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...jecture, Taniyama–Shimura conjecture, Taniyama–Weil conjecture, modularity conjecture'' A conjecture that postulates a deep connection between elliptic curves (cf. [[Elliptic c

...A beautiful application of this connection with the Birch–Swinnerton-Dyer conjecture to the classical problem of computing the set of areas of rational right tr

...ise to major advances, for example, results on the [[Birch–Swinnerton-Dyer conjecture]] [[#References|[a3]]], [[#References|[a6]]], [[#References|[a16]]], [[#Ref ...nt $\mathbf{Z} _ { p }$-extensions of $k$ (cf. below, the section Leopoldt conjecture). Every $k$ has at least one $\mathbf{Z} _ { p }$-extension, namely the cyc

function]] of an algebraic variety at integer points. There is a conjecture about the connection between the values of the zeta-functions of algebraic ...particular cases, and they are complementary to the Birch–Swinnerton-Dyer conjecture (see [[Zeta-function|Zeta-function]] in algebraic geometry).

...ons on $F$). In addition, C. Greither [[#References|[a6]]] proved the main conjecture (using Kolyvagin's method) for all $F / \mathbf Q $ Abelian and all $p$, in ...e very small explicit factors. Rubin proved that the Birch–Swinnerton-Dyer conjecture holds unconditionally for curves $y ^ { 2 } = x ^ { 3 } - p ^ { 2 } x$ for

==Beilinson's first conjecture.== Beilinson's first conjecture reads as follows. Assume $ m < {i / 2 } $.

...benius endomorphism|Frobenius endomorphism]] of this variety. Weil's first conjecture says that the category of algebraic varieties over finite fields admits a c According to Weil's second conjecture, the function $\zeta_X(t)$ must satisfy a functional equation. For a smooth