关于塔库尔对卡利茨 - 维费里希素数猜想的一个反例
A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes
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中文总结 AI 辅助
研究塔库尔关于卡利茨 - 维费里希素数的猜想，通过构造在\(F_{19^3}\)上次数为\(5\)的不可约\(c -\)维费里希素数给出反例，还给出相关公因子封闭形式，表明次数\(5\)是最小反例次数，特定范围内素域\(F_p\)无反例。
中文摘要 AI 辅助
设\(A = F_q[T]\)，其中\(q\)是奇素数\(p\)的幂，\([n] = T^{q^n} - T\)，\(\rho\)是卡利茨模。\(A\)的一个首一素数\(P\)若满足\(\rho_P(1) = 1\pmod{P^2}\)，则称为以\(1\)为底的\(c -\)维费里希素数。2015年，塔库尔基于有限数据及2次和3次的证明，推测在奇特征下每个\(c -\)维费里希素数的次数可被\(p\)整除。2024年该问题被重述为开放问题，巴穆诺巴和伯格斯特龙经大量计算后认为该陈述在奇特征下成立。我们证明这是错误的：展示了一个在\(F_{19^3}\)上次数为\(5\)的明确不可约\(c -\)维费里希素数，其中\(19\)不整除\(5\)。我们还给出了\([5]\)和\(M_5\)所得公因子的封闭形式：它等于\(\mu(T^q - T)\)，其中\(\mu\)是一个在素域\(F_{19}\)上系数的明确五次式，次数为\(5×19^3\)且无平方因子。次数\(5\)是此类反例的最小可能次数，详尽计算表明在相当大的次数和特征范围内，素域\(F_p\)上不存在反例。完备性陈述的证明及找到该例子的方法见一篇配套论文。
英文摘要
Let A = F_q[T] with q a power of an odd prime p, let [n] = T^(q^n) - T, and let rho be the Carlitz module. A monic prime P of A is a c-Wieferich prime (to base 1) if rho_P(1) = 1 mod P^2. Thakur suggested in 2015, on the basis of limited data and of proofs in degrees 2 and 3, that in odd characteristic every c-Wieferich prime has degree divisible by p; the question was restated as open in 2024, and Bamunoba and Bergstrom, after extensive computations, expressed the belief that the statement holds in odd characteristic. We show that it is false: an explicit irreducible c-Wieferich prime of degree 5 over F_{19^3} is exhibited, with 19 not dividing 5. We further give a closed form for the resulting common factor of [5] and M_5: it equals mu(T^q - T) for an explicit quintic mu with coefficients in the prime field F_19, squarefree of degree 5*19^3, and it divides gcd([5], M_5); we conjecture equality, and verify it for the part of low degree over the prime field. Degree 5 is the least possible degree of such a counterexample, and exhaustive computations show that no counterexample exists over the prime fields F_p in a substantial range of degrees and characteristics. The proof that degree 5 is minimal, and the method by which the example was found, appear in a companion paper.
