论文标题
通用$ p $ - adiC Gross-Zagier公式
The universal $p$-adic Gross-Zagier formula
论文作者
论文摘要
令$ {\ mathrm g} $为组$（{\ rm gl} _ {2} \ times {\ rm gu}（1））/{\ rm gl} _ {1} $在完全真实的$ f $上，然后让$ \ mathscr {x} $ a a hida for $ f $ a n h.重新访问霍华德和福克的构造，我们构建了一个明确的部分，$ \ mathscr {p} $ y Mathscr {x} $的selmer群的捆绑。我们向回答霍华德的问题显示，$ \ mathscr {p} $是通用的Heegner类，从某种意义上说，它在$ \ Mathscr {x} $的所有相关经典点上都在几何定义的Heegner类中插值。我们还为在经典点的$ \ mathscr {p} $的领先术语提出了一个“ Bertolini-Darmon”的猜想。 然后，我们证明$ \ mathscr {p} $的$ p $ -ADIC高度由$ p $ -adic $ l $ function的环形衍生物给出。该公式超过$ \ MATHSCR {X} $（在某些通用普通自动形态表示的功能上的标识）专门针对所有Gross-Zagier公式，以$ {\ rm g} $从代表性的考虑中预期。 结合Fouquet的结果，该公式意味着分析等级中Beilinson-Bloch-kato猜想的$ p $ - 亚种类似物，因为Hilbert模块化形式所附加的自我动机及其曲折由CM Hecke角色曲折。这也意味着非亚洲iwasawa的第一个示例的一半是衍生物的主要猜想，价格为$ 2 [f：{\ bf q}] $变量。其他应用程序包括Heegner类别的两个不同的通用非逐步结果和$ p $ -Adic Heights。
Let ${\mathrm G}$ be the group $({\rm GL}_{2}\times {\rm GU}(1))/{\rm GL}_{1}$ over a totally real field $F$, and let $\mathscr{X}$ be a Hida family for ${\rm G}$. Revisiting a construction of Howard and Fouquet, we construct an explicit section $\mathscr{P}$ of a sheaf of Selmer groups over $\mathscr{X}$. We show, answering a question of Howard, that $\mathscr{P}$ is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of $\mathscr{X}$. We also propose a `Bertolini-Darmon' conjecture for the leading term of $\mathscr{P}$ at classical points. We then prove that the $p$-adic height of $\mathscr{P}$ is given by the cyclotomic derivative of a $p$-adic $L$-function. This formula over $\mathscr{X}$ (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross-Zagier formulas for ${\rm G}$ that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the $p$-adic analogue of the Beilinson-Bloch-Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in $2[F:{\bf Q}]$ variables. Other applications include two different generic non-vanishing results for Heegner classes and $p$-adic heights.
