Timeline for Hecke Operators for $\Gamma_1(N)$ *with* character?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 6, 2013 at 17:57 | answer | added | Marc Palm | timeline score: 2 | |
| Feb 6, 2013 at 17:25 | answer | added | Fabian Werner | timeline score: 1 | |
| Jan 20, 2012 at 10:29 | comment | added | Fabian Werner | 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 $\chi$) | |
| Jan 20, 2012 at 10:27 | comment | added | Fabian Werner | 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 $\gamma \in \Gamma_1(N)$. | |
| Jan 19, 2012 at 23:25 | comment | added | Ramsey | 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"? | |
| Jan 19, 2012 at 12:51 | history | asked | Fabian Werner | CC BY-SA 3.0 |