🔢 mathematics

Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$

This paper establishes an isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level 2, derives explicit Fourier coefficients for the corresponding Jacobi-Eisenstein series, and constructs a weight $3/2$ modular form from the Zagier-Eisenstein series to demonstrate a shared Hecke eigenvalue structure among four specific functions.

Original authors: Shuichi Hayashida

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shuichi Hayashida

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of mathematics as a vast, bustling city. In this city, there are different neighborhoods, each with its own unique architecture and rules. Some neighborhoods are built on Elliptic Modular Forms (let's call them the "City Center"), known for their rigid, symmetrical structures. Others are the Jacobi Forms (the "Garden District"), which are more complex, living in a space that combines time and space (or in math terms, a complex variable and a vector).

For a long time, mathematicians knew there was a secret tunnel connecting a specific part of the Garden District (Jacobi forms with a simple "index") to the City Center. But the map was incomplete. They suspected there were tunnels connecting the entire Garden District to the City Center, but they couldn't prove it, nor could they figure out exactly how to walk through them.

This paper by Shuichi Hayashida is like a master architect who finally draws the complete map, builds the bridges, and explains exactly how to travel between these two worlds.

Here is the breakdown of what the paper achieves, using simple analogies:

1. The Grand Connection (The Isomorphism)

The Problem: The author wanted to prove that the "Garden District" of Jacobi forms (specifically those with an index related to a root lattice called $D_{2n+1}$ ) is actually just a different way of looking at a specific part of the "City Center" (Elliptic Modular forms of Level 2).

The Analogy: Imagine you have a set of complex, multi-colored Lego sculptures (Jacobi forms). You also have a set of simpler, single-colored bricks (Modular forms). For years, people thought the complex sculptures were just random piles of bricks. Hayashida proves that every single complex sculpture is actually a perfect, one-to-one translation of a specific arrangement of simple bricks.

He doesn't just say they are similar; he builds a "dictionary" (an isomorphism) that translates every feature of the complex sculpture into the simple brick structure without losing any information. This confirms a conjecture made by another mathematician, A. Mocanu.

2. The Translation Guide (Explicit Formulas)

The Problem: Now that we know the two worlds are connected, how do we actually translate a specific number from one to the other? If you have a complex pattern in the Garden, what are the exact numbers in the City Center?

The Analogy: Think of the Jacobi forms as a secret code written in a complex cipher. The paper provides the decoder ring. It gives an explicit formula to take the "code" (Fourier coefficients) of the Jacobi forms and instantly convert it into the "plain text" of the Elliptic Modular forms.

Specifically, the author shows how to decode the "Jacobi-Eisenstein series" (a special type of pattern in the Garden) by breaking it down into a combination of "Cohen-Eisenstein series" (a known type of pattern). It's like saying, "To understand this complex song, just listen to these three simpler songs playing together."

3. The Special Bridge to Level 8 (Weight 3/2)

The Problem: There is a very special, rare type of building in the City Center called a "Level 8" structure. It's hard to build because it requires a very specific weight (3/2). The author wanted to show how to construct this rare building using a famous, slightly different structure called the "Zagier-Eisenstein series."

The Analogy: Imagine the Zagier-Eisenstein series is a famous, slightly wobbly tower. The author shows you how to take that tower, apply a specific "filter" (a mathematical operation called $U(4)$ ), and magically stabilize it into a perfect, new tower of Level 8.

The "Magic" Discovery: In doing this, the author discovers a strange coincidence. Four completely different mathematical objects (a specific theta function, the Zagier series, a non-holomorphic Eisenstein series, and a new modular form) all "sing" the same tune. When you test them with a specific mathematical probe (a Hecke operator), they all respond with the exact same number: $1 + p$ (where $p$ is a prime number). It's like four different instruments playing the exact same note at the same time, proving they are fundamentally linked.

4. Counting Squares (Arithmetic Applications)

The Problem: Math isn't just about abstract shapes; it's about counting things. One classic problem is: "In how many ways can you write a number as a sum of three squares?" (e.g., $5 = 2^2 + 1^2 + 0^2$ ).

The Analogy: The paper uses the new bridges it built to solve an old counting puzzle. By translating the complex Jacobi forms into the simpler modular forms, the author derives a new, clean formula for counting these squares.

It turns out that the number of ways to write a number as a sum of three squares is directly related to something called "Hurwitz class numbers" (a way of counting specific types of number systems). The paper provides a precise recipe:

Take the class number of 4 times the number, subtract twice the class number of the number, and multiply by 12.

This connects the abstract world of "modular forms" to the concrete world of "counting squares," showing that deep, invisible structures govern how numbers break down.

Summary

In short, this paper is a universal translator for a specific branch of number theory.

It connects two worlds: It proves that complex Jacobi forms are just disguised versions of simpler modular forms.
It provides the dictionary: It gives the exact formulas to translate between them.
It builds a new landmark: It constructs a rare mathematical object (Level 8, Weight 3/2) from existing pieces.
It solves a puzzle: It uses these connections to give a new, elegant formula for counting how numbers can be written as sums of squares.

Hayashida didn't just find a path; he paved a highway, complete with signposts and a guidebook, allowing mathematicians to travel freely between these two beautiful mathematical landscapes.

1. Problem Statement

The paper addresses the structural relationship between Jacobi forms of lattice index and elliptic modular forms. While the isomorphism between Jacobi forms of integer index $m$ and a subspace of elliptic modular forms of level $m$ was established by Eichler and Zagier (1988), the case of lattice index (specifically the root lattices $D_r$ for odd $r$ ) remained largely conjectural.

The primary objectives are:

To rigorously establish an isomorphism between the space of Jacobi forms of weight $k + \frac{r+1}{2}$ and index $D_r$ (where $r$ is an odd integer) and specific subspaces of elliptic modular forms of weight $2k$ and level 2.
To provide explicit formulas for the Fourier coefficients of Jacobi–Eisenstein series of index $D_r$ .
To construct specific holomorphic modular forms of weight $3/2$ and level 8 from Zagier–Eisenstein series and analyze their Hecke eigenvalues.

2. Methodology

The author employs a multi-step approach combining the theory of Siegel modular forms, theta decompositions, and Shimura correspondences:

Ikeda Lifting and Fourier-Jacobi Coefficients: The paper utilizes the Ikeda lift, which maps elliptic modular forms of weight $2k$ to Siegel modular forms of degree $r+1$ and weight $k + \frac{r+1}{2}$ . By extracting the Fourier-Jacobi coefficient corresponding to the maximal matrix associated with the lattice $D_r$ , the author constructs a map from elliptic modular forms to Jacobi forms. This image is identified as the space of "oldforms."
Theta Decomposition: The author decomposes Jacobi forms of index $D_r$ using theta functions associated with the discriminant module $D_r^\sharp / D_r$ . This decomposition allows the reduction of Jacobi forms to vector-valued modular forms of half-integral weight.
Shimura Correspondence: The paper leverages the Shimura correspondence (specifically the version for Kohnen plus spaces of level 8 established by Ueda and Yamauchi) to link the half-integral weight modular forms (derived from the theta decomposition) to integral weight elliptic modular forms of level 2.
Twisted Hecke Operators: To handle the specific symmetries of the lattices and the $\eta$ -type modular forms, the author defines and utilizes twisted Hecke operators ( $\tilde{T}(p^2)$ ) and standard Hecke operators ( $T(p^2)$ , $T^J(p)$ ) to prove that the constructed maps preserve the Hecke module structure.

3. Key Contributions and Results

A. Proof of the Isomorphism Conjecture

The paper proves Conjecture 1 (proposed by A. Mocanu), establishing that for any odd integer $r$ and weight $k$ , the space of Jacobi forms $J_{k+\frac{r+1}{2}, D_r}$ decomposes as a Hecke module:
$J_{k+\frac{r+1}{2}, D_r} \cong M_{2k}^{\text{new}, \epsilon_2}(2) \oplus M_{2k}^{\epsilon_1}(1)$
where:

$\epsilon_2 = -\left(\frac{-8}{r}\right)$ and $\epsilon_1 = -\left(\frac{-4}{r}\right)$ are signs determined by the Kronecker symbol.
$M_{2k}^{\text{new}, \epsilon_2}(2)$ is the space of newforms of level 2 with specific Atkin-Lehner eigenvalue.
$M_{2k}^{\epsilon_1}(1)$ corresponds to the space of "oldforms" lifted from level 1.
The isomorphism holds for both cusp forms and the full space, distinguishing between odd and even weights.

B. Explicit Fourier Coefficients

The author derives an explicit formula for the Fourier coefficients of the Jacobi–Eisenstein series $E_{k+\frac{r+1}{2}, D_r}$ .

The coefficients are expressed as linear combinations of the Fourier coefficients of Cohen–Eisenstein series ( $H_k$ ) of weight $k+1/2$ and level 4.
This provides a concrete computational tool for these series, generalizing previous results for general lattice indices.

C. Weight $3/2$ Modular Forms and Hecke Eigenvalues

A significant result concerns the case $k=1$ (weight $3/2$ for Jacobi forms). The paper shows that four distinct functions share the same Hecke eigenvalues ( $1+p$ for any odd prime $p$ ):

$\eta^3$ (Cube of the Dedekind eta function).
$\theta^3$ (Cube of the standard theta function).
$F$ (Zagier's non-holomorphic Eisenstein series of weight $3/2$ ).
$E_2^*$ (Non-holomorphic Eisenstein series of weight 2, modified).

Specifically, the author constructs a holomorphic modular form $E_{3/2}^{(8)} \in M_{3/2}^{+, -5}(8)$ (Kohnen plus space of level 8) from $\theta^3$ and $F$ . The paper proves:
$E_{3/2}^{(8)}(\tau) = (\theta^3 | U_1(4))(\tau) = 12(F | U_1(4) - 2F)(\tau)$
This leads to a new identity for the number of representations of an integer as a sum of three squares ( $r_3(N)$ ):
$r_3(N) = 12(H(4N) - 2H(N))$
where $H(N)$ is the Hurwitz class number.

D. Explicit Maps

The paper provides explicit formulas (Theorems 1.8 and 1.9) for linear maps $S_{d_0}$ from Jacobi forms to elliptic modular forms. These maps are defined via the Fourier coefficients of the Jacobi forms and involve Dirichlet characters and divisor sums, ensuring compatibility with Hecke operators.

4. Significance

Resolution of Conjecture: The work definitively resolves the conjecture regarding the isomorphism between Jacobi forms of odd lattice index $D_r$ and elliptic modular forms of level 2, extending the classical Eichler-Zagier theory.
Unified Framework: It unifies the treatment of Jacobi forms of odd and even weights, clarifying the role of "oldforms" and "newforms" in the context of lattice indices.
Arithmetic Applications: The derivation of the identity relating $r_3(N)$ to Hurwitz class numbers provides a new perspective on classical arithmetic functions, linking them directly to the theory of modular forms of half-integral weight and Zagier's work.
Computational Utility: The explicit formulas for Fourier coefficients allow for the practical computation of Jacobi–Eisenstein series, which are otherwise difficult to calculate directly.

In summary, Hayashida's paper bridges the gap between the theory of Jacobi forms on root lattices and classical elliptic modular forms, providing rigorous proofs, explicit constructions, and new arithmetic identities.