Questions tagged [j-invariant]

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strange factorization of a simple combination of j-invariant coefficients

I am just curious: is it accidental that $744^2+744\cdot 3-196884\cdot2=162000$ has only the small primes $2$, $3$ and $5$ in its factorization?
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7 votes
1 answer
327 views

Questions on the $j$-invariant

The j-invariant as a modular function is typically defined $$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$ since $E_4$ is a modular form of weight 4 and $\Delta$ has weight 12, it follows that $j$ is a ...
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  • 821
6 votes
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272 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 ...
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3 votes
1 answer
239 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(\...
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2 votes
1 answer
152 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 ...
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2 votes
0 answers
119 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 ...
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  • 2,255
11 votes
2 answers
464 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 ...
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2 votes
0 answers
269 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 ...
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  • 2,255
0 votes
0 answers
141 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_\...
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3 votes
1 answer
347 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 ...
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  • 2,255
3 votes
1 answer
260 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}}}{\...
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1 vote
1 answer
148 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 ...
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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 ...
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  • 597
52 votes
1 answer
5k 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 ...
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35 votes
2 answers
2k 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(\...
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