希达族中的庞加莱对偶
Poincaré duality in Hida families
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中文总结 AI 辅助
该研究针对p进上同调类族构造双线性配对，推广了模曲线的Atkin--Lehner对合，能交换上同调的普通与反普通部分，可恢复Ohta为模曲线构造的配对。
中文摘要 AI 辅助
我们针对p进上同调类族构造了一个双线性配对，该配对与p处具有特定水平的局部对称空间相关，且在特殊化映射下，我们得到了所构造配对的特殊化与自守层上同调的自然对偶配对之间的比较结果。我们证明，所选水平下的该局部对称空间允许一个自同构，其自然推广了模曲线的Atkin--Lehner对合，且能交换上同调的普通部分与反普通部分，从而使得对两个反普通类进行配对成为可能。我们的构造受Ohta为模曲线构造的配对启发，并能恢复该配对。
英文摘要
We construct a bilinear pairing for $p$-adic families of cohomology classes associated to locally symmetric spaces of a certain level at $p$ and obtain a comparison between the specialisations of constructed pairing and the natural duality pairings on cohomology of automorphic sheaves under specialisation maps. We show that the considered locally symmetric spaces with the chosen level admit an automorphism which naturally generalises the Atkin--Lehner involution for modular curves and interchanges the ordinary and anti-ordinary parts of the cohomology thereby making it possible to pair two anti-ordinary classes. Our construction is motivated by the pairing constructed by Ohta for modular curves and recovers it.
