论文标题
tamagawa的中央$ l $ l $ - 曲折椭圆曲线的划分
Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve
论文作者
论文摘要
给定任何整数$ n> 1 $ prime至$ 3 $，我们用$ c_n $表示椭圆曲线$ x^3+y^3 = n $。我们首先研究了$ 3 $ - adic的估值，该代数的估值是hasse-weil $ l $ l $ -function $ l（c_n，s）$ $ c_n $ of $ \ mathbb {q} $ at $ s = 1 $，我们展示了$ 3 $ $ 3 $ - $ 3 $的$ tivate $ prive distate-n. tivant-n. n difters n. n n nork n nork n nork n nork n nork n norking n nork n nork n nork n nork n nork n of n nork n nork n of n n divance nos n norking n of。假想二次字段$ k = \ mathbb {q}（\ sqrt {-3}）$。如果$ l（C_N，1）\ neq 0 $和$ n $是$ k $中的分裂素数的产物，我们表明泰特·夏法雷维奇集团的顺序是桦木和Swinnerton-Dyer的猜想所预测的，是一个完美的广场。
Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $\mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-3})$. In the case where $L(C_N,1)\neq 0$ and $N$ is a product of split primes in $K$, we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.
