On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety
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arXiv
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| Format: | Preprint |
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2023
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| author | Salazar, Daniel Barrera Graham, Andrew Williams, Chris |
| author_facet | Salazar, Daniel Barrera Graham, Andrew Williams, Chris |
| contents | Friedberg--Jacquet proved that if $π$ is a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A})$, then $π$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H = \mathrm{GL}_n \times \mathrm{GL}_n$ is non-vanishing on $π$. We conjecture a $p$-refined analogue: that any $P$-parahoric $p$-refinement $\tildeπ^P$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a $P$-twisted version of $Z_H$ is non-vanishing on the $\tildeπ^P$-eigenspace in $π$. This twisted $Z_H$ appears in all constructions of $p$-adic $L$-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the $\mathrm{GL}_{2n}$ eigenvariety, and -- by proving upper bounds on the dimensions of such families -- obtain various results towards the conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02649 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety Salazar, Daniel Barrera Graham, Andrew Williams, Chris Number Theory 11F33, 11F67 (Primary) 11R23, 11G22 (Secondary) Friedberg--Jacquet proved that if $π$ is a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A})$, then $π$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a global zeta integral $Z_H$ over $H = \mathrm{GL}_n \times \mathrm{GL}_n$ is non-vanishing on $π$. We conjecture a $p$-refined analogue: that any $P$-parahoric $p$-refinement $\tildeπ^P$ is a functorial transfer from $\mathrm{GSpin}_{2n+1}$ if and only if a $P$-twisted version of $Z_H$ is non-vanishing on the $\tildeπ^P$-eigenspace in $π$. This twisted $Z_H$ appears in all constructions of $p$-adic $L$-functions via Shalika models. We connect our conjecture to the study of classical symplectic families in the $\mathrm{GL}_{2n}$ eigenvariety, and -- by proving upper bounds on the dimensions of such families -- obtain various results towards the conjecture. |
| title | On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety |
| topic | Number Theory 11F33, 11F67 (Primary) 11R23, 11G22 (Secondary) |
| url | https://arxiv.org/abs/2308.02649 |