5^{15}}{h_5^5}.$$ | |
| Jul 31, 2011 at 4:32 | comment | added | Noam D. Elkies | @D.Kneezel, cont'd: these are $e=2,3,4,5,7,9,13,25$. For example, the first of these, $e=2$, makes $$ h_e = q^{-1} - 24 + 276q - 2048q^2 - + \cdots,\phantom{and} h_e + e^{12/(e-1)}/h_e = q^{-1} - 24 + 4372*q + 96256q^2 + \cdots; $$ and for the last, $e=25$, it's $$ h_e = q^{-1} - 1 - q + q^4 + q^6 - q^{11} - q^{14} \cdots,\phantom{and} h_e + e^{12/(e-1)}/h_e = q^{-1} - 1 + 4q + 5q^2 + 10q^3 + 16q^4 + 25q^5 + 36q^6 + \cdots. $$ Any of this look familiar? [Nicely enough $e^{12/(e-1)}$ is an integer even in the two cases $e=9$, $e=25$ when the exponent $12/(e-1)$ isn't!] | |
| Jul 31, 2011 at 4:25 | comment | added | Noam D. Elkies | @D.Kneezel: all these series can be computed without invoking Moonshine, because they parametrize modular curves, but I don't know of a uniform formula. Nor do I know exactly which modular function each $T_g$ corresponds to, but there are uniform formulas in some cases, e.g. for the 8 cases of $(e-1) | 24$ there are series $h_e$ parametrizing $X_0(e)$, namely $(\eta(q)/\eta(q^e))^{24/(e-1)}$, and also $h_e + e^{12/(e-1)}/h_e$ parametrizing the quotient of $X_0(e)$ by the Atkin-Lehner involution; at least some of these should be within an additive constant of $T_g$ with $g$ of order $e$. | |
| Jul 31, 2011 at 4:14 | comment | added | Dan Kneezel | Can this approach be extended to Thompson series $T_g(q)$ away from $g = id$? | |
| Jul 31, 2011 at 3:55 | comment | added | Noam D. Elkies | I see that while I was typing this A.Lozano-Robledo expanded on his answer to include some of the above information. | |
| Jul 31, 2011 at 3:54 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |