Romik's Conjecture for the Jacobi Theta Function
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arXiv
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866912473736544256 |
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| author | Wakhare, Tanay |
| author_facet | Wakhare, Tanay |
| contents | Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point $i$. He then conjectured that the Taylor coefficients $d(n)$ either vanish or are periodic modulo any prime ${p}$; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, who considered arbitrary half integral weight modular forms. We refine previous work for $p \equiv 1 \pmod{4}$ by displaying a concise algebraic relation between $d\left( n+ \frac{p-1}{2} \right)$ and $d(n)$ related to the $p$-adic factorial, from which we can deduce periodicity with an effective period. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_01485 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Romik's Conjecture for the Jacobi Theta Function Wakhare, Tanay Number Theory Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point $i$. He then conjectured that the Taylor coefficients $d(n)$ either vanish or are periodic modulo any prime ${p}$; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, who considered arbitrary half integral weight modular forms. We refine previous work for $p \equiv 1 \pmod{4}$ by displaying a concise algebraic relation between $d\left( n+ \frac{p-1}{2} \right)$ and $d(n)$ related to the $p$-adic factorial, from which we can deduce periodicity with an effective period. |
| title | Romik's Conjecture for the Jacobi Theta Function |
| topic | Number Theory |
| url | https://arxiv.org/abs/1909.01485 |