# Tagged Questions

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### Proving the Uniformization Theorem for Elliptic Curves (An Exercise from Silverman's Advanced Topics on Elliptic Curves )

In Silverman's Advanced Topics in the Arithmetic of Elliptic Curves there are two demonstrations of the Uniformization Theorem for the Elliptic Curves (It says that, given an Elliptic Curve $E$, ...
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### How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?

the answer should be $$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$ But first how to prove $$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
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### When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
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### Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
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### how to calculate $q$-expansion coefficients of a modular form?

Can you please explain for me how we can calculate the coefficients of $q$-expansion series of a modular form function $f$? I am really confused.
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### What is the difference between the dimension of $\mathcal{M}_1(\Gamma)$ and $\mathcal{S}_1(\Gamma)$?

There is a formula given in the book saying that $$\dim{\mathcal{M}_1(\Gamma)}=\dim\mathcal{S}_1(\Gamma)+\frac{\varepsilon^{\text{reg}}_\infty}{2}$$ Where $\varepsilon^{\text{reg}}_\infty$ means ...
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### In one version of the Modularity Theorem, what does “arise from modular forms” mean?

One version of Modularity Theorem says that The elliptic curves with rational $j$-values arise from modular forms. Where $$j(\tau)=1728\frac{g_2^3(\tau)}{\Delta(\tau)}$$ I know every ...
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### Question about a property of elliptic function

Let $E=\mathbb{C}/\Lambda$ be a complex elliptic curve where $\Lambda=\omega_1\mathbb{Z}\oplus\omega_2\mathbb{Z}$ Let $f$ be a nonconstant elliptic function with respect to $\Lambda$. Let ...
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### A question about complex tori.

Let $\Lambda$,$\Lambda'$ be two lattice in $\mathbb{C}$ and $m\neq 0\in\mathbb{C}$ satisfying $$m\Lambda\subset\Lambda'$$ The, the book I'm reading says that by the theory of finite Abelian groups ...
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### Why the kernel of isogeny is finite?

It is said that the kernel of a isogeny is finite because it is discrete and complex tori are compact. I have some questions about this. 1. Following is my reason for the kernel is discrete. ...
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### Question about weight 2 Eisenstein series

I'm new to modular form, reading the book A First Course in Modular Forms We have the weight 2 Eisenstein series $$G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2}$$ ...
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