Zeros of theta functions associated with self-dual lattices
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917222433161216 |
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| author | Raveh, Roei |
| author_facet | Raveh, Roei |
| contents | We study the zeros of theta functions $Θ_{Γ_{4k}}$ associated with the lattices $Γ_{4k}$, a family of self-dual lattices generalizing the $\mathsf{E}_{8}$ lattice. Our results show two different behaviors of the zeros according to the lattice parity: When $Γ_{4k}$ is an even lattice, we show that the zeros all lie on the line $\Re z =\frac{1}{2}$ in the fundamental domain and prove that the zeros are equidistributed with respect to an explicit probability measure on the line $\Re z = \frac{1}{2}$. However, when the $Γ_{4k}$ is an odd lattice, there are no zeros on the line $\Re z =\frac{1}{2}$, only exponentially close to it. Our argument relies on representing $Θ_{Γ_{4k}}$ as a polynomial in the modular $λ$-function. We then study the zeros of this polynomial and exploit some conformal properties of $λ$ to get our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of theta functions associated with self-dual lattices Raveh, Roei Number Theory Classical Analysis and ODEs We study the zeros of theta functions $Θ_{Γ_{4k}}$ associated with the lattices $Γ_{4k}$, a family of self-dual lattices generalizing the $\mathsf{E}_{8}$ lattice. Our results show two different behaviors of the zeros according to the lattice parity: When $Γ_{4k}$ is an even lattice, we show that the zeros all lie on the line $\Re z =\frac{1}{2}$ in the fundamental domain and prove that the zeros are equidistributed with respect to an explicit probability measure on the line $\Re z = \frac{1}{2}$. However, when the $Γ_{4k}$ is an odd lattice, there are no zeros on the line $\Re z =\frac{1}{2}$, only exponentially close to it. Our argument relies on representing $Θ_{Γ_{4k}}$ as a polynomial in the modular $λ$-function. We then study the zeros of this polynomial and exploit some conformal properties of $λ$ to get our results. |
| title | Zeros of theta functions associated with self-dual lattices |
| topic | Number Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.06128 |