Toric periods for a $p$-adic quaternion algebra
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915435245469696 |
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| author | Anandavardhanan, U. K. Pattanayak, Basudev |
| author_facet | Anandavardhanan, U. K. Pattanayak, Basudev |
| contents | Let $G$ be a compact group with two given subgroups $H$ and $K$. Let $π$ be an irreducible representation of $G$ such that its space of $H$-invariant vectors as well as the space of $K$-invariant vectors are both one dimensional. Let $v_H$ (resp. $v_K$) denote an $H$-invariant (resp. $K$-invariant) vector of unit norm in a given $G$-invariant inner product $\langle ~,~ \rangle_π$ on $π$. We are interested in calculating the correlation coefficient \[c(π;H,K) = |\langle v_H,v_K \rangle_π|^2.\] In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the $p$-adic quaternion algebra with respect to any two tori. In particular, if $π$ is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori $H$ and $K$, then its root number $\varepsilon(π)$ is $\pm 1$ and $c(π; H, K)$ is non-vanishing precisely when $\varepsilon(π) = 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09765 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Toric periods for a $p$-adic quaternion algebra Anandavardhanan, U. K. Pattanayak, Basudev Number Theory Representation Theory 11F70, 22E50 Let $G$ be a compact group with two given subgroups $H$ and $K$. Let $π$ be an irreducible representation of $G$ such that its space of $H$-invariant vectors as well as the space of $K$-invariant vectors are both one dimensional. Let $v_H$ (resp. $v_K$) denote an $H$-invariant (resp. $K$-invariant) vector of unit norm in a given $G$-invariant inner product $\langle ~,~ \rangle_π$ on $π$. We are interested in calculating the correlation coefficient \[c(π;H,K) = |\langle v_H,v_K \rangle_π|^2.\] In this paper, we compute the correlation coefficient of an irreducible representation of the multiplicative group of the $p$-adic quaternion algebra with respect to any two tori. In particular, if $π$ is such an irreducible representation of odd minimal conductor with non-trivial invariant vectors for two tori $H$ and $K$, then its root number $\varepsilon(π)$ is $\pm 1$ and $c(π; H, K)$ is non-vanishing precisely when $\varepsilon(π) = 1$. |
| title | Toric periods for a $p$-adic quaternion algebra |
| topic | Number Theory Representation Theory 11F70, 22E50 |
| url | https://arxiv.org/abs/2304.09765 |