发表机构
Sun Yat-Sen University; Tsinghua University; Beijing Institute of Mathematical Sciences and Applications（中山大学; 清华大学; 北京数学科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明特殊函数模的Tate代数单位存在性等价于其秩一自由性，解决Gazda--Maurischat猜想；给出基本周期与Drinfeld对数的显式留数公式，并将Pellarin L(1)-级数表示为生成微分第一次Frobenius扭曲，适用于任意N≥1。
AI 中文摘要
设 \\(A\\) 为 \\(\mathbb{F}_q\\) 上光滑射影曲线的系数环，其无穷远闭点的次数为任意 \\(\mathrm{N}=°(\infty)\ge 1\\)，并设 \\(\varphi\\) 为秩一的Drinfeld \\(A\\)-模。本文证明了关于特殊函数模及其与Pellarin \\(\mathcal L(1)\\)-级数关系的三个主要结果。首先，特殊函数模 \\(\mathrm{sf}(\varphi)\\) 包含一个Tate代数单位当且仅当它是秩一自由的，等价地，当且仅当其周期格同构于正则微分模。这解决了Gazda--Maurischat猜想。证明通过计算Cauchy核完成；在周期格上所有这些求值一致，其公共特征留数恢复了周期。其次，对于到主周期目标 \\(\psi\\) 的适当同源数据，一个严格规范化的shtuka乘积连同单个特征留数给出了基本周期 \\(\tilde{\pi}\\)（自由格 \\(\Lambda_\psi\\) 的生成元）的显式公式。相同的亏量方程也导出了Drinfeld对数的留数公式。第三，规范化shtuka微分的第一次Frobenius扭曲被等同于由有限挠迹构造的配对，其系数是定义域上的有限étale迹。作为应用，Pellarin \\(\mathcal L(1)\\)-级数被精确地表示为基本周期处生成微分的第一次Frobenius扭曲。这些结果的一个显著特点是它们对任意 \\(\mathrm{N} \geq 1\\) 成立，且源周期格不必是自由的。
英文摘要
Let \(A\) be the coefficient ring of a smooth projective curve over \(\mathbb{F}_q\) with a closed point at infinity of arbitrary degree \(\mathrm{N}=°(\infty)\ge 1\), and let \(φ\) be a rank-one Drinfeld \(A\)-module. In this paper, we prove three main results concerning the module of special functions and its relation to Pellarin's \(\mathcal L(1)\)-series. First, the module \(\mathrm{sf}(φ)\) of special functions contains a Tate-algebra unit exactly when it is free of rank one, equivalently when its period lattice is isomorphic to the module of regular differentials. This settles the Gazda--Maurischat conjecture. The proof evaluates Cauchy kernels; on the period lattice all such evaluations agree, and their common characteristic residue recovers the period. Second, for suitable isogeny data to a principal-period target \(ψ\), a strictly normalized shtuka product, together with one characteristic residue, gives an explicit formula for the fundamental period \(\tildeπ\) (the generator of the free lattice \(Λ_ψ\)). The same defect equation also yields the residue formula for the Drinfeld logarithm. Third, the first Frobenius twist of the normalized shtuka differential is identified with a pairing built from finite torsion traces, whose coefficients are finite étale traces over the field of definition. As an application, Pellarin's \(\mathcal L(1)\)-series is expressed exactly as the first Frobenius twist of the generating differential at a fundamental period. A notable feature of these results is that they hold for arbitrary $\mathrm{N} \geq 1$, and the source period lattice need not be free.
Comments1 figure, 70 pages
