几乎所有的素数都是部分规则的
Almost all primes are partially regular
AI总结：
该研究证明几乎所有素数在特定条件下满足部分规则性，通过Leopoldt反射得出部分Vandiver定理，并影响p-adic L-函数、模形式同余及K-群中的p- torsion。
摘要：
对于奇素数p，我们令K_p:=$\mathbb{Q}(ζ_p)$为p次循环数域，并令ω表示其Teichmuller特征标。对于α>1/2，我们说一个奇素数p是部分规则的，如果p-阶Sylow子群的Cl(K_p)的特征空间在Galois作用下，对于所有特征标ω^{p-2k}（其中2≤2k≤√p/（log p）^α）都消失。等价地，p不整除B_{2k}的分子在整个范围内。我们证明了一组密度为一的素数集合是部分规则的。通过Leopoldt反射，这导致了一个部分Vandiver定理：对于一组密度为一的素数p，所有满足上述不等式的偶数2k的偶特征空间A_p(ω^{2k})都消失。该结果对Kubota-Leopoldt p-adic L-函数、模形式与 Eisenstein 系列之间的同余关系以及代数K-群中的p- torsion有影响。证明几乎所有p的部分规则性的定理在Lean/Mathlib中完全形式化，并由AxiomProver从自然语言陈述的猜想自动产生。
英文摘要：
For odd primes $p$, we let $K_p:=\mathbb{Q}(ζ_p)$ be the $p$th cyclotomic field and let $ω$ denote its Teichmuller character. For $α>1/2$, we say that an odd prime $p$ is partially regular if the eigenspaces of the $p$-Sylow subgroup of $\operatorname{Cl}(K_p)$ under the Galois action vanish for all characters $ω^{p-2k}$ with \[ 2\le 2k \le \frac{\sqrt{p}}{(\log p)^α}. \] Equivalently, $p\nmid \operatorname{num}(B_{2k})$ throughout this range. We prove that a density-one subset of primes is partially regular in this sense. By Leopoldt reflection, this yields a partial Vandiver Theorem: for a density-one set of primes $p$, the even eigenspaces $A_p(ω^{2k})$ vanish for all even $2k$ satisfying the inequality above. This result has consequences for Kubota-Leopoldt $p$-adic $L$-functions, congruences between cusp forms and Eisenstein series, and $p$-torsion in algebraic $K$-groups. The theorem proving partial regularity for almost all $p$ is fully formalized in Lean/Mathlib and was produced automatically by AxiomProver from a natural-language statement of the conjecture.
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