Horizontal norm compatibility of cohomology classes for $\mathrm{GSp}_{6}$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917769919856640 |
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| author | Shah, Syed Waqar Ali |
| author_facet | Shah, Syed Waqar Ali |
| contents | We establish abstract horizontal norm relations involving the unramified Hecke-Frobenius polynomials that correspond under the Satake isomorhpism to the degree eight spinor $L$-factors of $ \mathrm{GSp}_{6} $. These relations apply to classes in the degree seven motivic cohomology of the Siegel modular sixfold obtained via Gysin pushforwards of Beilinson's Eisenstein symbol pulled back on one copy in a triple product of modular curves. The proof is based on a novel approach that circumvents the failure of the so-called multiplicity one hypothesis in our setting, which precludes the applicability of an existing technique. In a sequel, we combine our result with the previously established vertical norm relations for these classes to obtain new Euler systems for the eight dimensional Galois representations associated with certain non-endoscopic cohomological cuspidal automorphic representations of $ \mathrm{GSp}_{6} $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_03738 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Horizontal norm compatibility of cohomology classes for $\mathrm{GSp}_{6}$ Shah, Syed Waqar Ali Number Theory Representation Theory 11R23, 11F70 (Primary) 20E42, 20G25, 22D99 (Secondary) We establish abstract horizontal norm relations involving the unramified Hecke-Frobenius polynomials that correspond under the Satake isomorhpism to the degree eight spinor $L$-factors of $ \mathrm{GSp}_{6} $. These relations apply to classes in the degree seven motivic cohomology of the Siegel modular sixfold obtained via Gysin pushforwards of Beilinson's Eisenstein symbol pulled back on one copy in a triple product of modular curves. The proof is based on a novel approach that circumvents the failure of the so-called multiplicity one hypothesis in our setting, which precludes the applicability of an existing technique. In a sequel, we combine our result with the previously established vertical norm relations for these classes to obtain new Euler systems for the eight dimensional Galois representations associated with certain non-endoscopic cohomological cuspidal automorphic representations of $ \mathrm{GSp}_{6} $. |
| title | Horizontal norm compatibility of cohomology classes for $\mathrm{GSp}_{6}$ |
| topic | Number Theory Representation Theory 11R23, 11F70 (Primary) 20E42, 20G25, 22D99 (Secondary) |
| url | https://arxiv.org/abs/2409.03738 |