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- Tags: - #AG/elliptic-curves #open/conjectures - Refs: - Swinnerton-Dyer, Notes on elliptic curves. II - J. T. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. - Links: - elliptic curve - modularity - L function - motivic L function - Application: congruent number problem - Mazur’s theorem - Faltings theorem
Birch and Swinnerton-Dyer conjecture
On the free abelian group \(E(K) / E(K)_{\text {tor }}\), there is areal-valued positive definite bilinear form, the Néron-Tate height pairing. From this one can construct a regulator Reg \({ }_{E} / K\), which is the determinant of the matrix of this bilinear form with respect to a basis.
The Tate-Shafarevich group of \(E\) is a certain abelian torsion group \(\mathrm{\Pi}_{E / K}\) attached to \(E\), classifying locally solvable \(E{\hbox{-}}\)-torsors. This group is conjectured to be finite for every \(E\), but this has not been proved in general. If \(\Pi_{E / K}\) is finite, then its order is a square.
Weak BSD
Relation to finiteness of Sha:
Birch and Swinnerton-Dyer conjecture
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Misc
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