Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910763293081600 |
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| author | Liu, Yifeng Tian, Yichao Xiao, Liang |
| author_facet | Liu, Yifeng Tian, Yichao Xiao, Liang |
| contents | In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for unitary groups. We prove one-side divisibility of the corresponding Iwasawa main conjecture: when the global root number is $1$, the $p$-adic $L$-function belongs to the characteristic ideal of the Iwasawa Bloch--Kato Selmer group; when the global root number is $-1$, the square of the characteristic ideal of a certain Iwasawa module is contained in the characteristic ideal of the torsion part of the Iwasawa Bloch--Kato Selmer group (analogous to Perrin-Riou's Heegner point main conjecture). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_00624 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case Liu, Yifeng Tian, Yichao Xiao, Liang Number Theory 11F33, 11G05, 11G18, 11G40, 11R34 In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for unitary groups. We prove one-side divisibility of the corresponding Iwasawa main conjecture: when the global root number is $1$, the $p$-adic $L$-function belongs to the characteristic ideal of the Iwasawa Bloch--Kato Selmer group; when the global root number is $-1$, the square of the characteristic ideal of a certain Iwasawa module is contained in the characteristic ideal of the torsion part of the Iwasawa Bloch--Kato Selmer group (analogous to Perrin-Riou's Heegner point main conjecture). |
| title | Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case |
| topic | Number Theory 11F33, 11G05, 11G18, 11G40, 11R34 |
| url | https://arxiv.org/abs/2406.00624 |