代数函数方程与完全忠实Selmer群
Algebraic functional equations and completely faithful Selmer groups
摘要：
设$E$是定义在数域$K$上的椭圆曲线，不需要复乘，并且在有理素数$p \geq 5$以上的所有素数处都有良好的普通约简。在强容许$p$ -adic Lie扩张$K_\infty/K$上的$E$的对偶$p^\infty$ -Selmer群上构造了一个配对，假设它是Galois群$G=\operatorname{Gal}(K_\infty/K)$的Iwasawa代数上的挠模.在适当的附加假设下，给出了构造的$p$ -adic L-函数的代数函数方程.作为应用，我们构造了$p$ -adic Lie扩张是通过邻接另一条非CM椭圆曲线$A$的$p$ -幂除点得到的完全忠实Selmer群.
英文摘要：
Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p \geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_\infty/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group $G=\operatorname{Gal}(K_\infty/K)$. Under some mild additional hypotheses this gives an algebraic functional equation of the conjectured $p$-adic L-function. As an application we construct completely faithful Selmer groups in case the $p$-adic Lie extension is obtained by adjoining the $p$-power division points of another non-CM elliptic curve $A$.
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