The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields
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arXiv
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| Format: | Preprint |
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2023
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| author | Tan, Ki-Seng Trihan, Fabien Tsoi, Kwok-Wing |
| author_facet | Tan, Ki-Seng Trihan, Fabien Tsoi, Kwok-Wing |
| contents | We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $μ$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $δ_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $δ_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $μ$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09073 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields Tan, Ki-Seng Trihan, Fabien Tsoi, Kwok-Wing Number Theory Algebraic Geometry 11R23, 11G10, 11S40, 14J27 We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $μ$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $δ_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $δ_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $μ$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results. |
| title | The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields |
| topic | Number Theory Algebraic Geometry 11R23, 11G10, 11S40, 14J27 |
| url | https://arxiv.org/abs/2301.09073 |