1
vote
0answers
39 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
0
votes
1answer
46 views

Proving addition and multiplication

(1)Show that addition and multiplication mod n are associative operations. (2)Show that there are both an additive and a multiplicative identity. (3)Show that multiplication distributes over ...
4
votes
2answers
159 views

Order of special linear group $SL_2 (F)$ [duplicate]

In ex. 1.2.3 on p.21 of Diamond's A first course in modular forms, I was trying to show that $$|SL_2(\mathbb{Z}/N \mathbb{Z})|=N^3 \prod_{p|N}(1-1/p^2).$$ First of all, I knew that If $F$ is a field ...
3
votes
1answer
140 views

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 ...
1
vote
1answer
77 views

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. ...
2
votes
1answer
236 views

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 $$ G_2(\tau)=\sum_{c\in\mathbb{Z}}\sum_{d\in\mathbb{Z}_c'}\frac{1}{(c\tau+d)^2} $$ ...
4
votes
1answer
61 views

Why is this map a translation?

Let $\Gamma$ be a congruence subgroup, $s$ be a cusp of $\Gamma$, $\sigma \in SL_{2}(\mathbb{Z})$ be the map which sends $\infty$ to $s$, and $\Gamma_{s} = \{\gamma \in \Gamma \mid \gamma(s) = s\}$. ...