In the paper *p-adic L-functions and p-adic periods of modular forms*, Greenberg/Stevens assert that if $\sigma\_l:=\begin{pmatrix}l&0\\0&1\end{pmatrix}$ is the usual Hecke operator at $l$ double coset representative, and $\Gamma$ is the congruence group $\Gamma\_1(N)$, then $g\sigma\_lg^{-1}$ continues to lie in the same double coset class if $l\equiv 1 \pmod{N}$, for any $g\in \text{SL}\_2(\mathbb{Z})$. I thought this would be straightforward to see via some sort of restricted row operations/Euclidean algorithm argument, but am struggling. What is the argument here, and what goes wrong when $N$ does not divide $l-1$?



When does the double coset representative for a congruence subgroup contain a $\text{SL}_2(\mathbb{Z})$-conjugacy class?
*Why the condition $\ell = 1 \bmod N$ is necessary.* Suppose $g \sigma\_\ell g^{-1}$ lies in the double coset $\Gamma \sigma\_\ell \Gamma$. Then $g \sigma\_\ell g^{-1} = r \sigma\_\ell s$ for some $r, s \in \Gamma\_1(N)$; but $r \sigma\_\ell s$ is the product of three matrices which are upper-triangular mod $N$, so it must be upper-triangular mod $N$.


If we take $g = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$, this amounts to requiring that $\begin{pmatrix} \ell & 0 \\ \ell-1 & 1 \end{pmatrix}$ be upper-triangular mod $N$, which is precisely the condition that $\ell = 1\bmod N$.




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*Why the condition is sufficient.* One checks easily that if $\ell \nmid N$ then the double coset $\Gamma\_1(N) \sigma\_\ell \Gamma\_1(N)$ contains every matrix of determinant $\ell$ which is congruent to $\begin{pmatrix} \* & \* \\ 0 & 1 \end{pmatrix}$ mod $N$. If $\ell = 1 \bmod N$, then $g \sigma\_\ell g^{-1}$ satisfies these conditions, so it lies in the double coset $\Gamma\_1(N) \sigma\_\ell \Gamma\_1(N)$.




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*Remark*. For a general prime $\ell$ (even assuming $\ell \nmid N$), the double coset $\Gamma\_1(N) \sigma\_\ell \Gamma\_1(N) = \Gamma\_1(N) \begin{pmatrix} \ell & 0 \\ 0 & 1 \end{pmatrix}\Gamma\_1(N)$ is **not** what is normally called $T\_\ell$. It's sometimes known as $T\_\ell^\*$; and when $\ell \nmid N$ (as here), one has $T\_\ell^\* = \langle \ell \rangle^{-1} T\_\ell$. 


So there are really two assertions being blended together here, both of which need $\ell = 1 \bmod N$ to work: firstly, that the double coset $T\_\ell^\*$ is stable under conjugation; secondly, that it coincides with the double coset $T\_\ell$.


