Basis for $M_k(\Gamma(N))$ with Fourier coeffs in $\mathbb{Z}[\Zeta_N]$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:29:59Z http://mathoverflow.net/feeds/question/87672 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87672/basis-for-m-k-gamman-with-fourier-coeffs-in-mathbbz-zeta-n Basis for $M_k(\Gamma(N))$ with Fourier coeffs in $\mathbb{Z}[\Zeta_N]$? Fabian Werner 2012-02-06T14:52:49Z 2012-02-07T11:12:54Z <p>Hi all.</p> <p>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 $\zeta_N = e^{2 \pi i / N}$.</p> <p>Can somebody point out a reference for this?</p> <p>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</p> <p>$\sum_{n \in \mathbb{Z} \setminus \{0\}, n \equiv v_1 \mod N} \frac{1}{n^k}$</p> <p>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?</p> <p>Note that i am aware of this post: <a href="http://mathoverflow.net/questions/78043/is-there-a-miller-basis-for-m-kn" rel="nofollow">http://mathoverflow.net/questions/78043/is-there-a-miller-basis-for-m-kn</a> but i was not able to locate the result in these books.</p> <p>best and thanks!</p> <p>Fabian Werner</p> http://mathoverflow.net/questions/87672/basis-for-m-k-gamman-with-fourier-coeffs-in-mathbbz-zeta-n/87789#87789 Answer by François Brunault for Basis for $M_k(\Gamma(N))$ with Fourier coeffs in $\mathbb{Z}[\Zeta_N]$? François Brunault 2012-02-07T11:12:54Z 2012-02-07T11:12:54Z <p>The constant term of the Eisenstein series $G_k^{0,v}$ in Diamond-Shurman is, up to a factor $N^k$, given by</p> <p>$$\zeta(k,\frac{v}{N}) + (-1)^k \zeta(k,-\frac{v}{N})$$</p> <p>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.</p> <p>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 <a href="http://en.wikipedia.org/wiki/Hurwitz_zeta_function" rel="nofollow">Wikipedia</a>.</p> <p>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, <em>Introduction to the arithmetic theory of automorphic functions</em>, 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, <em>$p$-adic Hodge theory and values of zeta functions of modular forms</em>, Astérisque 295, section 3.</p>