Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| author | Cauchi, Antonio Lemma, Francesco Jacinto, Joaquín Rodrigues |
| author_facet | Cauchi, Antonio Lemma, Francesco Jacinto, Joaquín Rodrigues |
| contents | We study instances of Beilinson-Tate conjectures for automorphic representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at $s=1$. We construct algebraic cycles of codimension three in the Siegel-Shimura variety of dimension six and we relate its regulator to the residue at $s=1$ of the $L$-function of certain cuspidal forms of $\mathrm{PGSp}_6$. Using the exceptional theta correspondence between the split group of type $G_2$ and $\mathrm{PGSp}_6$ and assuming the non-vanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank $7$ motives of type $G_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_09394 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$ Cauchi, Antonio Lemma, Francesco Jacinto, Joaquín Rodrigues Number Theory We study instances of Beilinson-Tate conjectures for automorphic representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at $s=1$. We construct algebraic cycles of codimension three in the Siegel-Shimura variety of dimension six and we relate its regulator to the residue at $s=1$ of the $L$-function of certain cuspidal forms of $\mathrm{PGSp}_6$. Using the exceptional theta correspondence between the split group of type $G_2$ and $\mathrm{PGSp}_6$ and assuming the non-vanishing of a certain archimedean integral, this allows us to confirm a conjecture of Gross and Savin on rank $7$ motives of type $G_2$. |
| title | Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2202.09394 |