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 $l-1$?


1 Answer 1


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 $\ell = 1\bmod N$.

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 $\Gamma_1(N) \sigma_\ell \Gamma_1(N)$.

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 $T_\ell$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.