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

There are three aims in this paper: (i) We show an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) We give an explicit formula of Fourier coefficients of Jacobi-Eisenstein series of index $D_{2n+1}$. (iii) We construct a holomorphic modular form of weight $3/2$ of level $8$ from the Zagier Eisenstein series $\mathscr{F}$ of weight $3/2$ of level $4$. Moreover, we show that the four functions $E^_2$, $η^3$, $θ^3$ and $\mathscr{F}$ have essentially the same Hecke eigenvalue $1+p$ for any odd prime $p$, where $E^_2$ is the non-holomorphic Eisenstein series of weight $2$, $η$ is the Dedekind eta-function and $θ$ is the usual theta function. This fact follows from a special case of the isomorphism of (i). As an application, we give a formula for a sum of the numbers $r_3(n)$, where $r_3(n)$ is the number of representations of an integer $n \geq 0$ as a sum of $3$ squares.

💡 Research Summary

The paper establishes a precise Hecke‑module isomorphism between Jacobi forms of lattice index D_{2n+1} (the odd‑dimensional root lattice) and elliptic modular forms of level 2. Building on the classical Shimura‑Ikeda correspondence for integer index, the author proves A. Mocanu’s conjecture that for every integer k ≥ 2 the space
J_{k+r+1, D_r} ≅ M_{new, ε₂}^{2k}(2) ⊕ M_{ε₁}^{2k}(1)
holds when r is odd, where ε₁ and ε₂ are explicit signs determined by Kronecker symbols. The proof proceeds by composing two maps: an Ikeda lift from degree‑r Siegel modular forms to Jacobi forms, followed by extraction of the Fourier‑Jacobi coefficient, which lands in a subspace of level‑2 modular forms. The “old” part of the Jacobi space is shown to correspond exactly to the one‑dimensional space M_{ε₁}^{2k}(1), while the orthogonal complement (“new”) consists of cusp forms that match the Kohnen plus space S_{new, ε₂}^{2k}(2).

A second major contribution is an explicit formula for the Fourier coefficients of the Jacobi‑Eisenstein series E_{k+r+½, D_r}. The author introduces the Cohen Eisenstein series H_k of weight k+½ and level 4, then applies the operator U_k(4) to produce a level‑8 form H_k^. By proving that E_{k+r+½, D_r} coincides with H_k^, the coefficients of the Jacobi‑Eisenstein series are expressed as linear combinations of the well‑known coefficients of H_k. This yields a compact, computable description of these coefficients.

The third focus is the construction of a holomorphic weight 3/2 modular form of level 8, denoted E(8){3/2}, from Zagier’s weight 3/2 level 4 Eisenstein series 𝔽 and the cube of the theta function θ³. Using the operator U₁(4) the author shows
E(8){3/2}(τ) = (θ³ | U₁(4))(τ) = ∑{n≥0, n≡0,3 (mod 4)} r₃(n) qⁿ,
where r₃(n) counts representations of n as a sum of three squares. Moreover, E(8){3/2} satisfies the Hecke eigenvalue relation T_p E(8){3/2} = (1 + p) E(8){3/2} for every odd prime p, the same eigenvalue enjoyed by the non‑holomorphic Eisenstein series E₂* (weight 2), the Dedekind eta‑cube η³, and the theta‑cube θ³. This unifies four apparently unrelated objects under a single Hecke‑theoretic framework.

The paper also provides explicit linear maps S_{d₀} from Jacobi forms to the direct sum M_{new, ε₂}^{2k}(2) ⊕ M_{ε₁}^{2k}(1). These maps involve twisted L‑values, divisor sums, and the Kronecker symbols, and they commute with all Hecke operators. When the weight k + r + ½ is even, the maps send the old part of the Jacobi space to M_{ε₁}^{2k}(1) and the new part to M_{new, ε₂}^{2k}(2); when the weight is odd, a slightly different construction yields maps into the Kohnen plus space.

As an arithmetic application, the author derives a new identity for the three‑square representation numbers:
∑{0≤s≤√N, N−s²≡0,3 (mod 4)} δ̃_s r₃(N−s²) = σ(N) − 3σ(N/2) + 14σ(N/4) − 24σ(N/8),
where δ̃_s = ½ if s = 0 and 1 otherwise, and σ denotes the divisor‑sum function. This formula follows from the product θ · E(8){3/2} and the known basis of M₂(8). It provides a more efficient way to compute sums involving r₃ than the classical relation obtained from r₄.

In summary, the paper achieves three intertwined goals: (1) it proves the full conjectural isomorphism between Jacobi forms of index D_{2n+1} and level‑2 modular forms; (2) it supplies explicit Fourier coefficient formulas for Jacobi‑Eisenstein series via Cohen Eisenstein series; (3) it constructs a weight 3/2 level‑8 modular form linking theta‑cubes, Zagier’s Eisenstein series, and three‑square representations, all sharing the Hecke eigenvalue 1 + p. These results deepen the connections among Jacobi forms, Siegel modular forms, and classical modular forms, and they open new avenues for arithmetic applications involving representation numbers and class‑number formulas.