# Questions tagged [j-invariant]

The j-invariant tag has no usage guidance.

13
questions

**6**

votes

**0**answers

220 views

### Geometric interpretation of j-invariants of supersingular elliptic curves

In the classical theory of Complex Multiplication, one considers elliptic curves with an endomorphism ring larger than the integers. In this theory, it's possible to determine the j-invariants of all ...

**2**

votes

**1**answer

146 views

### Twisted modular equation

Let $\gamma_2(\tau)=j(\tau)^{1/3}$. The modular equation shows that the functions $$j\left(\frac{a\tau+b}{c\tau+d}\right),\qquad ad-bc=n$$
are integral over $\mathbf Z[j]$. Under what conditions is ...

**3**

votes

**1**answer

215 views

### How do modular functions of level $N>1$ transform under the full modular group?

Let $f$ be a modular function of level $N>1$ and let $\gamma\in SL_2(\mathbf Z)$. Write $$f(\gamma\tau)=j(\gamma,\tau)f(\tau).$$
First question
What can we say in general about the factor $j(\...

**2**

votes

**0**answers

117 views

### Expressing modular functions of level 9 and 32 as rational functions

Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...

**10**

votes

**2**answers

434 views

### Effective bound on the expansion of the $j$-invariant

The $j$-ivariant has the following Fourier expansion
$$j(\tau)=\frac 1q +\sum_{n=0}^{\infty}a_nq^n=\frac{1}{q}+744+196884q+21493760q^2+\cdots.$$
Here is $q=e^{2\pi i \tau}$.
Is there some simple ...

**2**

votes

**0**answers

265 views

### Zagier's algebraicity of singular moduli

Let $\mathcal M_m\subset M(2,\mathbb Z)$ be the set matrices with determinant $n$. The modular group $\Gamma$ acts on $\mathcal M_m$ from the left and we have the following finite set as a set of ...

**0**

votes

**0**answers

118 views

### Why the cofficients of the $j$-invariant polynomial are integers and how to compute them?

In the Page 333 of the book “Elliptic curves number theory and cryptogarphy” Lawrence C. Washington.
Given $\tau_i\in H$(up-half plane),and $j(\tau_i)$ is the j-invariant of the lattice $\mathbb Z_\...

**3**

votes

**1**answer

312 views

### The degree of the cube root of the $j$-invariant

I have a question which is fairly elementary, but first I must provide relevant context. Without it, my question would seem rather arbitrary and scarcely interesting. Note also that my question can be ...

**52**

votes

**1**answer

4k views

### There's something strange about $\sqrt{\big(j(\tau)-1728\big)d}$

Given discriminant $d$ and j-function $j(\tau)$, I was looking at,
$$F(\tau) = \sqrt{\big(j(\tau)-1728\big)d}$$
which appears in Ramanujan-type pi formulas. Let $C_d$ be the odd prime factors of the ...

**3**

votes

**1**answer

239 views

### Asymptotic Formula of the coefficients of the q-expansion of the J-Invariant

I'm currently writing my master thesis about the j-invariant and his q-expansion.
Now i have the result that the growth of the coefficients is asymptotically
$$c(n) \sim \frac{e^{4\pi \sqrt{n}}}{\...

**1**

vote

**1**answer

144 views

### From an eigenfom with $\mathbf{Q}$-coefficients to $j$-invariants

Given a cuspidal, classical eigenform $f\in S_2(\Gamma_0(N))$ of weight $2$ and with $\mathbf{Q}$-coefficients is there a way of describing the set $J_f$ of $j$-invariants of the elliptic curves lying ...

**12**

votes

**3**answers

2k views

### $j$-invariants of elliptic curves over finite fields

Let $K$ be a finite field, and $\overline{K}$ its algebraic closure. It is well known that two curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant. If two such ...

**32**

votes

**2**answers

1k views

### Difference of j-invariant values and the abc conjecture

I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual $j$-invariant, and $\alpha = (-1+i\sqrt{163})/2$, then
\begin{align*}
\frac{j(i)}{1728} &= 1 \\
\frac{-j(\...