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 is finitely generated over its prime subfield, if is a smooth projective variety over , if is a prime number different from the characteristic of the field , if

is the natural -adic representation, and , then the -space , the space of elements of annihilated by , is generated by the homology classes of algebraic cycles of codimension on (cf. also Algebraic cycle).

Conjecture 2. The rank of the group of classes of algebraic cycles of codimension on modulo homology equivalence coincides with the order of the pole of the function at the point .

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

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

Comments

In conjecture 2 above is the -series of , defined by

where the product is over all primes where has good reduction and where is the -th polynomial factor appearing in the zeta-function of the variety over the residue field of at ,

In the case , with and Abelian varieties, conjecture 1 takes for (i.e. for divisors) the following form: The natural homomorphism

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