论文标题
部分可观测时空混沌系统的无模型预测
A $p$-adic arithmetic inner product formula
论文作者
论文摘要
修复质数$ p $，让$ e/f $是数字字段的cm扩展，其中$ p $相对拆分。让$π$是关于$ e/f $的准单行组的自动形式表示，因此，$π$相对于Siegel抛物线子组是普通的$ p $。我们构建了$π$的环形$ P $ -ADIC $ L $ - 功能，并在某些条件下表明，如果其在琐碎角色的消失顺序为$ 1 $，那么与$ e $相关的$ e $的Selmer群体等级至少为$ 1 $。此外，在某些模块化假设下，我们在单一的Shimura品种上使用特殊周期来构建名为Selmer Theta Lifts的Selmer组中的一些明确元素。我们证明了一个精确的公式，将其$ p $ -Adic Heights与$ p $ -Adic $ l $ unction的衍生产品联系起来。与Perrin-riou的$ P $ -ADIC类似物 - Zagier公式的类似物，我们的公式是Chao〜i Li和第二作者最近建立的算术内部产品公式的$ P $ ADIC类似物。
Fix a prime number $p$ and let $E/F$ be a CM extension of number fields in which $p$ splits relatively. Let $π$ be an automorphic representation of a quasi-split unitary group of even rank with respect to $E/F$ such that $π$ is ordinary above $p$ with respect to the Siegel parabolic subgroup. We construct the cyclotomic $p$-adic $L$-function of $π$, and show, under certain conditions, that if its order of vanishing at the trivial character is $1$, then the rank of the Selmer group of the Galois representation of $E$ associated with $π$ is at least $1$. Furthermore, under a certain modularity hypothesis, we use special cycles on unitary Shimura varieties to construct some explicit elements in the Selmer group called Selmer theta lifts; and we prove a precise formula relating their $p$-adic heights to the derivative of the $p$-adic $L$-function. In parallel to Perrin-Riou's $p$-adic analogue of the Gross--Zagier formula, our formula is the $p$-adic analogue of the arithmetic inner product formula recently established by Chao~Li and the second author.
