$L$-function for $\mathrm{Sp}(4)\times\mathrm{GL}(2)$ via a non-unique model
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866911425081901056 |
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| author | Yan, Pan |
| author_facet | Yan, Pan |
| contents | In this paper we prove a conjecture of Ginzburg and Soudry on an integral representation for the $L$-function $L^S(s, π\times τ)$ attached to a pair $(π, τ)$ of irreducible automorphic cuspidal representations of $\mathrm{Sp}_4({\mathbb A})$ and $\mathrm{GL}_2({\mathbb A})$, which is derived from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis. Two applications are given. First, we relate the existence of the poles of $L^S(s,π\timesτ)$ to the non-vanishing of certain period integrals. Second, for certain family of cuspidal representations, we prove that $L^S(s, π\times τ)$ is holomorphic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_05693 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $L$-function for $\mathrm{Sp}(4)\times\mathrm{GL}(2)$ via a non-unique model Yan, Pan Number Theory Representation Theory 11F70 (Primary) 11F55, 22E50, 22E55 (Secondary) In this paper we prove a conjecture of Ginzburg and Soudry on an integral representation for the $L$-function $L^S(s, π\times τ)$ attached to a pair $(π, τ)$ of irreducible automorphic cuspidal representations of $\mathrm{Sp}_4({\mathbb A})$ and $\mathrm{GL}_2({\mathbb A})$, which is derived from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis. Two applications are given. First, we relate the existence of the poles of $L^S(s,π\timesτ)$ to the non-vanishing of certain period integrals. Second, for certain family of cuspidal representations, we prove that $L^S(s, π\times τ)$ is holomorphic. |
| title | $L$-function for $\mathrm{Sp}(4)\times\mathrm{GL}(2)$ via a non-unique model |
| topic | Number Theory Representation Theory 11F70 (Primary) 11F55, 22E50, 22E55 (Secondary) |
| url | https://arxiv.org/abs/2110.05693 |