On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911377883398144 |
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| author | Salazar, Daniel Barrera Dimitrov, Mladen Graham, Andrew Jorza, Andrei Williams, Chris |
| author_facet | Salazar, Daniel Barrera Dimitrov, Mladen Graham, Andrew Jorza, Andrei Williams, Chris |
| contents | In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is étale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at $p$, and give $p$-adic variation of $L$-values (of regular algebraic cuspidal automorphic representations of $\mathrm{GL}(2n)$ admitting Shalika models) over the whole pure weight space. In the case of $\mathrm{GL}(4)$, these results have been used by Loeffler and Zerbes to prove cases of the Bloch--Kato conjecture for $\mathrm{GSp}(4)$.
Our main innovations are: (a) the introduction and systematic study of `Shalika refinements' of local representations of $\mathrm{GL}(2n)$, and evaluation of their attached local twisted zeta integrals; and (b) the $p$-adic interpolation of representation-theoretic branching laws for $\mathrm{GL}(n) \times \mathrm{GL}(n)$ inside $\mathrm{GL}(2n)$. Using (b), we give a construction of multi-variable $p$-adic functionals on the overconvergent cohomology groups for $\mathrm{GL}(2n)$, interpolating the zeta integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our main arithmetic applications. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_08126 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions Salazar, Daniel Barrera Dimitrov, Mladen Graham, Andrew Jorza, Andrei Williams, Chris Number Theory Primary 11F33, 11F67, Secondary 11R23, 11G22 In this paper, we prove that a $\mathrm{GL}(2n)$-eigenvariety is étale over the (pure) weight space at non-critical Shalika points, and construct multi-variable $p$-adic $L$-functions varying over the resulting Shalika components. Our constructions hold in tame level 1 and Iwahori level at $p$, and give $p$-adic variation of $L$-values (of regular algebraic cuspidal automorphic representations of $\mathrm{GL}(2n)$ admitting Shalika models) over the whole pure weight space. In the case of $\mathrm{GL}(4)$, these results have been used by Loeffler and Zerbes to prove cases of the Bloch--Kato conjecture for $\mathrm{GSp}(4)$. Our main innovations are: (a) the introduction and systematic study of `Shalika refinements' of local representations of $\mathrm{GL}(2n)$, and evaluation of their attached local twisted zeta integrals; and (b) the $p$-adic interpolation of representation-theoretic branching laws for $\mathrm{GL}(n) \times \mathrm{GL}(n)$ inside $\mathrm{GL}(2n)$. Using (b), we give a construction of multi-variable $p$-adic functionals on the overconvergent cohomology groups for $\mathrm{GL}(2n)$, interpolating the zeta integrals of (a). We exploit the resulting non-vanishing of these functionals to prove our main arithmetic applications. |
| title | On the GL(2n) eigenvariety: branching laws, Shalika families and $p$-adic $L$-functions |
| topic | Number Theory Primary 11F33, 11F67, Secondary 11R23, 11G22 |
| url | https://arxiv.org/abs/2211.08126 |