Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914432230096896 |
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| author | Hayashida, Shuichi |
| author_facet | Hayashida, Shuichi |
| contents | This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of Jacobi--Eisenstein series of index $D_{2n+1}$. (iii) To construct a holomorphic modular form of weight $3/2$ and level $8$ (and $4$) from the Zagier--Eisenstein series $\mathscr{F}$ of weight $3/2$ and 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 arises as a special case of the isomorphism of (i). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$ Hayashida, Shuichi Number Theory Primary 11F50, Secondary 11F37, 11F20, 11F46 This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of Jacobi--Eisenstein series of index $D_{2n+1}$. (iii) To construct a holomorphic modular form of weight $3/2$ and level $8$ (and $4$) from the Zagier--Eisenstein series $\mathscr{F}$ of weight $3/2$ and 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 arises as a special case of the isomorphism of (i). |
| title | Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$ |
| topic | Number Theory Primary 11F50, Secondary 11F37, 11F20, 11F46 |
| url | https://arxiv.org/abs/2512.15012 |