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 ...
Community wiki
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 ...
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 ...
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 ...
Community wiki
11
votes
Accepted
Relationship between Tate-Shafarevich group and the BSD conjecture
Firstly, the functional field result your state is due to Tate in his Bourbaki talk. In fact he proves that the finiteness of the $p$-primary part of Sha is enough for $p$ different from the ...
10
votes
Accepted
Does Chabauty-Coleman method give an algorithm for finding rational points?
Conjecturally, yes. Check out Section 4.4 of this paper by Nils Bruin and myself.
The point is to combine Chabauty-Coleman with the "Mordell-Weil Sieve".
In the following, I will assume for ...
10
votes
Clarification on the weak BSD conjecture
I believe that the original experimental observation was that the product seems to converge to $\infty$ if $E(\mathbb Q)$ is infinite, and to a finite value if $E(\mathbb Q)$ is finite. Also, I may be ...
9
votes
What is the smallest positive integer for which the congruent number problem is unsolved?
Kazuo Matsuno writes (personal communication):
"I verified (10 years ago) by using mwrank and magma that E_N:y^2=x^3-N^2x
has a non-torsion point if N<=10^6 is congruent to 1,2,3 modulo 8 and
the ...
7
votes
Accepted
Clarification on the weak BSD conjecture
In regards to question 2, in 1982 Goldfeld proved that if $f_{E}(x) \sim C (\log x)^{r}$, then (i) $L(E,s)$ has no zeroes with ${\rm Re}(s) > 1$, and (ii) the order of vanishing at $L(E,s)$ is ...
6
votes
What is the smallest positive integer for which the congruent number problem is unsolved?
This is a practical question rather than a theoretical one. My suspicion is that the smallest $N$ for which it is not feasible to determine whether $N$ is congruent is probably very large (maybe $N \...
6
votes
Clarification on the weak BSD conjecture
I recommend you to look at Birch and Swinnerton-Dyer's second paper (Notes on elliptic curves II). They explain beautifully the background for their conjecture.
The basic idea is that if you can ...
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_{...
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 $E$ ...
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.
1
vote
BSD and generalisation of Gross-Zagier formula
Note that the Gross-Zagier theorem only yields the rank inequality for rank $0$ and $1$.
But the full rank inequality is alredy a theorem (of Tate and Milne) in the function fields case.
So perhaps ...
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