论文标题
较高的总Zagier公式和Selmer组的结构
A higher Gross-Zagier formula and the structure of Selmer groups
论文作者
论文摘要
我们通过将Heegner Point Kolyvagin Systems与Kurihara编号进行比较时，描述了Koloss-Zagier公式的Kolyvagin系统理论改进，当时有理理性的椭圆曲线$ e $ y $ $ $ k $是$ -1 $ -1 $。当$ e $ $ k $的$ e $的根号为1时，我们首先建立$ p^\ infty $ -selmer组的结构定理，$ e $ a $ a $ a $ y $ k $。该描述由某些Quaternionic自动形式的家族的价值给出，这是双方Euler系统的一部分。通过比较具有kurihara数字的两分欧拉系统，我们还获得了Waldspurger公式的类似细化。在两种改进中都没有施加低分析等级假设。 We also prove the equivalence between the non-triviality of various ``Kolyvagin systems" and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of $p^\infty$-Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner点主要的猜想位于增强理想中，这意味着强度的排名是Gross-Zagier和Kolyvagin定理的一个$ p $ converse。
We describe a Kolyvagin system-theoretic refinement of Gross--Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve $E$ over an imaginary quadratic field $K$ is $-1$. When the root number of $E$ over $K$ is 1, we first establish the structure theorem of the $p^\infty$-Selmer group of $E$ over $K$. The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various ``Kolyvagin systems" and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of $p^\infty$-Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one $p$-converse to the theorem of Gross-Zagier and Kolyvagin.
