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Results tagged with birch-swinnerton-dyer
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user 5015
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.
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 character …
- 8,945
2
votes
Accepted
integral basis for the Lie algebra of the Neron model of an abelian variety
The same problem already appears in the formulation of the Birch and Swinnerton-Dyer conjecture for say an elliptic curve over a number field. When there is no longer a global minimal Weierstrass mode …
- 8,945
12
votes
Accepted
Is the leading Taylor coefficient at $s = 1$ of the $L$-series of an elliptic curve over $\m...
Let me summarise the comments above that give a full answer (correct me if I am wrong).
The analytic continuation of $L(E,\bar{s})=\overline{L(E,s)}$ shows that $c\in\mathbb{R}$.
If $r=0$, the fact …
- 8,945
10
votes
Deducing BSD from Gross-Zagier and Kolyvagin
Let $E/\mathbb{Q}$ be an elliptic curve of analytic rank $0$ or $1$. Then indeed the rank part of the BSD conjecture is known, but the exact formula for the leading term is not yet fully proven. The $ …
- 8,945