次数不超过199的奇次数费马四重曲面的霍奇猜想
The Hodge conjecture for Fermat fourfolds of odd degree at most 199
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中文总结 AI 辅助
该研究通过计算机辅助方法，结合几何闭包准则与霍奇轨道普查，证明了次数不超过199的奇次数费马四重曲面的霍奇猜想，分离出13个特殊轨道并验证了相关代数性。
中文摘要 AI 辅助
设$X^4_m = \{x_0^m + \dots + x_5^m = 0\} \subset \mathbb{P}^5$为次数$m$的费马四重曲面。我们对所有满足$m \le 199$的奇数$m$，给出$X^4_m$的霍奇猜想的计算机辅助证明：结合三个几何闭包准则与霍奇$(2,2)$轨道的穷举机器普查，该证明具有完备性且可复现验证证书。准则如下：(1) 若特征多重集拆分为两个零和三元组，则有理霍奇块从费马曲线乘积的$(1,1)$子结构传输而来，故为代数的；(2) 若特征在添加两个消失对后分解为Aoki标准六元组与2级霍奇四元组，则可推出其代数性；(3) $m=33$处的例外类具有到66级的拟分解提升，其代数性沿$X^4_{66} \to X^4_{33}$下降。该普查覆盖$21 \le m \le 199$且$m \ne 23$的89个次数，对全部78299个伽罗瓦轨道代表元分类，分离出13个超出可分解性、拟分解性与Aoki标准循环的轨道：6个通过$*$-拆分准则闭合，7个通过两对与级提升闭包闭合，最终无剩余。每个轨道均带有机器验证的见证（终端轨道的见证为负筛选），且该普查由算法独立实现复现，经$m=143$的穷举验证。13个轨道中有7个是Aoki格演算之外的间隙类，据作者所知为新类；其余6个的代数性也可从该演算导出，显式表示为新内容。在$p=67$处的精确雅可比和计算表明，不存在定义在$\u211a(\zeta_{33})$上的循环非平凡投影到例外$m=33$块；更一般地，在任何带有验证循环的$\u211a(\zeta_{33})$有限扩域上，所有高于67的剩余次数均被6整除。
英文摘要
Let $X^4_m=\{x_0^m+\dots+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold of degree $m$. We give a computer-assisted proof of the Hodge conjecture for $X^4_m$ for every odd $m\le 199$: three geometric closure criteria combined with an exhaustive machine census of the Hodge $(2,2)$-orbits, with a completeness proof and re-verifiable certificates. Criteria: (1) if the character multiset splits into two zero-sum triples, the rational Hodge block is transported from a $(1,1)$-substructure of a product of Fermat curves, hence algebraic; (2) algebraicity follows for characters that, after adjoining two vanishing pairs, decompose into an Aoki standard sextuple and a grade-$2$ Hodge quadruple; (3) the exceptional class at $m=33$ has a quasi-decomposable lift to level $66$, whose algebraicity descends along $X^4_{66}\to X^4_{33}$. The census covers the $89$ levels $21\le m\le 199$, $m\ne 23$, classifies all $78{,}299$ Galois-orbit representatives, and isolates thirteen orbits beyond decomposability, quasi-decomposability and Aoki's standard cycles: six close by the $*$-split criterion, seven by the two-pair and level-lifted closures, leaving none. Every orbit carries a machine-checked witness (negative screenings for the terminal ones), and the census is reproduced by an algorithmically independent implementation and brute force through $m=143$. Seven of the thirteen are gap classes outside Aoki's lattice calculus, new to the author's knowledge; for the other six, algebraicity is also derivable from that calculus, the explicit presentations being the new content. An exact Jacobi-sum computation at $p=67$ shows no cycle defined over $\mathbb{Q}(ζ_{33})$ projects nontrivially onto the exceptional $m=33$ block; more generally, over any finite extension of $\mathbb{Q}(ζ_{33})$ carrying a certifying cycle, every residue degree above $67$ is divisible by $6$.
