# Questions tagged [birch-swinnerton-dyer]

Questions related to the Birch and Swinnerton-Dyer conjecture about the vanishing order and first Taylor coefficient of the L-functions of elliptic curves at the point 1.

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### Rank 1 curves with prime conductor have trivial torsion. Why?

In the LMFDB database, there are 337912 elliptic curves over $\mathbb{Q}$ for which the rank is 1 and the conductor is a prime number.
All of these curves have trivial torsion group.
Is there a known ...

3
votes

0
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### 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{...

9
votes

1
answer

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### Analogue of the original Birch–Swinnerton-Dyer conjecture for abelian varieties

$\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\F}{\Bbb F}
\newcommand{\p}{\mathfrak{p}}
$
Let $A$ be an abelian ...

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### 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 ...

14
votes

3
answers

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### 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

17
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2
answers

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### 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

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### 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 ...

6
votes

1
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### 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....

12
votes

5
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### 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 ...

17
votes

2
answers

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### 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 ...

5
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### 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 ...

1
vote

0
answers

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### The Birch-Swinerton-Dyer conjecture for rank 1 [closed]

I'm reading the kolinvagin's work about the Birch Swinnerton-Dyer of elliptic curves woose rational points have rank one. I want to ask about the prerequisites to understand the Kolinvagin's work on ...

2
votes

1
answer

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### 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{...

2
votes

2
answers

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### 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{\...

9
votes

1
answer

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### Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\mathbb{Q}$ positive, as predicted by BSD?

Let $E$ be an elliptic curve over $\mathbb{Q}$. As proved by Wiles et al., its $L$-series $L(E, s)$ is entire. Set $r := \mathrm{ord}_{s = 1} L(E, s)$, a value conjecturally equal to $\mathrm{dim}_{\...

7
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### 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 =...

2
votes

3
answers

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### 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 ...

12
votes

3
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### 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 ...

7
votes

2
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### 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/...

61
votes

3
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### 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 ...

6
votes

1
answer

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### 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}...

17
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6
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### 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 ...