AI 中文总结
研究将Pila-Zannier策略扩展到Drinfeld模，证明了相关定理类似物。在特征零中，用刚性解析Pila-Wilkie定理替代基于\(o\)-极小性的计数步骤，通过点数论证建立函数超越性输入，实现该策略在函数域的应用。
AI 中文摘要
我们将Pila-Zannier策略扩展到Drinfeld模：证明了等秩两个Drinfeld模乘积的Manin-Mumford定理类似物，以及两个Drinfeld模曲线乘积的André-Oort定理类似物。在特征零中，该策略的几个步骤基于\(o\)-极小性，在函数域上无对应物；我们用Binyamini-Kato的刚性解析Pila-Wilkie定理代替计数步骤，这似乎是其首个算术应用。通过独立的点数论证建立了函数超越性输入，即在两种情况下Ax-Lindemann定理的类似物。
英文摘要
We extend the Pila-Zannier strategy to Drinfeld modules: we prove analogues of the Manin-Mumford theorem for a product of two Drinfeld modules of equal rank, and of the André-Oort theorem for a product of two Drinfeld modular curves. In characteristic zero, several steps of this strategy rest on $o$-minimality, which has no counterpart over a function field; we replace the counting step by the rigid analytic Pila-Wilkie theorem of Binyamini-Kato, and this appears to be its first arithmetic application. The functional transcendence input, namely an analogue of the Ax-Lindemann theorem in both settings, is established here by an independent point counting argument.
Comments41 pages, comments welcome
