Hecke Operators for $\Gamma_1(N)$ *with* character?

Question

Hello.

I wonder whether there are hecke operators for modular forms for $\Gamma = \Gamma_1(N)$ with additive character $\chi : \mathbb{Z}_N \mapsto \mathbb{C}^{\times}$. There is a somewhat reasonable abstract Hecke algebra for $\Gamma_1(N)$, namely the free $\mathbb{Z}$-module generated by those double cosets $\Gamma \alpha \Gamma \in \Gamma \backslash \Delta/\Gamma$ where $\Delta = \{ \alpha = \begin{pmatrix}a & b \\ c & d\end{pmatrix} \in \mathbb{Z}^{2 \times 2} : c \equiv 0 \mod N, a \equiv 1 \mod N,$ $\det(\alpha) \in \mathbb{N}\}$. In order to let this algebra act on the space of modular forms one has to construct a continuation of the character $\chi$ to a semigroup homomorphism $\tilde{\chi} : \Delta \mapsto \mathbb{C}^\times$ such that $$\alpha \gamma \alpha^{-1} \in \Gamma \Rightarrow \tilde{\chi}(\alpha \gamma \alpha^{-1}) = \chi(\gamma)$$ (see e.g. Miyake, Modular Forms, formula (2.8.1)). Tried though i have, i have been unable even to find a continuation of the character. For example, for $N=3$ i think that i was able to show that there is no continuation at all that satisfies $$\alpha \equiv \beta \mod N \Rightarrow \tilde{\chi}(\alpha) = \tilde{\chi}(\beta)$$ (which is a reasonable assumption). I am sure that i am not the first person ever who tried this. Are there Hecke operators on modular forms for $\Gamma_1(N)$ with character? If so, do they arise as actions of an abstract Hecke algebra as above? Does one have to choose another $\Delta$ maybe?

Best regards,

Fabian Werner

Answer 1

If $N$ is square free, you obtain only modular forms, which is supercuspidal at $p$ diving $N$ and unramfied at those $p$ not diving $N$, so no non-trivial Hecke theory at those $p$ diving $N$ exists! Any nontrivial character will produce the same result.

If $N$ is non-square free, you will probably have to understand $Ind_{\Gamma_1(N)}^{\Gamma(1)} \psi$. This depends heavily on the conductor of $\psi$. This route is pretty messy.

This is of course only a partial answer, but the most useful thing is to look at are primitive characters $\psi$ mod $N$ on $\Gamma_0(N)$ $$ \gamma \in \Gamma_0(N) \mapsto \psi(a).$$ See Casselman's annals-paper "Restriction of GL(2,F)-reps to GL(2,o)-reps" (or something title close to this), but this is a very representation theoretic approach.

Answer 2

I think, the reason why one cannot define these Hecke operators is that they somehow 'switch' the characters, i.e. if one decomposes

$$M_k(\Gamma(N)) = \bigoplus_{\chi} M_k(\Gamma_1(N), \chi)$$

where now, $\chi$ runs through the additive characters, one could expect that for a modular form $f \in M_k(\Gamma_1(N), \psi)$, interpreting this modular form as a modular form for $\Gamma(N)$ and then applying the Hecke operator for $\Gamma(N)$ should be the same as applying the $\Gamma_1(N)$ Hecke operator on this $f$. For this reason, i believe that one should define the Hecke operator on the whole space $\oplus_{\chi} M_k(\Gamma_1(N), \chi)$ as the pullback of the $\Gamma(N)$-Hecke operator. This means that the $f$ above is interpreted as a vector $(0, ..., 0, f, 0, ..., 0)$ and the Hecke operator 'mixes up', i.e. it is possible that if $g = T(m)f$ as $\Gamma(N)$-modular form, the result $g$ has a decomposition that corresponds to some vector $(g_{\chi})_{\chi}$ where some of the $g_\chi$ for $\chi \neq \psi$ are nonzero.