27
votes
Accepted
12
votes
Accepted
Effective bound on the expansion of the $j$-invariant
Once you know that the coefficients are all positive (see postscript),
it's easy to get an effective upper bound that grows as $\exp(4\pi \sqrt{n})$,
which is within a factor $O(\sqrt n)$ of the ...
- 79.9k
9
votes
Effective bound on the expansion of the $j$-invariant
By a variation of Elkies's answer we can even get $a_n<e^{4\pi\sqrt{n}}$ without using $j(i)=1728$.
For $n=1$ the claim is clear. Now let $0<t<1$ and use the identity $j(it)=j(i/t)$. After ...
- 105k
5
votes
Accepted
Uniqueness of the $J$ invariant
Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...
- 37k
4
votes
Arguments where the $j$-invariant has non-trivially real values — and a conjectured duality
Recall that the $j$-invariant identifies $\Bbb{H}/{\rm{PSL}}_2(\Bbb{Z})$ with $\Bbb{C}$; and a fundamental domain for the action of ${\rm{PSL}}_2(\Bbb{Z})$ on $\Bbb{H}$ is $D:=\left\{z\in\Bbb{H}\,\big|...
- 3,599
4
votes
Questions on the $j$-invariant
The first question comes naturally under Schwarz's theory of uniformization of
hyperbolic triangles: let $\tau_0=(1+\sqrt{-3})/2$, $\tau_1=i$, and
$\tau_\infty=i\infty$, and denote by $\Delta$ the ...
- 13.1k
3
votes
How do modular functions of level $N>1$ transform under the full modular group?
In general, there is little that one can say about the function $j(\gamma,\tau)$ in the formula $f(\gamma \tau) = j(\gamma,\tau) f(\tau)$, and in general, it is not easy to take a modular function of ...
- 20.4k
3
votes
Accepted
The degree of the cube root of the $j$-invariant
Let $N=\frac {9-D}{4}$ and let $\bar\omega$ be the complex conjugate of $\omega$. In particular $\omega+\bar\omega = -3$ and $\omega\bar\omega=N$.
For every $A=\bigl(\begin{smallmatrix}a & b\\c&...
- 1,384
2
votes
Accepted
Asymptotic Formula of the coefficients of the q-expansion of the J-Invariant
There are lots of reasons why it's interesting to know the growth rate of modular functions. For the $j$-invariant, the coefficients are closely related to the dimensions of the irreducible ...
- 47.4k
2
votes
Twisted modular equation
Since nobody has answered yet, I'll try it. I think that if $j(M \cdot \tau)$ satisfies the $n$-th modular equation $\Phi_n(X) = 0$ over $\mathbb{Z}[j]$ (where $\mathrm{det}(M) = n$) then $\gamma_2(M \...
- 21
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