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

Comments

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 75 (1984), 381) MR0718935 MR0732554 Zbl 0588.14026
[a4]	N.O. Nygaard, "The Tate conjecture for ordinary $K3$-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