# Tate conjectures

Conjectures expressed by J. Tate (see [1]) 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 $V$.

#### References

 [1] J.T. Tate, "Algebraic cycles and poles of zeta-functions" D.F.G. Schilling (ed.) , Arithmetical Algebraic geometry (Proc. Purdue Conf. 1963) , Harper & Row (1965) pp. 93–110 MR0225778 Zbl 0213.22804

In conjecture 2 above $L _ {i} ( V, s )$ is the $L$- series of $V$, defined by

$$L _ {i} ( V, s) = \prod _ {\mathfrak p } \{ P _ {i} ( q ^ {-} s ) \} ^ {-} 1 ,$$

where the product is over all primes $\mathfrak p$ where $V$ has good reduction and where $P _ {i} ( q ^ {-} s )$ is the $i$- th polynomial factor appearing in the zeta-function of the variety $V \mathop{\rm mod} \mathfrak p$ over the residue field $\mathbf F _ {q}$ of $k$ at $\mathfrak p$,

$$\zeta _ {V \mathop{\rm mod} \mathfrak p } ( s) = \ \frac{P _ {1} ( q ^ {-} s ) \dots P _ {2d-} 1 ( q ^ {-} s ) }{P _ {0} ( q ^ {-} s ) \dots P _ {2d} ( q ^ {-} s ) } .$$

In the case $V = A \times \widehat{B}$, with $A$ and $B$ Abelian varieties, conjecture 1 takes for $i = 1$( i.e. for divisors) the following form: The natural homomorphism

$$\mathop{\rm Hom} _ {k} ( A, B) \otimes \mathbf Z _ {l} \rightarrow \ \mathop{\rm Hom} _ { \mathop{\rm Gal} ( \overline{k}\; / k ) } ( T _ {l} ( A), T _ {l} ( B) )$$

is an isomorphism (where $T _ {l} (-)$ is the Tate module of the Abelian variety) (see [1]). This case of the conjecture has been proved: i) $k$ is a finite field by J. Tate [a1]; ii) if $k$ is a function field over a finite field by J.G. Zarkin [a2]; and iii) if $k$ is a number field by G. Faltings [a3].

For examples of particular cases where the Tate conjecture has been proved see, e.g., [a4] for ordinary $K3$- surfaces over finite fields and [a5] for Hilbert modular surfaces.

#### References

 [a1] J. Tate, "Endomorphisms of Abelian varieties over finite fields" Invent. Math. , 2 (1966) pp. 104–145 MR0206004 Zbl 0147.20303 [a2] J.G. Zarking, "A remark on endomorphisms of Abelian varieties over function fields of finite characteristic" Math. USSR Izv. , 8 (1974) pp. 477–480 Izv. Akad. Nauk SSSR , 38 : 3 (1974) pp. 471–474 [a3] G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" Invent. Math. , 73 (1983) pp. 349–366 (Erratum: Invent. Math (1984), 381) MR0718935 MR0732554 Zbl 0588.14026 [a4] N.O. Nygaard, "The Tate conjecture for ordinary -surfaces over finite fields" Invent. Math. , 74 (1983) pp. 213–237 MR723215 [a5] G. van der Geer, "Hilbert modular surfaces" , Springer (1987) Zbl 0634.14022 Zbl 0511.14021 Zbl 0483.14009 Zbl 0418.14021 Zbl 0349.14022 [a6] G. Wüstholz (ed.) , Rational points , Vieweg (1984) MR0766568 Zbl 0588.14027
How to Cite This Entry:
Tate conjectures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_conjectures&oldid=48950
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article