On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions

Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes

Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ be a prime of good supersingular reduction for $E$. Let $K$ be an imaginary quadratic field satisfying a modified "Heegner hypothesis" in which $p$ splits, write $K_\infty$ for the anticyclotomic $\mathbb Z_p$-extension of $K$ and let $Λ$ denote the Iwasawa algebra of $K_\infty/K$. By extending to the supersingular case the $Λ$-adic Kolyvagin method originally developed by Bertolini in the ordinary setting, we prove that Kobayashi's plus/minus $p$-primary Selmer groups of $E$ over $K_\infty$ have corank $1$ over $Λ$. As an applicat…

Selmer groups and anticyclotomic $\mathbb{Z}_p$-extensions

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 give a new proof to a theorem of Bertolini which determines the value of the $Λ$-corank of $\text{Sel}(E/K_{\infty})$ in the case where $E$ has ordinary reduction at $p$. In the case where $E$ has supersingular reduction at $p$ we make a conjecture about the structure of the module of Heegner points mod $p$. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines…

关于 $p$-adic 超奇异情况下的 $p$-adic Rankin-Selberg $L$-函数因子分解

On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case

本文研究了超奇异情况下 $p$-adic Rankin-Selberg $L$-函数的因子分解，通过构造双变量解析 $p$-adic $L$-函数，证明了其与对称平方 $L$-函数和 Kubota-Leopoldt $L$-函数的乘积关系。

Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture

We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p = \mathrm{corank}_{\mathbb{Z}_p}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})$, we show that $r_p \le 1 \implies \mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q}) = \mathrm{ord}_{s = 1}L(E/\mathbb{Q},s) = r_p$ and $\#\mathrm{Sha}(E/\mathbb{Q}) < \infty$. In particular, this has applications to two classical Diophantine problems. First, it resolves Sylvester's conjecture on rational sums of cubes…

$Λ$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Let $p$ be an odd prime and $K$ an imaginary quadratic field where $p$ splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a $p$-ordinary elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$ does not admit any proper $Λ$-submodule of finite index, where $Λ$ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of $p$-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $Λ$-corank one, so they are not $Λ$-cotorsion. As an application of our ma…

椭圆曲线的反cyclotomic Selmer群

On anticyclotomic Selmer groups of elliptic curves

本文在温和假设下证明超奇异椭圆曲线反cyclotomic扩张上符号Selmer群对偶无有限子模，推广Bertolini结果并改进Hatley-Lei-Vigni定理。