On the Birch-Swinnerton-Dyer quotients modulo squares

By Tim Dokchitser, Vladimir Dokchitser. 2006

152 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/988c3e1175e187bc78fbae263af286c5cbe28f4a

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Citations in the corpus

The p-adic Gross-Zagier formula for elliptic curves at supersingular primes (2013, unknown): “次に Heegner 点 z_{K,E}\in E(K)\otimes \mathbb{Q} を z_{K,E}=\mathrm{T}\mathrm{r}_{H/K} $\pi$(z_{H})\otimes c_{ $\pi$}^{-1}\in E(K)\otimes \mathbb{Q} 1近年 Nékovár \breve{} [16] やDokchitser 兄弟 [8] らにより, P‐Selmer 群の階数の偶奇と関数等式の符号 (root number) が一致することは,かなり一般的な代数体上の楕円曲線に対して証明されている.”
Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 (2010, methodology): “First, by combining our counting techniques with the remarkable recent results of Dokchitser and Dokchitser [17] on the parity of p -ranks of Selmer groups, we prove Theorem 4.”
Root numbers, Selmer groups, and non-commutative Iwasawa theory (2010, background): “We also note that [ 18 ] proves that the parity conjecture (1) holds for all elliptic curves E over Q, and for all primes p, generalizing the works in [34], [35], [28].”
The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, methodology): “To the special family F constructed in Theorem 6, we may then apply Theorem 2, along with the aforementioned theorem of Dokchitser–Dokchitser [24].”