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 […]

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?

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 […]

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 $j(\tau)=j(\mathbb{Z} \oplus \tau \mathbb{Z}) = 1728 \frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}$ […]

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?

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 $s\to\infty$ […]

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 […]

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