All Questions
Tagged with birch-swinnerton-dyer nt.number-theory
21 questions
4
votes
0
answers
110
views
Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
- 531
7
votes
1
answer
233
views
Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at most $1$
For an elliptic curve, $E$, defined over $\mathbb{Q}$ and with the Mordell-Weil group, $E(\mathbb{Q})$, having rank, $r$, the Birch Swinnerton-Dyer (BSD) conjecture states that
$$
c_{E} = \lim_{s \...
- 73
8
votes
1
answer
325
views
3-divisibility of Manin constant for elliptic curves with 3-torsion
Let $E/\mathbb{Q}$ be an elliptic curve with $E(\mathbb{Q}) \cong \mathbb{Z}/3\mathbb{Z}$ (not necessarily $\Gamma_0$-optimal). Does $3$ necessarily divide one of: the Manin constant (not necessarily $...
- 91
3
votes
0
answers
171
views
Rank and Taylor coefficient in Birch and Swinnerton–Dyer
I am trying to get a better understanding of the Birch and Swinnerton–Dyer conjecture. I have two questions
Why might one expect that the analytic rank of $L(E,s)$ is equal to the rank of $E(\mathbb{...
- 901
4
votes
0
answers
8k
views
Which proportion of elliptic curves over Q is proved to satisfy BSD conjecture?
In 2014, Bhargava and other authors proved that more than 66 % of all elliptic curves defined over $ \mathbb{Q} $ satisfy the Birch and Swinnerton-Dyer conjecture. Tonight I stumbled upon an article ...
- 6,982
14
votes
3
answers
4k
views
Recent progress toward Birch and Swinnerton-Dyer conjecture
Has there been any progress toward the Birch and Swinnerton-Dyer conjecture after
The current status of the Birch & Swinnerton-Dyer Conjecture
- 141
17
votes
2
answers
3k
views
Consequences of the Birch and Swinnerton-Dyer Conjecture?
Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following
What are the consequences of the Birch and ...
7
votes
1
answer
557
views
Does Chabauty-Coleman method give an algorithm for finding rational points?
Let $X$ be a curve of genus $g\geq 2$ over a number field $K$. If $\mathrm{rk} \,\mathrm{Jac}\, X$ is less than $g$ there is a $p$-adic method of bounding $\# X(K)$ due to Chabauty and Coleman (see ...
- 7,377
6
votes
1
answer
937
views
Relationship between Tate-Shafarevich group and the BSD conjecture
The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}_{q},$ this result is due to Kato and Trihan if I am not mistaken....
- 1,805
12
votes
5
answers
2k
views
Clarification on the weak BSD conjecture
It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f_E(x) = \prod_{p \le x}\frac{|E(\mathbb{F}_p)|}{p}
$$
as $x$ tends to ...
user85435
17
votes
2
answers
2k
views
What is the smallest positive integer for which the congruent number problem is unsolved?
The congruent number problem is the problem of figuring out whether a given positive integer $N$ is the area of a right-angled triangle with all side lengths rational. According to Dickson's "History ...
- 41.4k
5
votes
2
answers
1k
views
BSD and generalisation of Gross-Zagier formula
The classical Gross-Zagier formula and the modularity theorem leads to a proof of half-BSD (i.e. an inequality and not equality) for elliptic curves of analytic rank 0.
The Gross-Zagier formula gives ...
- 329
2
votes
1
answer
463
views
integral basis for the Lie algebra of the Neron model of an abelian variety
Let $A$ be an abelian variety over a number field $K$. Let $\mathcal{A}$ be the Neron model of $A$ over $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ be the sheaf of invariant differential forms on $\mathcal{...
- 213
2
votes
2
answers
306
views
BSD leading-term coefficient in terms of places without distinction
After reading this blog post, I learned the BSD conjectural formula for the coefficient of the leading term $a_0$ of the L-function of an elliptic curve $E$, namely
$$
a_0 \stackrel{?}{=} \frac{\...
- 35.5k
7
votes
2
answers
874
views
BSD conjecture for X_0(17)
I use Magma to calculate the L-value, yields
E:=EllipticCurve([1, -1, 1, -1, 0]);
E;
Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E);
Elliptic Curve defined by y^2 + x*y + y =...
- 73
2
votes
3
answers
1k
views
A generalisation of the Birch and Swinnerton-Dyer conjecture
We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch ...
- 1,197
14
votes
3
answers
2k
views
Deducing BSD from Gross-Zagier and Kolyvagin
Does anyone know which papers deduce BSD for elliptic curves $E/\mathbb{Q}$ of rank 0 or 1 from the papers by Gross and Zagier and Kolyvagin? If I understand these theories right, there is still a ...
- 993
7
votes
2
answers
818
views
Power series expansions of $L$-series
Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/...
- 1,022
63
votes
3
answers
8k
views
Is there a "Basic Number Theory" for elliptic curves?
Tate's thesis showed how to profitably analyze $\zeta$ functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ...
- 156k
6
votes
1
answer
3k
views
How did Birch and Swinnerton Dyer arrive at their conjecture?
I suspect that they knew that the $L-$function is defined only for $Re(s) \gt 3/2$. Did they attempt to evaluate the $L-$function at $s=1$ by plugging $s=1$ in the infinite product $\prod_p (\frac{1}...
Trust God
17
votes
6
answers
5k
views
A non-technical account of the Birch—Swinnerton-Dyer Conjecture
I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birch—Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was ...
- 489