A $p$-adic arithmetic inner product formula
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917595781791744 |
|---|---|
| author | Disegni, Daniel Liu, Yifeng |
| author_facet | Disegni, Daniel Liu, Yifeng |
| contents | Fix a prime number $p$ and let $E/F$ be a CM extension of number fields in which $p$ splits relatively. Let $π$ be an automorphic representation of a quasi-split unitary group of even rank with respect to $E/F$ such that $π$ is ordinary above $p$ with respect to the Siegel parabolic subgroup. We construct the cyclotomic $p$-adic $L$-function of $π$, and show, under certain conditions, that if its order of vanishing at the trivial character is $1$, then the rank of the Selmer group of the Galois representation of $E$ associated with $π$ is at least $1$. Furthermore, under a certain modularity hypothesis, we use special cycles on unitary Shimura varieties to construct some explicit elements in the Selmer group called Selmer theta lifts; and we prove a precise formula relating their $p$-adic heights to the derivative of the $p$-adic $L$-function. In parallel to Perrin-Riou's $p$-adic analogue of the Gross--Zagier formula, our formula is the $p$-adic analogue of the arithmetic inner product formula recently established by Chao~Li and the second author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_09239 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A $p$-adic arithmetic inner product formula Disegni, Daniel Liu, Yifeng Number Theory Algebraic Geometry Representation Theory 11G18, 11G40, 11G50, 11R34 Fix a prime number $p$ and let $E/F$ be a CM extension of number fields in which $p$ splits relatively. Let $π$ be an automorphic representation of a quasi-split unitary group of even rank with respect to $E/F$ such that $π$ is ordinary above $p$ with respect to the Siegel parabolic subgroup. We construct the cyclotomic $p$-adic $L$-function of $π$, and show, under certain conditions, that if its order of vanishing at the trivial character is $1$, then the rank of the Selmer group of the Galois representation of $E$ associated with $π$ is at least $1$. Furthermore, under a certain modularity hypothesis, we use special cycles on unitary Shimura varieties to construct some explicit elements in the Selmer group called Selmer theta lifts; and we prove a precise formula relating their $p$-adic heights to the derivative of the $p$-adic $L$-function. In parallel to Perrin-Riou's $p$-adic analogue of the Gross--Zagier formula, our formula is the $p$-adic analogue of the arithmetic inner product formula recently established by Chao~Li and the second author. |
| title | A $p$-adic arithmetic inner product formula |
| topic | Number Theory Algebraic Geometry Representation Theory 11G18, 11G40, 11G50, 11R34 |
| url | https://arxiv.org/abs/2204.09239 |