Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series

This paper establishes the meromorphic continuation and, in the specific case of the $E_8$ lattice, a precise functional equation for a Dirichlet series involving Fourier-Jacobi coefficients of cusp forms on orthogonal groups of signature $(2,n+2)$ by utilizing an integral representation derived from Klingen-type orthogonal Eisenstein series and their connections to Epstein and Siegel Eisenstein series.

The Gist

Imagine you are trying to understand the hidden music of a complex, multi-dimensional instrument. In the world of mathematics, this instrument is a Dirichlet series. Think of this series as a long, infinite song made up of numbers. The author of this paper, Rafail Psyroukis, wants to know: Does this song go on forever without stopping, or does it eventually hit a wall? And if we play it "backwards" (a process called a functional equation), does it sound the same?

To answer this, the author uses a clever set of mathematical tools, which we can explain through a few creative analogies.

1. The Problem: A Song That Might Be Broken

The paper starts with a specific type of mathematical "song" (a Dirichlet series) built from the "Fourier-Jacobi coefficients" of two special shapes called cusp forms.

• The Analogy: Imagine two complex, 3D sculptures (the cusp forms). If you shine a light on them from different angles, you get shadows (the coefficients). The author is trying to combine these shadows into a single, long melody (the Dirichlet series).
• The Issue: We know the song sounds good for the first few notes, but we don't know if it continues smoothly into infinity or if it crashes and burns. We need to prove it can be played all the way through the universe of numbers.

2. The Strategy: The "Bridge" (Integral Representation)

To fix the song, the author builds a bridge. In math, this is called an Integral Representation.

• The Analogy: Instead of trying to walk across a deep canyon (the infinite series) directly, the author builds a bridge using a different structure called a Klingen Eisenstein series.
• How it works: The author proves that the "song" (Dirichlet series) is actually just a shadow cast by this bridge. If we can understand the bridge, we understand the song.

3. The Twist: The "Epstein Zeta" Transformation

The bridge (the Eisenstein series) is still too complicated to analyze directly. So, the author performs a magic trick: they rewrite the bridge to look like an Epstein zeta function.

• The Analogy: Imagine the bridge is made of a tangled knot of ropes. The author unties the knot and rearranges the ropes into a neat, orderly grid (the Epstein zeta function). This grid is much easier to study because it has a very regular pattern.
• The Catch: This only works if the "rope" (the underlying lattice) has a very specific shape—specifically, if it has only one "end" (a 1-dimensional cusp). It's like saying, "This trick only works if the bridge has exactly one exit door."

4. The Heavy Lifting: Differential Operators as "Noise Cancelers"

Now the author wants to connect this grid to a famous, well-understood object: a Siegel Eisenstein series (think of this as a "Master Song" from a different musical family, the symplectic group).

• The Problem: If you try to mix the "Grid" with the "Master Song," they don't fit. The Grid has some "static noise" (terms that cause the math to explode or diverge).
• The Solution: The author uses Differential Operators.
• The Analogy: Imagine the Grid is a radio signal full of static. The author uses a special Noise-Canceling Headphone (the differential operator). This device doesn't just turn down the volume; it surgically removes the specific frequencies that cause the static, leaving only the pure, clean signal.
• The Condition: This headphone only works if the signal is tuned to a specific frequency. In math terms, the author requires that the number $n$ (related to the dimension of the shape) is divisible by 4.

5. The Grand Connection: Theta Correspondence

Once the noise is gone, the author reveals the Theta Correspondence.

• The Analogy: This is the "Aha!" moment. The author shows that the "Grid" (from the orthogonal group) and the "Master Song" (from the symplectic group) are actually two different languages describing the exact same underlying reality. By translating one into the other, the author can use the known properties of the Master Song to prove the properties of the original Dirichlet series.

6. The Grand Finale: The E8 Lattice

Finally, the author applies this whole process to a very special, famous shape called the E8 lattice.

• The Analogy: The E8 lattice is like the "Holy Grail" of 8-dimensional shapes. It is perfectly symmetrical and unique. Because it fits all the strict rules the author set (it has one exit door, and its dimensions work with the noise-canceling headphones), the author can write down the exact functional equation for the song.
• The Result: They prove that if you play the song "backwards" (changing $s$ to $2k - 9 - s$), it sounds exactly the same. This is a perfect symmetry, a beautiful mathematical truth.

Summary

In short, this paper is a journey of translation and cleanup:

1. Take a mysterious, infinite song (Dirichlet series).
2. Build a bridge to a known structure (Eisenstein series).
3. Untangle the structure into a neat grid (Epstein zeta).
4. Use a special tool (differential operators) to remove the mathematical "static" that breaks the math.
5. Translate the result into a famous "Master Song" (Siegel Eisenstein series) to prove the original song is safe, sound, and perfectly symmetrical.

The author has successfully shown that these complex mathematical objects, which seem chaotic at first, actually follow a strict, beautiful, and predictable rhythm.

Technical Summary

Here is a detailed technical summary of the paper "Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series" by Rafail Psyroukis.

1. Problem Statement

The paper investigates the analytic properties (specifically meromorphic continuation and functional equations) of a Dirichlet series constructed from the Fourier-Jacobi coefficients of two cusp forms associated with orthogonal groups of signature $(2, n+2)$ where $n \ge 1$.

Let $F$ and $G$ be orthogonal cusp forms of weight $k$ on the group $G = SO(2, n+2)$. Their Fourier-Jacobi expansion is given by:
$$F(Z) = \sum_{m \ge 1} \phi_m(\tau, z) e^{2\pi i m \omega}$$
where $Z = (\omega, z, \tau)$ lies in a tube domain $H_S \subset \mathbb{C}^{n+2}$. The coefficients $\phi_m$ are Jacobi forms of index $m$. The object of study is the Dirichlet series:
$$D_{F,G}(s) = \sum_{m=1}^{\infty} \langle \phi_m, \psi_m \rangle m^{-s}$$
where $\langle \cdot, \cdot \rangle$ is the Petersson inner product on the space of Jacobi forms.

The author aims to generalize previous results by Kohnen and Skoruppa (for Siegel modular forms) and Gritsenko (for Hermitian forms) to the orthogonal case. The specific challenges in the orthogonal setting involve the divergence of certain inner-product integrals and the complexity of the underlying lattice structures.

2. Methodology

The proof follows a classical "Rankin-Selberg" style approach adapted for orthogonal groups, consisting of three main stages:

A. Integral Representation

The author constructs an integral representation of $D_{F,G}(s)$ by pairing the cusp forms with a non-holomorphic Eisenstein series of Klingen type, denoted $E(W, s)$.

• Proposition 4.4: Establishes that the inner product $\langle F(\cdot)E(\cdot, s), G(\cdot) \rangle$ is proportional to $D_{F,G}(s + k - n - 1)$.
• This reduces the problem of analyzing the Dirichlet series to analyzing the analytic properties of the Klingen Eisenstein series $E(W, s)$.

B. Rewriting as an Epstein Zeta Function

To analyze $E(W, s)$, the author rewrites it in the form of an Epstein zeta function.

• Condition: This step requires that the lattice associated with the quadratic form has exactly one 1-dimensional cusp (i.e., $\#C_1(\Gamma_S) = 1$).
• Proposition 5.3: Under this condition, $E(W, s)$ is expressed as a sum over primitive vectors in the lattice, resembling an Epstein zeta function involving a majorant $R_W$ of the quadratic form.

C. Theta Correspondence and Differential Operators

The core innovation involves establishing a theta correspondence between the orthogonal group $SO(2, n+2)$ and the symplectic group $Sp_2(\mathbb{R})$.

• The Obstacle: Directly integrating the theta series $\Theta(Z, W)$ against a Siegel Eisenstein series for $Sp_2$ leads to divergence because the theta series contains terms of non-full rank (rank $< 2$).
• The Solution: The author employs a two-step differential operator strategy to eliminate these divergent terms:
  1. Maass-Shimura Operator ($\delta_k^{(r)}$): Applied first to increase the weight of the theta series. This requires the condition **$4 \mid n$** (since the initial weight is$-n/2$and the operator increases weight by$2r$, requiring$r = n/4$ to reach weight 0).
  2. Invariant Differential Operator ($R$): A specific $Sp_2(\mathbb{R})$-invariant operator (based on Deitmar and Krieg) is applied to the result. This operator annihilates the terms corresponding to non-full-rank matrices, rendering the inner product integral convergent.

3. Key Contributions and Results

Main Theorem (Theorem 8.2)

Under the assumptions that $n$ is divisible by 4 ($4 \mid n$) and the lattice has a single 1-dimensional cusp, the author proves a precise theta correspondence:
$$\langle \tilde{E}(Z, \chi, (s+1)/2 - r), R \delta_k^{(r)} \Theta(Z, W) \rangle_{\Gamma_0^{(2)}(q)} = \xi(s)\xi(s-1)\gamma_S(s) E(W, s)$$
where:

• $\tilde{E}$ is a Siegel Eisenstein series for $Sp_2$.
• $\xi(s)$ is the completed Riemann zeta function.
• $\gamma_S(s)$ is an explicit gamma factor.
• $R$ and $\delta_k^{(r)}$ are the differential operators described above.

Meromorphic Continuation (Corollary 8.4)

As a direct consequence of the main theorem and the known analytic properties of Siegel Eisenstein series, the Dirichlet series $D_{F,G}(s)$ admits a meromorphic continuation to the entire complex plane $\mathbb{C}$.

Functional Equation for the $E_8$ Lattice (Theorem 9.1)

The author applies the general theory to the specific case where the underlying lattice is the $E_8$ lattice.

• The $E_8$ lattice is unimodular, even, and satisfies the "single cusp" condition.
• Because the level $q=1$, the character $\chi$ is trivial, and the symplectic Eisenstein series corresponds to the full modular group $Sp_2(\mathbb{Z})$.
• Result: The paper derives a precise functional equation for the completed Dirichlet series $D^*_{F,G}(s)$:
  $$D^*_{F,G}(s) = D^*_{F,G}(2k - 9 - s)$$
  This provides a complete analytic description of the series in this specific, highly symmetric case.

4. Significance

• Generalization: This work extends the Kohnen-Skoruppa theory from Siegel and Hermitian modular forms to the orthogonal case, filling a gap in the literature regarding Fourier-Jacobi coefficients for $SO(2, n+2)$.
• Methodological Advance: The paper demonstrates a robust method for handling divergent inner products in theta correspondences for orthogonal groups. By combining the Maass-Shimura operator with the invariant differential operator $R$ (originally developed by Deitmar and Krieg), the author bypasses the need for explicit construction of operators in every new case, showing their existence and applicability in the orthogonal setting.
• Lattice Theory Connection: The results highlight the deep connection between the arithmetic properties of lattices (specifically the number of cusps and the genus class number) and the analytic behavior of associated automorphic L-functions. The $E_8$ case serves as a concrete, computable example where these abstract properties yield a clean functional equation.
• Future Applications: The meromorphic continuation and functional equation are prerequisites for studying special values of these series and their potential relation to spinor L-functions or other arithmetic invariants, a direction hinted at in the author's previous work.