Tate Conjecture
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In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
and algebraic geometry, the Tate conjecture is a 1963
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...
of John Tate that would describe the
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the al ...
s on a
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ...
in terms of a more computable invariant, the Galois representation on
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the
Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjec ...
.


Statement of the conjecture

Let ''V'' be a smooth
projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...
over a field ''k'' which is finitely generated over its
prime field In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive id ...
. Let ''k''s be a
separable closure In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
of ''k'', and let ''G'' be the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...
Gal(''k''s/''k'') of ''k''. Fix a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
â„“ which is invertible in ''k''. Consider the â„“-adic cohomology groups (coefficients in the â„“-adic integers Zâ„“, scalars then extended to the â„“-adic numbers Qâ„“) of the base extension of ''V'' to ''k''s; these groups are
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of ''G''. For any ''i'' ≥ 0, a
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equal ...
-''i'' subvariety of ''V'' (understood to be defined over ''k'') determines an element of the cohomology group : H^(V_,\mathbf_(i)) = W which is fixed by ''G''. Here Qâ„“(''i'' ) denotes the ''i''th
Tate twist In number theory and algebraic geometry, the Tate twist, 'The Tate Twist', https://ncatlab.org/nlab/show/Tate+twist named after John Tate, is an operation on Galois modules. For example, if ''K'' is a field, ''GK'' is its absolute Galois grou ...
, which means that this representation of the Galois group ''G'' is tensored with the ''i''th power of the cyclotomic character. The Tate conjecture states that the subspace ''W''''G'' of ''W'' fixed by the Galois group ''G'' is spanned, as a Qâ„“-vector space, by the classes of codimension-''i'' subvarieties of ''V''. An algebraic cycle means a finite linear combination of subvarieties; so an equivalent statement is that every element of ''W''''G'' is the class of an algebraic cycle on ''V'' with Qâ„“ coefficients.


Known cases

The Tate conjecture for
divisors In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
(algebraic cycles of codimension 1) is a major open problem. For example, let ''f'' : ''X'' → ''C'' be a morphism from a smooth projective surface onto a smooth projective curve over a finite field. Suppose that the generic fiber ''F'' of ''f'', which is a curve over the function field ''k''(''C''), is smooth over ''k''(''C''). Then the Tate conjecture for divisors on ''X'' is equivalent to the
Birch and Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory ...
for the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
of ''F''. By contrast, the Hodge conjecture for divisors on any smooth complex projective variety is known (the Lefschetz (1,1)-theorem). Probably the most important known case is that the Tate conjecture is true for divisors on
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
. This is a theorem of Tate for abelian varieties over finite fields, and of Faltings for abelian varieties over number fields, part of Faltings's solution of the Mordell conjecture. Zarhin extended these results to any finitely generated base field. The Tate conjecture for divisors on abelian varieties implies the Tate conjecture for divisors on any product of curves ''C''1 × ... × ''C''''n''. The (known) Tate conjecture for divisors on abelian varieties is equivalent to a powerful statement about homomorphisms between abelian varieties. Namely, for any abelian varieties ''A'' and ''B'' over a finitely generated field ''k'', the natural map : \text(A,B)\otimes_\mathbf_ \to \text_G \left (H_1 \left (A_,\mathbf_ \right), H_1 \left (B_,\mathbf_ \right) \right ) is an isomorphism. In particular, an abelian variety ''A'' is determined up to
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism of the underlyin ...
by the Galois representation on its Tate module ''H''1(''A''''k''s, Zâ„“). The Tate conjecture also holds for
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected ...
s over finitely generated fields of characteristic not 2. (On a surface, the nontrivial part of the conjecture is about divisors.) In characteristic zero, the Tate conjecture for K3 surfaces was proved by André and Tankeev. For K3 surfaces over finite fields of characteristic not 2, the Tate conjecture was proved by Nygaard, Ogus, Charles, Madapusi Pera, and Maulik. surveys known cases of the Tate conjecture.


Related conjectures

Let ''X'' be a smooth projective variety over a finitely generated field ''k''. The semisimplicity conjecture predicts that the representation of the Galois group ''G'' = Gal(''k''s/''k'') on the ℓ-adic cohomology of ''X'' is semisimple (that is, a direct sum of irreducible representations). For ''k'' of characteristic 0, showed that the Tate conjecture (as stated above) implies the semisimplicity of :H^i \left (X \times_k \overline, \mathbf_\ell(n) \right ). For ''k'' finite of order ''q'', Tate showed that the Tate conjecture plus the semisimplicity conjecture would imply the strong Tate conjecture, namely that the order of the pole of the zeta function ''Z''(''X'', ''t'') at ''t'' = ''q''−''j'' is equal to the rank of the group of algebraic cycles of codimension ''j'' modulo numerical equivalence.J. Tate. Motives (1994), Part 1, 71-83. Theorem 2.9. Like the Hodge conjecture, the Tate conjecture would imply most of Grothendieck's
standard conjectures on algebraic cycles In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications of these conjectures, envisaged by Alexander Gr ...
. Namely, it would imply the Lefschetz standard conjecture (that the inverse of the Lefschetz isomorphism is defined by an algebraic correspondence); that the Künneth components of the diagonal are algebraic; and that numerical equivalence and homological equivalence of algebraic cycles are the same.


Notes


References

* * * * * * * * *{{Citation, last=Totaro, first=Burt, author-link=Burt Totaro, title=Recent progress on the Tate conjecture, journal=Bulletin of the American Mathematical Society , series=New Series, volume=54, issue=4, pages=575–590, year=2017, doi=10.1090/bull/1588, doi-access=free


External links

* James Milne
The Tate conjecture over finite fields (AIM talk)
Topological methods of algebraic geometry Diophantine geometry Conjectures Unsolved problems in number theory