AI 中文总结
研究椭圆曲线\(E_{p,q}:y^{2}=x^{3}-pqx\)，通过刻画与2次同源相关塞尔默群及相关上核，建立\(\rank E_{p,q}(\bbQ)\)和\(\dim_{\mathbb{F}_{2}}\Sha(E_{p,q}/\bbQ)[2]\)都为\(0\)的充要条件。
AI 中文摘要
对于椭圆曲线\(E_{p,q}:y^{2}=x^{3}-pqx\)（其中\(p\)和\(q\)是不同的奇素数），我们建立了\(\rank E_{p,q}(\bbQ)\)和\(\dim_{\mathbb{F}_{2}}\Sha(E_{p,q}/\bbQ)[2]\)都为\(0\)的充要条件。通过对与2次同源\(\phi\)及其对偶\(\widehat{\phi}\)相关的塞尔默群何时都为最小规模进行类似刻画，并结合相关正合序列产生的上核的结果来实现。
英文摘要
For the elliptic curves $E_{p,q}: y^{2}=x^{3}-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which rank$\,E_{p,q}(\mathbb{Q})$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$. We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $ϕ$ and its dual $\widehatϕ$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.
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