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  1. Let $f$ be modular of level $p^nN$, $(p,n) = 1$, $p > 2$ with character $\chi\psi\eta$, where $\chi$ has conductor dividing $N$, $\psi$ conductor power of $p$ and order power of $p$, and $\eta$ conductor $p$ and order dividing $p-1$. Since $p$ is odd, one can write $\psi = \xi^{-2}$. (i) Why is the character of $f \otimes \xi$ equal to $\chi\eta$; (ii) why is the reduction mod $p$ of this equal to the reduction of $f$; (iii) why is for $r \gg n$ $f \otimes \xi$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(pN)$? Can this be proven using the converse theorem?

2.a Why is the twisted Eisenstein series $G = a_0 + \sum_{n=1}^\infty\sum_{d \mid n}\eta^{-1}(d)d^{i-1}q^n$ of Nebentypus $\eta^{-1}$ with respect to $\Gamma_0(p)$?

2.b Why is $fG$ modular with respect to $\Gamma_0(p^r) \cap \Gamma_1(N)$, wenn $f \in S_k(\Gamma_0(p^r) \cap\ \Gamma_1(pN), \chi\eta)$ ist?

(The article is here: math.berkeley.edu/~ribet/Articles/motives.pdf )

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Most of these questions would be answered by a solid overview article or basic textbook on the theory of modular forms. There are lots of these. I recall being fond of Miyake's book ("Modular Forms") when I first learned about this stuff, for example (though I don't know if he covers twisting). The article by Diamond and Im ("Modular forms and modular curves") does a pretty impressive job in a small space, but I think that this is best read after something a little slower paced. I suggest spending some time with a couple of reference like this first. –  Ramsey Feb 8 '11 at 22:45
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3 Answers 3

up vote 4 down vote accepted

William Stein has answered your question (i). As for your question (ii), since $\xi$ has $p$-power order, and since any $p$-power root of unity is congruent to $1$ modulo the unique prime ideal lying over $p$ in $\mathbb Q$ adjoin the $p$-power roots of unity, we see that $f\otimes \xi$ is congruent to $f$ modulo any prime ideal lying over $p$ in the field of definition of $f$ adjoin the $p$-power roots of unity.

Finally, you have already noted in (i) that the character of $f\otimes \xi$ is just $\chi\eta$. Since the conductor of $\xi$ divides $N$ and the conductor of $\eta$ divides $p$, we see that the conductor of $\chi\eta$ divides $p N$. This gives (iii). (Note that (iii) is simply a statement about the conductor of the character of $f\otimes \xi$: In general, a modular form $g$ of level $C$ is modular on $\Gamma_0(C)\cap \Gamma_1(D)$, rather than just $\Gamma_1(C)$, for some integer $D$ dividing $C$, if the character of $g$, a priori a character of $(\mathbb Z/C)^{\times}$, actually factors through $(\mathbb Z/D)^{\times}$. In your particular case, $f$ has conductor a power of $p$ times $N$, and $\xi$ has conductor a power of $p$, so $f\otimes \xi$ has conductor a power of $p$ times $N$. The conductor of its character divide $p N$, as already noted, and so (iii) follows.)

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This question (at least when I read it right now) seems loaded with typos... In any case, regarding (i) it's a general fact that if $f$ is a modular form with character $\varepsilon$ and $\chi$ is any Dirichlet character, then $f \otimes \chi$ is a modular forms with character $\varepsilon \cdot \chi^2$. See Proposition 3.64 of Shimura's book Introduction to the arithmetic theory of automorphic functions. This answers your question (i).

I'm curious what motivated this question. It seems like perhaps the person asking it was reading a paper, and wanted to check some claims in the paper, but didn't know how. If so, what paper was it?

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The article is here: math.berkeley.edu/~ribet/Articles/motives.pdf page 5. –  user12832 Feb 8 '11 at 19:29
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Regarding your question (2.a), I believe that your 'twisted' Eisenstein series is a special case of a very classical (and well known) construction.

Let $k\geq 2$. Suppose that you have two primitive Dirichlet characters $\chi_1$ and $\chi_2$ modulo $N_1$ and $N_2$ respectively (if $k=2$ then suppose also that both characters are not trivial). We construct a modular form as follows.

Set $$a_0=- L(0,\chi_1)L(1-k,\chi_2).$$

For $n>0$ define $$a_n = \sum_{d\mid n} \chi_1(\frac{n}{d})\chi_2(d) d^{k-1}.$$

Its worth noting that $$L(s,\chi_1)L(1+s-k, \chi_2)=\sum_{n=1}^\infty \frac{a_n}{n^s}.$$

Theorem: Let $F_{\chi_1,\chi_2}=\sum a_n q^n$. Then $F_{\chi_1,\chi_2}\in M_k(N_1N_2,\chi_1\chi_2)$. In fact, $F_{\chi_1,\chi_2}$ is an Eisenstein series.

Your question follows from taking $\chi_1=1$ and $\chi_2=\eta^{-1}$.

As I said, this is a very classical construction and should be in any book introductory text (e.g. Miyake, Diamond and Shurman, Shimura).

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