B
- Bad reduction
- See good reduction.
- Birch and Swinnerton-Dyer conjecture
- The Birch and Swinnerton-Dyer conjecture on elliptic curves postulates a connection between the rank of an elliptic curve and the order of pole of its Hasse-Weil L-function. It has been an important landmark in Diophantine geometry since the mid-1960s, with results such as the Coates–Wiles theorem, Gross–Zagier theorem and Kolyvagin's theorem.
- Bombieri–Lang conjecture
- Enrico Bombieri, Serge Lang and Paul Vojta have conjectured that algebraic varieties of general type do not have Zariski dense subsets of K-rational points, for K a finitely-generated field. This circle of ideas includes the understanding of analytic hyperbolicity and the Lang conjectures on that, and the Vojta conjectures. An analytically hyperbolic algebraic variety V over the complex numbers is one such that no holomorphic mapping from the whole complex plane to it exists, that is not constant. Examples include compact Riemann surfaces of genus g > 1. Lang conjectured that V is analytically holomorphic if and only if all subvarieties are of general type.
Read more about this topic: Glossary Of Arithmetic And Diophantine Geometry