Fourier Coefficients of Hilbert-Eisenstein Series

• Fourier coefficients of Hilbert-Eisenstein series are explicit arithmetic invariants constructed via adelic methods and metaplectic covers, extending Cohen’s classical formulas.
• Local Whittaker integrals and the Kohnen plus congruence precisely govern these coefficients by linking quadratic Hecke characters with Euler factorization.
• The construction bridges local automorphic representations and global L-functions, offering deep insights into modular forms over totally real fields.

The Fourier coefficients of Hilbert-Eisenstein series in the Kohnen plus space constitute a central topic in the arithmetic theory of modular forms of half-integral weight over totally real fields. The generalized construction by Hiraga and Ikeda extends the classical half-integral weight Eisenstein series, originally introduced by Cohen over $\mathbb{Q}$, to Hilbert modular forms defined over an arbitrary totally real field $F$. This theory reveals deep connections between the adelic representation of automorphic forms, explicit Fourier expansions, local Whittaker functions, and global $L$-functions, all governed by arithmetic congruence conditions determined by the Kohnen plus space paradigm (Su, 2014).

1. Adelic Construction of Hilbert-Eisenstein Series

Let $F$ be a totally real field of degree $n$, $\mathbb{A}_F$ its adele ring, and $D_F$ the field discriminant. The construction of Eisenstein series of parallel half-integral weight relies on the metaplectic double cover $\widetilde{\mathrm{SL}}_2(\mathbb{A}_F)$ determined by the Kubota cocycle.

For each integer $\kappa\geq 1$, define the parallel weight $K = (\kappa, \ldots, \kappa)$. Given a Hecke character $x'$ of the (wide) ideal class group, the global section $f_\kappa$ is constructed as a restricted tensor product of local vectors $f_{\kappa,v}$, where:

• For non-archimedean $v$, $f_{\kappa,v}$ is the unique function in the local principal series with $f_{\kappa,v}(1) = 1$ and Iwahori invariance,
• For archimedean $v$, $f_{\kappa,v}$ is the holomorphic section of weight $\kappa+\frac{1}{2}$.

The metaplectic Eisenstein series is constructed as

$$\widetilde{E}(g;\kappa,x') := \sum_{\gamma\in B(F)\backslash \mathrm{SL}_2(F)} x'(d_\gamma) f_\kappa(\gamma g),$$

where $B(F)$ is the upper triangular Borel subgroup and $d_\gamma$ denotes the lower-right entry of $\gamma$. Upon pulling back to the Hilbert upper half-space $\mathfrak{H}^n$, this defines the holomorphic Hilbert-Eisenstein series $E(z;\kappa,x')$ of weight $\kappa+\frac{1}{2}$, as a well-defined element of the generalized Kohnen plus space $M^+_{\kappa+\frac{1}{2}}(1)$ (Su, 2014).

2. Fourier Expansion and Kohnen-Plus Congruence

The Hilbert-Eisenstein series admits an adelic Fourier expansion of the form

$$E(z; \kappa, x') = \sum_{\xi\in F} c(\xi; \kappa, x')\, e^{2\pi i\,\mathrm{Tr}_{F/\mathbb{Q}}(\xi z)},$$

with Fourier coefficients given by the integral

$$c(\xi; \kappa, x') = \int_{\mathbb{A}_F/F} \widetilde{E}(n(x)g_z; \kappa, x') \psi(-\xi x)\,dx.$$

These coefficients are governed by the Kohnen-plus condition:

$$c(\xi; \kappa, x') = 0 \quad \text{unless} \quad (-1)^\kappa \xi \equiv \text{square} \pmod{4\mathfrak{O}_F}.$$

This selects those $\xi\in F$ such that, for every finite place $v$, the local invariant $f_{\xi,v} \geq 0$ in the sense detailed by Hiraga and Ikeda (Definition 2.4 of (Su, 2014)). Thus, the support of the Fourier coefficients reflects deep congruence properties characterizing the Kohnen plus space.

3. Local Whittaker Integrals and Euler Factorization

The Fourier coefficient $c(\xi; \kappa, x')$ factors as a product over all places $v$:

$$c(\xi; \kappa, x') = \prod_{v \leq \infty} W_v(\xi; \kappa, x'),$$

where $W_v$ denotes the local Whittaker integral:

$$W_v(\xi; \kappa, x') = \int_{F_v} f_{\kappa,v}(w n(x))\,\psi_v(-\xi x) dx.$$

Table: Local Whittaker Factors

Place $v$	$W_v(\xi; \kappa, x')$ Form	Support Condition
Non-archimedean	$0$ if $(-1)^\kappa\xi \not\in (F_v^\times)^2$ or $2\nmid\mathrm{ord}_v(\xi)$;<br>otherwise<br>$$L_v(1-\kappa, x_{(-1)^\kappa\xi} x')/L_v(1-2\kappa, x'^2)\cdot P_v(q_v^{-\kappa}; x_{(-1)^\kappa\xi}, x')$$	$(-1)^\kappa\xi$ square mod $4 \mathfrak{O}_F$
Archimedean	$$( -2\pi i )^{\kappa + 1/2} \xi^{\kappa - 1/2} e^{2 \pi i \xi z_v} / \Gamma(\kappa+1/2)$$	$\xi > 0$

Here, $x_{(-1)^\kappa \xi}$ is the quadratic Hecke character of $F(\sqrt{(-1)^\kappa \xi})$, and $P_v$ is a degree-two Euler factor:

$$P_v(t; \chi, X) = \frac{1 - \chi(\pi_v) q_v^{-1} t}{1 - X(\pi_v)^2 q_v^{-2\kappa} t^2}.$$

4. Global $L$-Functions and Explicit Fourier Coefficient Formula

Combining the local data yields the explicit formula for $\kappa \geq 2$:

$$c(\xi; \kappa, x') = D_F^{-\kappa/2 - 1/4} \, \frac{(2\pi)^{n\kappa}}{\Gamma(\kappa+1/2)^n} \, N_{F/\mathbb{Q}}(\xi)^{\kappa-1/2} \, x_{(-1)^\kappa\xi}((-1)^n) \, \frac{L_F(1-\kappa, x_{(-1)^\kappa\xi} x')}{L_F(1-2\kappa, x'^2)} \, \prod_{v < \infty} P_v(q_v^{-\kappa}; x_{(-1)^\kappa\xi}, x').$$

An equivalent “divisor-sum” formulation for the arithmetic part, denoted $C_\kappa(\xi)$, is

$$C_\kappa(\xi) = \sum_{\mathfrak{a} | \mathfrak{g}} \mu(\mathfrak{a})\, x(\mathfrak{a})\, x'(\mathfrak{a})\, N(\mathfrak{a})^{\kappa-1}\,\sigma_{2\kappa-1,x}(\mathfrak{g}/\mathfrak{a}),$$

where $$4\mathfrak{g} = (-1)^\kappa \xi \cdot \mathfrak{O}_F$$, $\mu$ is the ideal-theoretic Möbius function, $x$ the relevant quadratic character, and

$$\sigma_{2\kappa-1,x}(\mathfrak{b}) = \sum_{\mathfrak{d} | \mathfrak{b}} x(\mathfrak{d}) N_{F/\mathbb{Q}}(\mathfrak{d})^{2\kappa-1}.$$

5. Specialization to the Rational Field and Cohen’s Classical Formula

For $F = \mathbb{Q}$, the construction specializes to the classical setting introduced by Cohen. Setting $n = 1$ and $D_F = 1$, the quadratic character $x$ becomes the Dirichlet character $\chi_{(-1)^\kappa n}$. The Fourier expansion takes the form

$$H_{\kappa+\frac{1}{2}}(z) = \sum_{n \equiv 0,1 \bmod 4} H_{\kappa + \frac{1}{2}}(n)\, q^n,$$

where

$$H_{\kappa+\frac{1}{2}}(0) = \frac{1 - 2\kappa}{2}\zeta(1-2\kappa), \quad H_{\kappa+\frac{1}{2}}(n) = L(1-\kappa, \chi_{(-1)^\kappa n}) \sum_{d|f} \mu(d) \chi_{(-1)^\kappa n}(d) d^{\kappa-1} \sigma_{2\kappa-1}(f/d),$$

with $n = D f^2$, $D$ being the fundamental discriminant and $\sigma_{2\kappa-1}(m) = \sum_{d|m} d^{2\kappa - 1}$. This recovers exactly the classical formula for the Fourier coefficients of half-integral weight Eisenstein series as described by Cohen (Su, 2014).

6. Arithmetic and Representation-Theoretic Significance

The explicit determination of the Fourier coefficients via Hecke $L$-values, discriminants, and local Euler factors grants deep arithmetic insight into the algebraic and analytic structure of Hilbert modular forms of half-integral weight. The precise dependence on quadratic Hecke characters and the universality of the Kohnen plus congruence, together with the Euler product expansion, tie the existence and vanishing of coefficients to foundational properties of quadratic extensions and ideal theory in totally real fields.

Furthermore, the generation of the Kohnen plus space by Eisenstein series and cusp forms (over $\mathbb{C}$) implies that these explicit Eisenstein series provide both spanning sets for automorphic forms and a testing ground for conjectures concerning the arithmeticity and modularity phenomena in higher rank settings.

7. Research Context and Generalizations

Hiraga and Ikeda's construction (Su, 2014) provides a comprehensive generalization of the classical results of Kohnen and Cohen from $\mathbb{Q}$ to general totally real base fields. The explicit local-global factorization, the connection with metaplectic covers, and the adelic formalism employed here form the analytic backbone underlying many subsequent advances in the theory of Hilbert modular forms, their arithmetic subspaces, and the theory of $L$-values.

These results open avenues toward investigating the arithmetic geometry inherent in half-integral weight modular forms over number fields, the interaction with Siegel and orthogonal modular forms, and deepening connections with automorphic $L$-functions, algebraic cycles, and trace formulae.