等特征情形下的有界导子有限性猜想
On the bounded-conductor finiteness conjecture in equal characteristic
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中文总结 AI 辅助
本文研究等特征整体函数域上模p Galois表示的有界导子有限性猜想，建立了带提升假设的有限性定理，并构造反例证明无提升时猜想一般不成立，从而完全解决二维情形。
中文摘要 AI 辅助
我们研究了整体函数域上模$p$ Galois表示的Moon--Taguchi有界导子有限性猜想的等特征情形。我们首先建立一个条件性有限性定理：对于任意特征为$p$的整体函数域$K$和任意整数$n \ge 1$，存在有限多个连续、半单、处处非分歧且几何的表示$\rho: G_K \to \mathrm{GL}_n(\overline{\mathbf{F}}_p)$的同构类，这些表示允许一个处处非分歧的特征零提升。此外，我们证明在没有此提升假设的情况下，该猜想一般不成立。具体地，我们构造了无限多个特征为$p$的整体函数域$K$，对于每个整数$r \ge 4$，存在一个连续、满射、绝对不可约、处处非分歧且几何的表示$\rho_r: G_K \twoheadrightarrow \mathrm{SL}_2(\mathbf{F}_{p^r})$。作为推论，我们构造了一个处处非分歧的Galois扩张$L/K$，在$\mathbf F_p$上正则，其Galois群为$\mathrm{Gal}(L/K)\cong \prod_{r\geq4}\mathrm{PSL}_2(\mathbf F_{p^r})$。结合已知的跨特征有限性定理，这完全解决了Moon和Taguchi在二维情形提出的问题：这样的扩张存在于特征为$p$的整体函数域上，而不能存在于特征不同于$p$的整体函数域上。
英文摘要
We investigate the equal-characteristic case of the Moon--Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n \ge 1$, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations $ρ: G_K \to \mathrm{GL}_n(\overline{\mathbf{F}}_p)$ that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields $K$ of characteristic $p$ admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation $ρ_r: G_K \twoheadrightarrow \mathrm{SL}_2(\mathbf{F}_{p^r})$ for each integer $r \ge 4$. As a corollary, we construct an everywhere unramified Galois extension \(L/K\), regular over \(\mathbf F_p\), with Galois group \[ \mathrm{Gal}(L/K)\cong \prod_{r\geq4}\mathrm{PSL}_2(\mathbf F_{p^r}). \] Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic \(p\), whereas they cannot exist over global function fields of characteristic different from \(p\).
发表机构
- Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）
- Research Institute of Intelligent Complex Systems, Fudan University（复旦大学复杂系统智能研究院）
- Fakultät 8 – Mathematik und Physik, Universität Stuttgart（斯图加特大学第八学院——数学与物理系）
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