Deligne曲线Jacobian周期猜想改进的数值证据
Numerical Evidence for a refinement of Deligne's Period Conjecture for Jacobians of Curves
摘要：
设$A/\mathbb{Q}$是一个Jacobian簇，$F$是一个全实的、弱分歧的阿贝尔数域.给定$F/\mathbb{Q}$的一个特征符$ψ$，Deligne的周期猜想断言在$A$的$ψ$ -扭曲的Hasse-Weil-Artin $L$ -函数的$z=1$处适当归一化的值$\mathcal{L}(A,ψ,1)$的代数性。我们制定了一个猜想，关于家庭的正规化$L$ -值$(\mathcal{L}(A,ψ,1))_ψ$的完整性，以及它的关系的Tate-Shafarevich群的$A$在$F$。我们通过这些值之间的$p$ -adic同余关系数值研究了我们的猜想。
英文摘要：
Let $A/\mathbb{Q}$ be a Jacobian variety and let $F$ be a totally real, tamely ramified, abelian number field. Given a character $ψ$ of $F/\mathbb{Q}$, Deligne's Period Conjecture asserts the algebraicity of the suitably normalised value $\mathcal{L}(A,ψ,1)$ at $z=1$ of the Hasse-Weil-Artin $L$-function of the $ψ$-twist of $A$. We formulate a conjecture regarding the integrality properties of the family of normalised $L$-values $(\mathcal{L}(A,ψ,1))_ψ$, and its relation to the Tate-Shafarevich group of $A$ over $F$. We numerically investigate our conjecture through $p$-adic congruence relations between these values.
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