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

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

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 […]

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 $q=e^{2\pi iz}$, […]

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 $a(2015)$ ?

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 […]

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 $11$ […]

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 $ […]

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 $\mathbb{Q}$.

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 […]

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