PlanetMath: Birch and Swinnerton-Dyer conjecture

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Birch and Swinnerton-Dyer conjecture

(Conjecture)

Let be an elliptic curve over $\mathbb{Q}$ , and let be the L-series attached to .

Conjecture 1 (Birch and Swinnerton-Dyer)

has a zero at of order equal to the rank of $E(\mathbb{Q})$ .
Let $R=\operatorname{rank} (E(\mathbb{Q}))$ . Then the residue of at , i.e. $\lim_{s\to 1}(s-1)^{-R} L(E,s)$ has a concrete expression involving the following invariants of : the real period, the Tate-Shafarevich group, the elliptic regulator and the Neron model of .

J. Tate said about this conjecture: “This remarkable conjecture relates the behavior of a function at a point where it is not at present known to be defined to the order of a group (Sha) which is not known to be finite!” The precise statement of the conjecture asserts that:

$\displaystyle \lim_{s\to 1} \frac{L(E,s)}{(s-1)^R}=\frac{\vert\operatorname{Sha... ...mathbb{Q}) \cdot \prod_p c_p}{\vert E_{\operatorname{tors}}(\mathbb{Q})\vert^2}$

where

is the rank of $E/\mathbb{Q}$ .
$\Omega$ is either the real period or twice the real period of a minimal model for , depending on whether $E(\mathbb{R})$ is connected or not.
$\vert\operatorname{Sha}\vert$ is the order of the Tate-Shafarevich group of $E/\mathbb{Q}$ .
$\operatorname{Reg}(E/\mathbb{Q})$ is the elliptic regulator of $E(\mathbb{Q})$ .
$\vert E_{\operatorname{tors}}(\mathbb{Q})\vert$ is the number of torsion points on $E/\mathbb{Q}$ (including the point at infinity ).
is an elementary local factor, equal to the cardinality of $E(\mathbb{Q}_p)/E_0(\mathbb{Q}_p)$ , where $E_0(\mathbb{Q}_p)$ is the set of points in $E(\mathbb{Q}_p)$ whose reduction modulo is non-singular in $E(\mathbb{F}_p)$ . Notice that if is a prime of good reduction for $E/\mathbb{Q}$ then , so only $c_p\neq 1$ only for finitely many primes . The number is usually called the Tamagawa number of at .

The following is an easy consequence of the B-SD conjecture:

Conjecture 2 (Parity Conjecture) The root number of , denoted by , indicates the parity of the rank of the elliptic curve, this is, if and only if the rank is even.

There has been a great amount of research towards the B-SD conjecture. For example, there are some particular cases which are already known:

Theorem 1 (Coates, Wiles) Suppose is an elliptic curve defined over an imaginary quadratic field , with complex multiplication by , and is the L-series of . If $L(E,1)\neq 0$ then is finite.

Bibliography

1: Claymath Institute, Description, online.
2: J. Coates, A. Wiles, On the Conjecture of Birch and Swinnerton-Dyer, Inv. Math. 39, 223-251 (1977).
3: Keith Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, 189 - 212, Perseus Books Group, New York (2002).
4: James Milne, Elliptic Curves, online course notes.
5: Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
6: Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.

"Birch and Swinnerton-Dyer conjecture" is owned by alozano.

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Other names:

BS-D conjecture

Also defines:

Birch and Swinnerton-Dyer conjecture, parity conjecture

Keywords:

Birch, Swinnerton, Dyer, L-series, rank, elliptic curve

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Cross-references: finite, complex multiplication, imaginary quadratic field, even, parity, root number, consequence, good reduction, prime, non-singular, reduction, cardinality, factor, infinity, points, torsion, number, connected, minimal model, conjecture, elliptic regulator, Tate-Shafarevich group, period, real, invariants, expression, residue, rank, order, elliptic curve
There are 7 references to this entry.

This is version 13 of Birch and Swinnerton-Dyer conjecture, born on 2003-08-06, modified 2007-04-08.
Object id is 4561, canonical name is BirchAndSwinnertonDyerConjecture.
Accessed 9419 times total.

Classification:

AMS MSC:

14H52 (Algebraic geometry :: Curves :: Elliptic curves)

Pending Errata and Addenda

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