AI 中文总结
本文研究秩二吸引子引发的p-adic刚性，提出上同调判据证明Frobenius不动点与超同余，并应用于Hulek--Verrill A_4族及镜像四次K3铅笔。
AI 中文摘要
我们研究由秩二吸引子所暗示的p-adic刚性现象。在Calabi--Yau族的一个普通特殊纤维附近，Beukers和Vlasenko的优良Frobenius作用于参数圆盘。Achinger--Zdanowicz构造了Frobenius扭曲的典范W_2-提升，而Brantner--Taelman构造了典范形式提升。我们猜想Dwork与Achinger--Zdanowicz的一阶障碍映射一致，并且优良Frobenius的不动点恰为Brantner--Taelman典范形式提升的参数。我们的主定理给出了任意维数下的上同调判据。在其假设下，p-adic上同调中包含全纯类的Frobenius稳定秩二因子迫使该类张成Cartier稳定直线，并对每个s≥1给出模p^{2s}的超同余。此外，若Beukers--Vlasenko优良提升定理的假设成立，则优良Frobenius固定相应参数。我们将该判据应用于Hulek--Verrill A_4族在t_*=-1/7处，该参数被预测为秩二吸引子。假设Dummigan猜想1.1成立，我们证明优良Frobenius固定t_*，超同余成立，并且在所有满足所需Dwork正规性条件的素数处，参数为t_* mod p^2的纤维是Achinger--Zdanowicz典范提升。镜像四次K3铅笔的一个CM纤维给出第二个应用。我们还证明了在W(F_q)[1/p]的每个有限扩张上，分裂形式环面上的q次幂映射在单位残差圆盘中具有唯一周期点即单位截面，其中q为残差域的基数。
英文摘要
We study a $p$-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi--Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger--Zdanowicz construct a canonical $W_2$-lift of the Frobenius twist, while Brantner--Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger--Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner--Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the $p$-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo $p^{2s}$ for every $s\geq1$. Moreover, if the hypotheses of the Beukers--Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek--Verrill $A_4$ family at $t_*=-1/7$, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes $t_*$, the supercongruences hold, and the fiber with parameter $t_*\bmod p^2$ is the Achinger--Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the $q$-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of $W(\mathbb{F}_q)[1/p]$, where $q$ is the cardinality of the residue field.
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