AI 中文总结
本文构造了 Carlitz 模的一族显式扭变，证明其解析秩无界，并给出了回路与轨道的一一对应及秩的精确公式。
AI 中文摘要
Grishkov 和 Logachev 在近期综述中提出问题（问题 1.4）：对于固定的有限域 F_q，Carlitz 模的扭变的解析秩是否有界。我们对每个 q >= 3 给出否定回答。设 q = p^s 且 n >= 1，令 P_{q,n} = (theta^(q^n) - theta)^(q-2)。我们证明 Grishkov-Logachev 的矩阵 M(P_{q,n},1,k) 的有向图具有完全显式的回路集：它们两两不相交，没有一个回路遇到涉及 t 的项，并且它们与 E^n 在循环旋转下的轨道一一对应，其中 E 是 [0,q-2] 中满足 p 不整除 l+1 的 l 的集合。因此 L(C_P, t, T) 不依赖于 t，它分解为每个轨道一项 1 - wt(nu) T^per(nu) 的乘积，解析秩等于权重为 1 的轨道上 p^(v_p(per(nu))) 之和。当 n 是 p 的幂时，这等于 (q/p)^n (p-1)^(n-1)，这是无界的。该论证也回答了该综述中关于此族的问题 3.6.1 至 3.6.4。
英文摘要
Grishkov and Logachev have asked (Problem 1.4 of their recent survey) whether, for a fixed finite field F_q, the analytic ranks of the twists of the Carlitz module are bounded. We answer this in the negative for every q >= 3. For q = p^s and n >= 1 put P_{q,n} = (theta^(q^n) - theta)^(q-2). We show that the directed graph of the matrix M(P_{q,n},1,k) of Grishkov-Logachev has a completely explicit set of circuits: they are pairwise disjoint, none of them meets an entry involving t, and they are in bijection with the orbits of E^n under cyclic rotation, where E is the set of l in [0,q-2] with p not dividing l+1. Consequently L(C_P, t, T) does not depend on t, it factors as a product of one term 1 - wt(nu) T^per(nu) per orbit, and the analytic rank equals the sum of p^(v_p(per(nu))) over the orbits of weight 1. When n is a power of p this equals (q/p)^n (p-1)^(n-1), which is unbounded. The argument also answers questions 3.6.1 to 3.6.4 of the survey for this family.
Comments7 pages. Ancillary file: verify_carlitz.py, an independent check of the rank formula for 37 pairs (q,n) in characteristics 2, 3, 5, 7, 11, 13
