论文标题
CM椭圆形曲线的广义Kato类别2
Generalised Kato classes on CM elliptic curves of rank 2
论文作者
论文摘要
令$ e/\ mathbf {q} $为CM椭圆曲线，让$ p \ geq 5 $为$ e $的良好普通降低。假设$ l（e，s）$在$ s = 1 $中消失，并具有符号$+1 $的功能方程式，因此尤其是$ {\ rm ord} _ {s = 1} l（e，s）\ geq 2 $。在本文中，我们稍微修改了darmon-rotger的结构，以定义{\ rm sel}中的广义$κ_p\ \ in {\ rm sel}（\ mathbf {q}，v_pe）$，并证明了以下等级的两个kolyvagin等级的结果：\ [κ__p\ neq 0 \ neq 0 dim} _ {\ Mathbf {q} _p} {\ rm sel}（\ Mathbf {q}，v_pe）= 2。相反，当$ {\ rm dim} _ {\ Mathbf {q} _p} {\ rm sel}（\ rm sel}（\ mathbf {q}，v_pe）= 2 $我们显示$κ_p\ neq neq 0 $ \ emph {if and if and if if} SEL}（\ Mathbf {Q}，V_PE）\ RightArrow E（\ MathBf {Q} _P）\ hat \ otimes \ otimes \ Mathbf {q} _p \]是非零的。这些结果的证明是在非CM情况下使用HSIEH扩展并延伸的相似结果，它利用了广义Kato类别的非趋势与反风速iwasawa理论的主要猜想之间的新联系。
Let $E/\mathbf{Q}$ be a CM elliptic curve and let $p\geq 5$ be a prime of good ordinary reduction for $E$. Suppose that $L(E,s)$ vanishes at $s=1$ and has sign $+1$ in its functional equation, so in particular ${\rm ord}_{s=1}L(E,s)\geq 2$. In this paper we slightly modify a construction of Darmon--Rotger to define a generalised Kato class $κ_p\in{\rm Sel}(\mathbf{Q},V_pE)$, and prove the following rank two analogue of Kolyvagin's result: \[ κ_p\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2. \] Conversely, when ${\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$ we show that $κ_p\neq 0$ \emph{if and only if} the restriction map \[ {\rm Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat\otimes\mathbf{Q}_p \] is nonzero. The proof of these results, which extend and strenghten similar results of the author with Hsieh in the non-CM case, exploit a new link between the nonvanishing of generalised Kato classes and a main conjecture in anticyclotomic Iwasawa theory.
