Dirichlet series associated with holomorphic cusp forms

Level one modular forms. A cusp form of weight $k$ for the full modular group is a holomorphic function $f$ on the upper half-plane which satisfies

$$f\bigg(\frac{az+b}{cz+d}\bigg) =(cz+d)^{k}f(z)$$

for all integers $a,b,c,d$ with $ad-bc=1$ and also has the property that $\lim _{y\to \infty} f(iy)=0$. Cusp forms for the whole modular group exist only for even integers $k=12$ and $k\ge 16$. The cusp forms of a given weight $k$ of this form make a complex vector space $S_k$ of dimension $[k/12]$ if $k \ne 2 \bmod 12$ and of dimension $[k/12]-1$ if $k=2\bmod 12$. Each such vector space has a special basis $H_k$ of Hecke eigenforms which consist of functions $f(z)=\sum_{n=1}^\infty a_f(n) e(nz)$ for which

$$a_f(m)a_f(n)=\sum_{d\mid (m,n)} d^{k-1} a_f(mn/d^2).$$

The Fourier coefficients $a_f(n)$ are real algebraic integers of degree equal to the dimension of the vector space $=\char93 H_k$. Thus, when $k=12, 16, 18, 20, 22, 26$ the spaces are one dimensional and the coefficients are ordinary integers. The L-function associated with a Hecke form $f$ of weight $k$ is given by

$$L_f(s)=\sum_{n=1}^\infty{a_f(n)/n^{(k-1)/2}}{n^s}=\prod_p \bigg(1-\frac{a_f(p)/p^{(k-1)/2}}{p^s}+\frac{1}{p^{2s}}\bigg)^{-1}.$$

By Deligne's theorem $a_f(p)/p^{(k-1)/2}=2\cos \theta_{f}(p)$ for a real $\theta_f(p)$. It is conjectured (Sato-Tate) that for each $f$ the $\{\theta_f(p): p\text{ prime}\}$ is uniformly distributed on $[0,\pi)$ with respect to the measure $\frac{2}{\pi}\sin^2\theta d\theta$. We write $\cos \theta_f(p) = \alpha_f(p)+\overline{\alpha_f(p)}$ where $\alpha_f(p)=e^{i\theta_f(p)}$; then

$$L_f(s)=\prod_p\left(1-\frac{\alpha_f(p)}{p^s}\right)^{-1}\left(1-\frac{\overline{\alpha_f(p)}}{p^s}\right)^{-1}.$$

The functional equation satisfied by $L_f(s)$ is

$$\xi_f(s)=(2\pi)^{-s}\Gamma(s+(k-1)/2)L_f(s)=(-1)^{k/2}\xi_f(1-s).$$

Higher level forms. Let $\Gamma_0(q)$ denote the group of matrices $\pmatrix a & b \cr c & d\endpmatrix$ with integers $a,b,c,d$ satisfying $ad-bc=1$ and $q\mid c$. This group is called the Hecke congruence group. A function $f$ holomorphic on the upper half plane satisfying

$$f\bigg(\frac{az+b}{cz+d}\bigg) =(cz+d)^{k}f(z)$$

for all matrices in $\Gamma_0(q)$ and $\lim _{y\to \infty} f(iy)=0$ is called a cusp form for $\Gamma_0(q)$; the space of these is a finite dimensional vector space $S_k(q)$. The space $S_k$ above is the same as $S_k(1)$. Again, these spaces are empty unless $k$ is an even integer. If $k$ is an even integer, then

$$\dim S_k(q)= \frac{(k-1)}{12}\nu(q)+\left(\left[\frac k4\rig... ... k3\right] -\frac{k-1}3\right)\nu_3(q)-\frac{\nu_\infty(q)}{2}$$

where $\nu(q)$ is the index of the subgroup $\Gamma_0(q)$ in the full modular group $\Gamma_0(1)$:

$$\nu(q)=q\prod_{p\mid q} \left(1+\frac 1p\right);$$

$\nu_\infty(q)$ is the number of cusps of $\Gamma_0(q)$:

$$\nu_\infty(q)=\sum_{d\mid q}\phi((d,q/d));$$

$\nu_2(q)$ is the number of inequivalent elliptic points of order 2:

$$\nu_2(q)=\begin{cases}0 &\text{ if $4\mid q$}\cr \prod_{p\mid q} (1+\chi_{-4}(p)) &\text{ otherwise} \end{cases}$$

and $\nu_3(q)$ is the number of inequivalent elliptic points of order 3:

$$\nu_3(q)=\begin{cases}0 & \text{ if $9\mid q$}\cr \prod_{p\mid q} (1+\chi_{-3}(p)) &\text{ otherwise}. \end{cases}$$

It is clear from this formula that the dimension of $S_k(q)$ grows approximately linearly with $q$ and $k$.

For the spaces $S_k(q)$ the issue of primitive forms and imprimitive forms arise, much as the situation with characters. In fact, one should think of the Fourier coefficients of cusp forms as being a generalization of characters. They are not periodic, but they act as harmonic detectors, much as characters do, through their orthogonality relations (below). Imprimitive cusp forms arise in two ways. Firstly , if $f(z)\in S_k( q)$, then $f(z)\in S_k( dq)$ for any integer $d > 1.$ Secondly, if $f(z)\in S_k( q)$, then $f(dz)\in S_k(\Gamma_0(dq))$ for any $d>1$. The dimension of the subspace of primitive forms is given by

$$\dim S^{\text{new}}_k(q)=\sum_{d\mid q}\mu_2(d)\dim S_k( q/d)$$

where $\mu_2(n)$ is the multiplicative function defined for prime powers by $\mu_2(p^e)=-2$ if $e=1$, $=1$ if $e=2$ , and $=0$ if $e>2$. The subspace of newforms has a Hecke basis $H_k(q)$ consisting of primitive forms, or newforms, or Hecke forms. These can be identified as those $f$ which have a Fourier series

$$f(z)=\sum_{n=1}^\infty a_f(n)e(nz)$$

where the $a_f(n)$ have the property that the associated L-function has an Euler product

$$L_f(s)=\sum_{n=1}^\infty \frac{a_f(n)/n^{(k-1)/2}}{n^s}=\prod... ...od_{p\mid q}\bigg(1-\frac{a_f(p)/p^{(k-1)/2}}{p^s}\bigg)^{-1}.$$

The functional equation satisfied by $L_f(s)$ is

$$\xi_f(s)=(2\pi/\sqrt{q})^{-s}\Gamma(s+(k-1)/2)L_f(s)=\pm (-1)^{k/2}\xi_f(1-s).$$

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