The paper I'm referring to is https://projecteuclid.org/euclid.dmj/1077303203. Here Shimura constructs an Hermitian Eisenstein series $E_m(z,k,s,\psi,\mathfrak{b})$ for, in the case I am interested in, a variable $z$ in the Hermitian upper half plane ($m\times m$ complex matrices $z$ for which $i(z-\overline{z}^t)$ is positive definite), an integer $k$, a complex $s$, an ideal $\mathfrak{b}$ of the ring of integers of a quadratic imaginary extension $\mathbb{L}$ of $\mathbb{Q}$ and a character $\psi$ modulo $\mathfrak{b}$, which is a modular form for (some subgroup of) $U(m,m)$, given by $$\left\{\gamma\in SL_{2m}(\mathbb{L}):\overline{\gamma}^t \left(\begin{matrix}0_m & -I_m\\I_m & 0_m\end{matrix}\right)\gamma=\left(\begin{matrix}0_m & -I_m\\I_m & 0_m\end{matrix}\right)\right\}$$

I am actually only interested in the simplest case, where $\mathfrak{b}$ is trivial (and hence so is $\psi$). My goal is to see whether the weight $4$ series $E_3(z,4,s)$ is well defined in $s=0$, and if $E_3(z,4,0)$ is a holomorphic function in $z$. I want to do this because as is well known the standard definition of Eisenstein series as $\sum j(\gamma,z)^{-k}$ is not absolutely convergent for $k=4$, so the idea is to analytically continue the non-holomorphic $\sum j(\gamma,z)^{-4}\det(\gamma(z-\overline{z}^t))^s$ towards $s=0$ and hope that everything works.

Shimura's paper deals exactly with this situation. Remember that in my case the genus is $m=3$ and the weight is $k=4$, and we are in what Shimura calls Case SU. Then, in theorem 7.1 pag 452, we have that $E_3(z,4,s)$ is holomorphic at $s=0$ since $k\ge m$, so $E_3(z,4,0)$ is well defined. Now, because $k=m+1$ over the field $\mathbb{Q}$, we fall in the exceptional case (ii); this is not a problem if our trivial character $\psi$ does not coincide with $\theta^{2}$, where $\theta$ is defined (formula 4.4) as $$\theta(\mathfrak{a})=\left(\frac{\mathbb{L}/\mathbb{Q}}{\mathfrak{a}}\right)$$ for an ideal $\mathfrak{a}$ of $\mathbb{Q}$ prime to the discriminant of $\mathbb{L}$ over the rationals.

Since the image of $\theta$ is $\pm 1$ where defined, $\theta^2=1$ where defined. Does $\theta^2$ coincide with the trivial character, the one being $1$ everywhere?

If that is not the case, then according to the theorem $E_3(z,4,0)$ is a nice holomorphic modular form of weight $4$. However, should $\theta^2$ equal the trivial character, does the paper suggest that $E_3(z,4,0)$ is not holomorphic in $z$?


1 Answer 1


The paper just states the fact that $E_n(z,n+1,0)$ is in general non holomorphic but has a somewhat explicit form.

Nagaoka, in Note on Siegel-Eisenstein series of low weight (https://link.springer.com/article/10.1007%2FBF02940802, theorem 2.2), shows that for some values of $n$ the Eisenstein series in question is indeed holomorphic (and has cyclotomic Fourier coefficients), namely if and only if $n\equiv 3\pmod{4}$. This is exactly the case for my question, so $E_4(z,3,0)$ is a holomorphic modular form.


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