Articles of modular forms

History of the Coefficients of Elliptic Curves — Why $a_6$?

This question already has an answer here: Reason behind standard names of coefficients in long Weierstrass equation 2 answers

What is the limit $\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$

If $\vartheta_{2}(q)$ is jacobi’s theta function, what is the limit $$\lim_{q\to 1} \vartheta_{2}^{2}(q)(1-q)$$ for the nome $q$. I would like to know whether the limit exists or not. If it does, please let me know and provide it’s evaluation

connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff $$\prod_{n=0}^{\infty}\frac{1}{(1-q^{4n+1})^p}=1+\sum_{n=1}^{\infty}\phi(n)\,q^n$$ such that $$\phi(n)\equiv 0\pmod{p}$$ is true for all natural numbers $n\in\mathbb{N}$ except at multiples of $p$. I’ve experimentally verified the […]

Show the Dedekind $\eta(\tau)$ function is a modular form of weight $\frac{1}{2}$ for $\Gamma_0(6)$

Given that $\Gamma_0(6)$ is generated by matrices in the set $$G = \left\{\left(\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right), \left(\begin{array}{cc} 7 & -3\\ 12 & -5 \end{array}\right),\left(\begin{array}{cc} 5 & -1\\ 6 & -1 \end{array}\right),\left(\begin{array}{cc} -1 & 0\\ 0 & -1 \end{array}\right) \right\}$$ Show that $$ \eta(\tau) := e^{\frac{\pi iz}{12}}\prod_{n=1}^{\infty} (1-q^n) $$ where $q=e^{2\pi iz}$, […]

Coefficient in the Fourier expansion of the cusp form

Ideal of cusp for $\Gamma_{0}(4)$ is principal and generated by $f(z)=η(2z)^{12}=q+\sum a(n)q^n $, this is discussed here. How one can compute the coefficient $a(n)$ when $n$ is rather large ? for example, what is the coefficient $a(2015)$ ?

Good applications of modular forms on $SL_2(\mathbb{Z})$

I’ve just read some materials of modular forms on $SL_2(\mathbb{Z})$, and find some interesting application. Deal with Ramanujan $\tau$ function. I saw it in Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$?. And I think is really a good example. Prove Ramanujan conjecture. I have read the proof of Ramanujan conjecture in Ahlgren S, Boylan M. Arithmetic […]

All $11$ Other Forms for the Chudnovsky Algorithm

Continued from this post Ramanujan found this handy formula for $\pi$$$\frac 1\pi=\frac {\sqrt8}{99^2}\sum_{k=0}^{\infty}\binom{2k}k\binom{2k}k\binom{4k}{2k}\frac {26390k+1103}{396^{4k}}\tag1$$ Which is related to Heegner numbers. Sometime after, the Chudnovsky brothers came up with another $\pi$ formula$$\frac 1\pi=\frac{12}{(640320)^{3/2}}\sum_{k=0}^\infty (-1)^k\frac {(6k)!}{(k!)^3(3k)!}\frac {545140134k+13591409}{640320^{3k}}\tag2$$ And according to Tito, $(2)$ has a total of $11$ other forms with integer denominators. Question: What are all $11$ […]

A special modular function: $ j $-invariant.

It is known that j invariant $$j(\tau)= 1728 \frac{g_2^3(\tau)}{\Delta(\tau)} $$ $\tau \in \mathbb{H}$ attains every complex value , Can someone guide me its proof.?? where $L(\tau ) = \{\tau m + n | \quad \tau \in \mathbb{C} ; m,n \in \mathbb{Z}\} $ $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ $g_2(\tau)=60G_4(\tau);g_3(\tau)=140G_6(\tau);\Delta = g_2^3 -27g_3^2 $ […]

Zeros of the second derivative of the modular $j$-function

I am interested in the zeros of $j”(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j”$ are algebraic over $\mathbb{Q}$, or, even better, quadratic over $\mathbb{Q}$.

Using Modularity Theorem and Ribet's Theorem to disprove existence of rational solutions

This is likely overly optimistic, but can one take the results from the Modularity theorem and Ribet’s theorem, and distill these down to an undergrad math level of a way to check if certain rational polynomials have no non-trivial solutions? For instance, I do not understand the details of either theorem, but if I can […]