arXiv 2301.090732023-01math.NTmath.AG

全局域的有限$p$-扩张上阿贝尔簇的$μ$-不变量变化

The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields

Ki-Seng Tan, Fabien Trihan, Kwok-Wing Tsoi

备注：: v3, 38 pages, basically identical to v2, with clarifications regarding certain citations from [LLSTT21]

AI总结：

该研究拓展前人工作，分析全局域$\bZ_p^d$-扩张上常模阿贝尔簇的$μ$-不变量在有限Galois $p$-扩张下的变化，得到特征$p>0$时超奇异位处局部Galois上同调大小的显式界，给出所有特征下的渐近增长与不变量变化下界，并提供数值验证。

摘要：

我们拓展了Lai、Longhi、Suzuki以及前两位作者的工作，研究全局域的$\bZ_p^d$-扩张$L/K$上的常模阿贝尔簇$A$的$μ$-不变量在有限Galois $p$-扩张$K'/K$下的变化，该扩张仅在$A$具有常模约化的有限个位处分歧。在特征$p>0$的情形下，我们得到了$A$的Mordell-Weil群关于在超奇异位$v$处分歧的$p$-扩张的局部Galois上同调大小$δ_v$的显式界。接下来，在所有特征下，我们描述了$δ_v$沿多重$\bZ_p$-扩张$L/K$的渐近增长，并给出了$A$的$μ$-不变量从塔$L/K$到塔$LK'/K'$的变化的下界。最后，我们给出了支撑这些结果的数值证据。

英文摘要：

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.

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摘要：

英文摘要：

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