# Tagged Questions

**5**

votes

**1**answer

147 views

### 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 ...

**7**

votes

**2**answers

542 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 + ...

**47**

votes

**3**answers

4k 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 ...

**14**

votes

**6**answers

2k views

### A non-technical account of the Birchâ€”Swinnerton-Dyer Conjecture

I was wondering whether anyone knows of any good non-technical or even popular expositions of the Birchâ€”Swinnerton-Dyer conjecture, for someone with minimal background in elliptic curves. I was ...