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Recently i read that the space of completely holomorphic (also at the cusps) modular forms $M_k(\Gamma(N))$ possesses a basis having Fourier coefficients in $\mathbb{Z}[\zeta_N]$ where $\zeta_N = e^{2 \pi i / N}$.

Can somebody point out a reference for this?

I already know the following things: At least for $k \geq 2$, $S_k(\Gamma(N))$ -- the subspace of cusp forms -- possesses a basis having Fourier coefficients in $\mathbb{Z}$ (see Shimura, Thm 3.52). What is missing is the Eisenstein series $G^{v}$ (see Diamond/Shurman, Thm 4.2.3). All the Fourier coiefficients except the first one do indeed lie inside $\mathbb{Z}[\zeta_N]$ (up to a constant in $\mathbb{Q}$) but the constant term of the Eienstein series is (in the case that $v_1 \equiv 0 \mod N$) the term

$\sum_{n \in \mathbb{Z} \setminus \{0\}, n \equiv v_1 \mod N} \frac{1}{n^k}$

This is the Hurwitz Zeta Function up to the term $N^{-k}$. The question here is: is this value in $\mathbb{Z}[\zeta_N]$ (up to some denominator) or is there a completely different way to see that such a basis with Fourier coeffs in $\mathbb{Z}[\zeta_N]$ exists?

Note that i am aware of this post: Is there a Miller basis for M_k(N)? but i was not able to locate the result in these books.

best and thanks!

Fabian Werner

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  • $\begingroup$ I guess you mean the value divided by $\pi^k$ ? $\endgroup$ Feb 6 '12 at 16:12
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    $\begingroup$ Fabian, don't forget that the higher Fourier coefficients are all multiplied by $\pi^k$ (look at the definition of $C_k$ in Diamond/Shurman Theorem 4.2.3). $\endgroup$
    – B R
    Feb 6 '12 at 20:52
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    $\begingroup$ (So, in fact, the Eisenstein series can be normalized.) $\endgroup$
    – B R
    Feb 6 '12 at 21:00
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    $\begingroup$ Fabian, I actually don't know any better methods than those Francois mentions in his answer. But, there is a general principle that the zero-th term in the Fourier expansion of a modular form lies in the field generated by the higher Fourier coefficients (see, e.g., math.umn.edu/~garrett/m/v/rationality_principle.pdf), so I knew it had to work even if I didn't know how! $\endgroup$
    – B R
    Feb 9 '12 at 3:21
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    $\begingroup$ Here is the reference to the rationality principle used by Klingen (in the more general context of Hilbert modular forms) : ams.u-strasbg.fr/mathscinet/search/…*&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=133304 (page 266). Note that in order to apply it here, you will need to know that $M_k(\Gamma(N))$ admits a basis having all Fourier coefficients in $\mathbf{Q}(\zeta_N)$, so this is a little bit circular... $\endgroup$ Feb 9 '12 at 10:49
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The constant term of the Eisenstein series $G_k^{0,v}$ in Diamond-Shurman is, up to a factor $N^k$, given by

$$\zeta(k,\frac{v}{N}) + (-1)^k \zeta(k,-\frac{v}{N})$$

where $\zeta(s,x) = \sum_{\substack{n \in \mathbf{Q}_{>0}, \\ n \equiv x \mod{1}}} \frac{1}{n^s}$ is the Hurwitz zeta function.

You can prove by hand that this constant term indeed lies in $\pi^k \cdot \mathbf{Q}(\zeta_N)$. This is a tedious exercise (which I admit I haven't done) using the functional equation of the Hurwitz zeta function linking $\zeta(s,\cdot)$ and $\zeta(1-s,\cdot)$ and the fact that $\zeta(1-k,x) \in \mathbf{Q}[x]$ for any $k \geq 1$ (it is given by a Bernoulli polynomial). For these two facts see for example Wikipedia.

The more conceptual explanation is that $\Gamma(N) \backslash (\mathcal{H} \cup \mathbf{P}^1(\mathbf{Q}))$ admits a canonical model $X(N)$ defined over $\mathbf{Q}(\zeta_N)$ (see Shimura, Introduction to the arithmetic theory of automorphic functions, Chapter 6). Moreover, there is a more conceptual definition of Eisenstein series of weight $k$ as sections of $\mathcal{L}^{\otimes k}$, where $\mathcal{L}$ is a certain line bundle on $X(N)$ (defined using the universal elliptic curve over $Y(N)$). Since the cusps of $X(N)$ are rational over $\mathbf{Q}(\zeta_N)$, the Fourier coefficients of these Eisenstein series belong automatically to $\mathbf{Q}(\zeta_N)$. It then suffices to check that these Eisenstein series coincide with $G_k^{0,v}$ (suitably divided by $(2\pi i)^k$). One reference I know for this point of view is Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295, section 3.

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    $\begingroup$ I was being somewhat sloppy. What you should consider is the vector of values $\zeta(k,v/N)+(-1)^k \zeta(k,-v/N)$, instead of $\zeta(k,v/N)$. Otherwise, you cannot get algebraic values (this is related to the following recent MO question : mathoverflow.net/questions/87348/…). $\endgroup$ Feb 8 '12 at 17:01
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    $\begingroup$ Ah, only now i have seen that the Hurwitz Zeta function only runs over positive numbers. Ok, Wikipedia tells me that the Hurwitz Zeta function is only defined by the sum for $Re(s)>0$ and $Re(x)>0$ but thats not a problem since for $s=k \in \mathbb{N}$, the sum actually converges for all $q \in \mathbb{C} \setminus -\mathbb{N}_0$ so that by the id-thm they coincide. But there is one problem: what to do with the additional $\zeta(k, -v/N)$? The functional equation does only work for positive rationals, right? $\endgroup$ Feb 9 '12 at 7:05
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    $\begingroup$ I am very thankful that you are so patient with me... I am sorry, now i am completely confused. Could you write down explicitely in what way i am supposed to use the functional equation? $\endgroup$ Feb 10 '12 at 8:18
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    $\begingroup$ Unbelievable, when one squares the matrices they become really nice and one can really see (after doing a long computation with a lot of roots of unity) that they are invertible... this is cool. Thanks once again François!!! $\endgroup$ Feb 23 '12 at 19:51
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    $\begingroup$ @Fabian : Good ! I agree the computations were somewhat tedious. I didn't know the matrices have a nice shape after squaring, this is interesting (probably related to the fact that the functional equation can be thought of as an "involution"). $\endgroup$ Feb 24 '12 at 9:53

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