Articles of modular forms

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 […]

Formula for index of $\Gamma_1(n)$ in SL$_2(\mathbf Z)$

Is there a precise formula for the index of the congruence group $\Gamma_1(n)$ in SL$_2(\mathbf Z)$? I couldn’t find it in Diamond and Shurman, and neither could I find an explicit formula with a simple google search. Certainly, there should be some explicit expression, no?

Is the derivative of a modular function a modular function

Fix a positive integer $n$. Let $f:\mathbf{H}\longrightarrow \mathbf{C}$ be a modular function with respect to the group $\Gamma(n)$. Is the derivative $$\frac{df}{d\tau}:\mathbf{H}\longrightarrow \mathbf{C}$$ also a modular function with respect to $\Gamma(n)$? I think it’s clear that $df/d\tau$ is meromorphic on $\mathbf{H}$ and that it is meromorphic at the cusp. I just don’t know why it […]

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus (0,0)} \frac{1}{(m+\tau n)^{2k}}.$$ Motivation Exercise 6.6 of Silverman’s “The Arithmetic of Elliptic Curves” asks to compute the special value: $$j(i)= j(\mathbb{Z} \oplus i\mathbb{Z})=1728.$$ Where $j(\tau)=j(\mathbb{Z} \oplus \tau \mathbb{Z}) = 1728 \frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$ […]

Agreement of $q$-expansion of modular forms

If I have modular functions $f$ and $g$ with $f = a_{1} + a_{2}q + \cdots$ and $g = b_{1} + b_{2}a + \cdots$ both $q$-expansions, why does/how does it follow $f = g$ after checking only finitely many terms?

Intermediate step in deducing Jacobi's triple product identity.

An intermediate step deduces Jacobi’s triple product identity by taking the $q$-binomial theorem $$ \prod_{i=1}^{m-1}(1+xq^i)=\sum_{j=0}^m\binom{m}{j}_q q^{\binom{j}{2}}x^j $$ and deducing $$ \prod_{i=1}^s(1+x^{-1}q^i)\prod_{i=0}^{t-1}(1+xq^i)=\sum_{j=-s}^t\binom{s+t}{s+j}_q q^{\binom{j}{2}}x^j $$ and then letting $s\to\infty$ and $t\to\infty$. I don’t follow the intermediate deduction, what’s the way to see it? (Thank you Colin McQuillan for pointing this out.) Much later edit: By letting $s\to\infty$ […]

Cusp forms' Fourier coefficients sign changes

I need some clarification on the following, if possible: I have seen in that for every $ f \in S_k$ which Fourier transform is $\sum_{n=1}^\infty a(n)q^n$ there is an upper bound $\sum_{n=1}^N \|a(n)\| \leq c_f \cdot N^{\frac{k+1}{2}} $. Now, somehow, using the theorem that states $ \| \sum_{n=1}^N a(n) \| \leq c_f \cdot N^{\frac{k}{2}} \cdot […]