# 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 ...
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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{\...
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 $\mathrm{dim}_{\... 7 votes 2 answers 819 views ### BSD conjecture for X_0(17) I use Magma to calculate the L-value, yields E:=EllipticCurve([1, -1, 1, -1, 0]); E; Evaluate(LSeries(E),1),RealPeriod(E),Evaluate(LSeries(E),1)/RealPeriod(E); Elliptic Curve defined by y^2 + x*y + y =... 2 votes 3 answers 1k views ### A generalisation of the Birch and Swinnerton-Dyer conjecture We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch ... 12 votes 3 answers 1k views ### Deducing BSD from Gross-Zagier and Kolyvagin Does anyone know which papers deduce BSD for elliptic curves$E/\mathbb{Q}$of rank 0 or 1 from the papers by Gross and Zagier and Kolyvagin? If I understand these theories right, there is still a ... 7 votes 2 answers 715 views ### Power series expansions of$L$-series Let$\zeta_K(s)$be the Dedekind zeta function for a number field$K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/... 61 votes 3 answers 7k views ### Is there a "Basic Number Theory" for elliptic curves? Tate's thesis showed how to profitably analyze$\zeta$functions of number fields in terms of adelic points on the multiplicative group. In particular, combining Fourier analysis and topology, Tate ... 6 votes 1 answer 3k views ### How did Birch and Swinnerton Dyer arrive at their conjecture? I suspect that they knew that the$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}... 