AI 中文总结
本文在正特征下证明了Greenberg的μ=0猜想的类比，针对全局函数域上的lisheaves，展示了Selmer群的结构及相关变形环的性质。
AI 中文摘要
设\(K\)是特征\(p>0\)的全局函数域，\(\ell\neq p\)是素数。我们研究\(\mathbb{Z}_\ell\) - 扩张\(K_\infty/K\)上的塞尔默群。对于一个光滑\(\mathbb{Z}_\ell\) - 层，我们证明相关塞尔默群的庞特里亚金对偶是岩泽代数上的有限生成挠模且\(\mu\) - 不变量等于零。这给出了格林伯格\(\mu = 0\)猜想在正特征且与\(p\)互素情形下的类似结果。该结果特别适用于阿贝尔簇、精细塞尔默群和伴随表示。我们还证明了在此情形下关于\(K_\infty\)的弱 Leopoldt 猜想的类似结果，并推断剩余表示的带框变形环是形式幂级数环。若剩余表示没有非标量自同态，无框变形环也有相同结论。
英文摘要
Let $K$ be a global function field of characteristic $p>0$ and $\ell\neq p$ be a prime number. We study Selmer groups over a $\mathbb{Z}_\ell$-extension $K_\infty/K$. For a lisse $\mathbb Z_\ell$-sheaf we prove that the Pontryagin dual of the associated Selmer group is a finitely generated torsion module over the Iwasawa algebra and has $μ$-invariant equal to zero. This gives a positive-characteristic, prime to $p$, analogue of Greenberg's $μ=0$ conjecture. Our result applies in particular to abelian varieties, fine Selmer groups, and adjoint representations. We also prove an analogue of the weak Leopoldt conjecture in this context over $K_\infty$, and deduce that the framed deformation ring of a residual representation is a formal power series ring. The same conclusion holds for the unframed deformation ring if the residual representation has no non-scalar endomorphisms.
CommentsVersion 2: 22 pages, updated references, added a section on Greenberg Selmer groups
