AI 中文总结
本文构造了一族无穷反例，否证了 Goss 关于分圆函数域 L-函数的猜想，通过精确同余式证明 deg_X g 可任意大。
AI 中文摘要
设 p 为素数，q = p，A = F_p[T]，并设 P 为次数为 d 的首一不可约多项式，其分圆函数域为 K_P，模 P 的 Teichmuller 特征为 omega_P。对于特征 chi = omega_P^i，Goss 定义了 g(X, chi)，即 Artin L-函数 L(X, chi) 的“同余于 1 模 p”的部分，并猜想对于每个 q-幻指数 i，有 deg_X g(X, omega_P^i) <= 1；这是函数域上 Vandiver 猜想的类似物，由 Angles 提出作为开放问题。我们否证了该猜想。对于每个 a 属于 F_p^*，令 P_a = T^p - T - a，其为次数为 p 的不可约多项式，并令 i = p^n - 1，其中 0 <= n <= p-1，这是一个 q-幻指数。我们证明了一个精确的同余式，表明 L-多项式模 P_a 约化为 (1-X)^n，从而 deg_X g = n-1。对于 p >= 5 且 3 <= n <= p-1，这给出 deg_X g >= 2，与 Goss 猜想矛盾；此外，deg_X g = n-1 无界，达到 p-2。
英文摘要
Let p be a prime, q = p, A = F_p[T], and let P be a monic irreducible of degree d with cyclotomic function field K_P and mod-P Teichmuller character omega_P. For a character chi = omega_P^i, Goss defined g(X, chi), the "congruent to one modulo p" part of the Artin L-function L(X, chi), and conjectured that deg_X g(X, omega_P^i) <= 1 for every q-magic index i; this is an analogue for function fields of Vandiver's conjecture, and was raised as an open problem by Angles. We disprove it. For every a in F_p^*, put P_a = T^p - T - a, irreducible of degree p, and let i = p^n - 1 with 0 <= n <= p-1, a q-magic index. We prove an exact congruence showing that the L-polynomial reduces to (1-X)^n modulo P_a, whence deg_X g = n-1. For p >= 5 and 3 <= n <= p-1 this gives deg_X g >= 2, contradicting Goss's conjecture; moreover deg_X g = n-1 is unbounded, attaining p-2.
Comments6 pages. Ancillary file: a self-contained verification program (pure Python, standard library only) reproducing both the mod-P theorem and the characteristic-zero Newton polygon computation
