On the Birch-Swinnerton-Dyer quotients modulo squares
On the Birch-Swinnerton-Dyer quotients modulo squares
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关于 Birch-Swinnerton-Dyer 商模平方
DOI：
10.4007/annals.2010.172.567
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发表时间：
2006
影响因子：
4.9
通讯作者：
V. Dokchitser
中科院分区：
文献类型：
--
作者：
T. Dokchitser;V. Dokchitser
令 A 为数域 K 上的阿贝尔簇。K 的外延 K i 的 L 函数 L(A/K i , s) 之间的恒等式引发了 Birch-Swinnerton-Dyer 商之间的猜想关系。我们证明了这些关系的 LII 模有限性，并对 Selmer 群给出了类似的陈述。基于此，我们开发了一种确定 A 的各种秩组合在 K 的扩展上的奇偶性的方法。作为应用之一，我们建立了椭圆曲线的奇偶性猜想，假设 III(E/K(E[2]))[6 ∞ ] 有有限性，并对 2 和 3 以上素数的约简进行一些限制：E/K 的 Mordell-Weil 秩的奇偶性与 分析等级，由根数确定。我们还证明了 Q 和所有素数 p 上的所有椭圆曲线的 p 宇称猜想：p ∞ -Selmer 秩和解析秩的宇称一致。
Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K i , s) for extensions K i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of LII, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of III(E/K(E[2]))[6 ∞ ] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p ∞ -Selmer rank and the analytic rank agree.