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arXiv 1503.06463math.NT
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精细Selmer群、Heegner点与反分圆$\mathbb{Z}_p$ -扩张

Fine Selmer Groups, Heegner points and Anticyclotomic $\mathbb{Z}_p$-extensions

Ahmed Matar

备注：
fixed mistake. This is the final version
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摘要：

设$E/\mathbb{Q}$是椭圆曲线，$p$是素数，$K_{\infty}/K$是满足Heegner假设的二次虚域$K$的反割圆$\mathbb{Z}_p$ -扩张.本文对$K_{\infty}$上的细Selmer群作了一个猜想.本文还对$E(K_{\mathfrak{p}_{\infty}})/p$中Heegner点模的结构作了一个猜想，其中$K_{\mathfrak{p}_{\infty}}$是域$K_n$在$p$之上的素数为$K_{\infty}$的补集的并.我们证明了这两个图是等价的。当$E$在$p$处有超奇异约化时，我们还证明了这些结果等价于我们早先工作中的猜想.当$E$在$p$上有超奇异约化时，我们证明了关于$K_{\infty}$上Selmer群结构的各种结果.

英文摘要：

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper we make a conjecture about the fine Selmer group over $K_{\infty}$. We also make a conjecture about the structure of the module of Heegner points in $E(K_{\mathfrak{p}_{\infty}})/p$ where $K_{\mathfrak{p}_{\infty}}$ is the union of the completions of the fields $K_n$ at a prime of $K_{\infty}$ above $p$. We prove that these conjectures are equivalent. When $E$ has supersingular reduction at $p$ we also show that these conjectures are equivalent to the conjecture in our earlier work. Assuming these conjectures when $E$ has supersingular reduction at $p$, we prove various results about the structure of the Selmer group over $K_{\infty}$.

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