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

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

• I don't think that I understand this question. Are you asking about constructing Hecke operators on a space of modular forms of level $N$ with a fixed nebentypus character mod $N$ (which is not an additive character...)? Or are you trying to construct something other than one of the usual Hecke operators that somehow depends on a character mod $N$? In other words what does the word "with" in the first sentence of you post apply to - "modular forms" or "Hecke operators"? Commented Jan 19, 2012 at 23:25
• I mean the following: when given a group homomorphism $\chi:(\mathbb{Z}_N, +) \mapsto (\mathbb{C}^\times, \cdot)$ then this extends to a character $\chi : \Gamma_1(N) \mapsto \C^\times$ by putting $\chi \begin{pmatrix}a & b \\ c & d \end{pmatrix} := \chi(b)$. Then one can define the space of modular forms that transform with this character, i.e. $f$ is holomorphic in $\mathbb{H}$ and all cusps and $f(\gamma.\tau) = \chi(\gamma) (c\tau + d)^k f(\tau)$ for all $\gamma \in \Gamma_1(N)$. Commented Jan 20, 2012 at 10:27
• One can construct the abstract Hecke algebra for $\Gamma_1(N)$ just as the one for $\Gamma_0(N)$ as the free $\mathbb{Z}$-module of the double cosets $\Gamma_1(N) \alpha \Gamma_1(N)$ with $\alpha \in \Delta$ but in order to let this abstract ring act on the space of modular forms as defined in the last comment one has to do the things mentioned in the question, so yes, it applies to both: modular forms and Hecke operators (both depend on this character $\chi$) Commented Jan 20, 2012 at 10:29

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.

• You actually obtain all modular forms with depth-0 supercuspidal in the first sentence. Commented Feb 6, 2013 at 18:01

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.