# Tate conjectures

Conjectures expressed by J. Tate (see ) and describing relations between Diophantine and algebro-geometric properties of an algebraic variety.

Conjecture 1. If the field $k$ is finitely generated over its prime subfield, if $V$ is a smooth projective variety over $k$, if $l$ is a prime number different from the characteristic of the field $k$, if

$$\rho _ {l} ^ {( i)} : \ \mathop{\rm Gal} ( \widetilde{k} /k) \rightarrow \ \mathop{\rm Aut} _ {\mathbf Q _ {l} } H _ {l} ^ {2i} ( V \otimes _ {k} \overline{k}\; ) ( i)$$

is the natural $l$- adic representation, and $g _ {l} ^ {( i)} = \mathop{\rm Lie} ( \mathop{\rm Im} ( \rho _ {l} ^ {( i)} ))$, then the $\mathbf Q _ {l}$- space $[ H _ {l} ^ {2i} ( V \otimes _ {k} \overline{k}\; ) ( i) ] ^ {g _ {l} ^ {( i)} }$, the space of elements of $H _ {l} ^ {2i} ( V \otimes _ {k} \overline{k}\; ) ( i)$ annihilated by $g _ {l} ^ {( i)}$, is generated by the homology classes of algebraic cycles of codimension $i$ on $V \otimes _ {k} \overline{k}\;$( cf. also Algebraic cycle).

Conjecture 2. The rank of the group of classes of algebraic cycles of codimension $i$ on $V$ modulo homology equivalence coincides with the order of the pole of the function $L _ {2i} ( V, s)$ at the point $s = \mathop{\rm dim} Y + i$.

These conjectures were verified for a large number of particular cases; restrictions are imposed both on the field $k$ and on the variety $V$.

How to Cite This Entry:
Tate conjectures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_conjectures&oldid=51033
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article