Glossary of Arithmetic and Diophantine Geometry - T

T

Tamagawa numbers

The direct Tamagawa number definition works well only for linear algebraic groups. There the Weil conjecture on Tamagawa numbers was eventually proved. For abelian varieties, and in particular the Birch–Swinnerton-Dyer conjecture (q.v.), the Tamagawa number approach to a local-global principle fails on a direct attempt, though it has had heuristic value over many years. Now a sophisticated equivariant Tamagawa number conjecture is a major research problem.

Tate conjecture

The Tate conjecture (John Tate, 1963) provided an analogue to the Hodge conjecture, also on algebraic cycles, but well within arithmetic geometry. It also gave, for elliptic surfaces, an analogue of the Birch–Swinnerton-Dyer conjecture (q.v.), leading quickly to a clarification of the latter and a recognition of its importance.

Tate curve

The Tate curve is a particular elliptic curve over the p-adic numbers introduced by John Tate to study bad reduction (see good reduction).

Tsen rank

The Tsen rank of a field, named for C. C. Tsen who introduced their study in 1936, is the smallest natural number i, if it exists, such that the field is of class T_i: that is, such that any system of polynomials with no constant term of degree d_j in n variables has a non-trivial zero whenever n > ∑ d_ji. Algebraically closed fields are of Tsen rank zero. The Tsen rank is greater or equal to the Diophantine dimension but it is not known if they are equal except in the case of rank zero.