21
votes
Recent progress toward Birch and Swinnerton-Dyer conjecture
No, the conjecture is still wide open for rank $r\geq 2$.
The closest thing to progress is the work of Bhargava and Shankar that quantifies the rank $0$ case and shows that BSD holds for a positive ...
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21
votes
Consequences of the Birch and Swinnerton-Dyer Conjecture?
There is a theorem of Michael Stoll that there are no $c\in\mathbb Q$ such that the polynomial $x^2+c$ admits a periodic 6-cycle starting at some $a\in\mathbb Q$, but the theorem is contingent on the ...
18
votes
Recent progress toward Birch and Swinnerton-Dyer conjecture
Benedict Gross recently gave a series of lectures here at the University of Virginia on things related to the Birch and Swinnerton-Dyer Conjecture. One of the recent notable developments he mentioned ...
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12
votes
Consequences of the Birch and Swinnerton-Dyer Conjecture?
Implicit in the BSD conjecture are two other basic conjectures about elliptic curves: the Parity Conjecture and the finiteness of the Tate-Shafarevich group. Most applications I know follow from the ...
4
votes
Accepted
Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties
$\newcommand{\p}{\mathfrak{p}}$By Theorem 6.3 of this paper by Keith Conrad, strong conjectures about $L(A,s)$ (stronger than GRH for this $L$-function, but still "believable"), imply that
$$
\prod_{...
- 5,392
4
votes
Accepted
Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at most $1$
I think that the answer to your questions depends in subtle ways on whether $r=0$ or $r=1$.
In full generality, I believe you are right that none of the properties you state are known for all elliptic ...
- 10.9k
1
vote
3-divisibility of Manin constant for elliptic curves with 3-torsion
Perhaps there is an elementary answer after all. Here is my attempt at a partial answer assuming BSD.
Let $c_0(E)$ denote the Manin constant of $E$, and let $L(E)$ denote the special L-value of $E$ ...
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1
vote
Recent progress toward Birch and Swinnerton-Dyer conjecture
Searching `Birch and Swinnerton-Dyer conjecture' by Google, there are two short preprints on this theme:Yongxiong Li, Yu Liu, Ye Tian and K.Morita.
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