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== Introduction ==

The Birch and Swinnerton-Dyer conjecture is a prominent unsolved problem in number theory, particularly in the field of [[Elliptic Curve|elliptic curves]]. It is one of the seven [[Millennium Prize Problems]], for which the [[Clay Mathematics Institute]] has offered a prize of one million dollars for a correct solution. The conjecture provides a deep connection between the number of rational points on an elliptic curve and the behavior of an associated [[L-function|L-function]] at a specific point.

== Historical Background ==

The conjecture was formulated in the early 1960s by British mathematicians [[Bryan Birch]] and [[Peter Swinnerton-Dyer]]. Their work was inspired by extensive numerical calculations performed on the [[EDSAC]] computer at the University of Cambridge. These calculations suggested a relationship between the rank of an elliptic curve and the order of the zero of its L-function at s = 1. This observation was later formalized into the conjecture that bears their names.

== Elliptic Curves ==

An elliptic curve is a smooth, projective algebraic curve of genus one, with a specified point defined over a field. In the context of the Birch and Swinnerton-Dyer conjecture, the field of interest is typically the field of rational numbers, \(\mathbb{Q}\). An elliptic curve over \(\mathbb{Q}\) can be expressed in the form:

\[ y^2 = x^3 + ax + b \]

where \(a\) and \(b\) are rational numbers such that the discriminant \(\Delta = 4a^3 + 27b^2 \neq 0\). The set of rational points on an elliptic curve forms a finitely generated abelian group, as established by the [[Mordell-Weil Theorem]].

== L-functions and the Conjecture ==

The L-function associated with an elliptic curve \(E\) over \(\mathbb{Q}\) is a complex analytic function \(L(E, s)\), defined as an Euler product over primes \(p\):

\[ L(E, s) = \prod_{p} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \]

where \(a_p\) is the number of solutions to the curve modulo \(p\), minus \(p\). The Birch and Swinnerton-Dyer conjecture posits that the rank of the group of rational points on an elliptic curve is equal to the order of the zero of its L-function at \(s = 1\).

== The Conjecture's Statement ==

The conjecture can be divided into two parts:

1. **Weak Form**: The rank of the elliptic curve \(E(\mathbb{Q})\) is equal to the order of vanishing of \(L(E, s)\) at \(s = 1\).
2. **Strong Form**: The leading coefficient of the Taylor series expansion of \(L(E, s)\) at \(s = 1\) is related to several arithmetic invariants of the curve, including the [[Tamagawa Number|Tamagawa numbers]], the regulator of the curve, and the order of the [[Shafarevich-Tate Group]].

== Numerical Evidence and Partial Results ==

Numerical evidence supporting the conjecture was first gathered by Birch and Swinnerton-Dyer through computational experiments. Since then, significant progress has been made in proving special cases of the conjecture. For instance, the conjecture is known to hold for elliptic curves with rank 0 or 1, thanks to the work of [[Andrew Wiles]] and others in proving the [[Modularity Theorem]], which was a crucial step in the proof of [[Fermat's Last Theorem]].

== The Role of the Shafarevich-Tate Group ==

The [[Shafarevich-Tate Group]] \(\Sha(E/\mathbb{Q})\) of an elliptic curve \(E\) is a mysterious and complex object that measures the failure of the local-global principle for the curve. The Birch and Swinnerton-Dyer conjecture implies that \(\Sha(E/\mathbb{Q})\) is finite, and its order is a key factor in the conjecture's strong form. Understanding \(\Sha(E/\mathbb{Q})\) is one of the major challenges in the study of elliptic curves.

== Current Research and Open Problems ==

Research on the Birch and Swinnerton-Dyer conjecture continues to be a vibrant area of mathematical inquiry. Efforts are focused on understanding the conjecture's implications for higher-rank elliptic curves and exploring its connections with other areas of mathematics, such as [[Algebraic Geometry]] and [[Arithmetic Geometry]]. Despite significant advances, a complete proof of the conjecture remains elusive.

== Image ==

[[Image:Detail-104425.jpg|thumb|center|Photograph of a chalkboard with complex mathematical equations related to elliptic curves written on it.]]

== See Also ==

* [[Riemann Hypothesis]]
* [[Modular Forms]]
* [[Hodge Conjecture]]

[[Category:Number Theory]]
[[Category:Elliptic Curves]]
[[Category:Unsolved Problems in Mathematics]]