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Conjectures expressed by J. Tate (see [[#References|[1]]]) and describing relations between Diophantine and algebro-geometric properties of an [[Algebraic variety|algebraic variety]].
 
Conjectures expressed by J. Tate (see [[#References|[1]]]) and describing relations between Diophantine and algebro-geometric properties of an [[Algebraic variety|algebraic variety]].
  
Conjecture 1. If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922501.png" /> is finitely generated over its prime subfield, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922502.png" /> is a smooth projective variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922503.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922504.png" /> is a prime number different from the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922505.png" />, if
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922506.png" /></td> </tr></table>
+
$$
 +
\rho _ {l}  ^ {( i)} : \
 +
\mathop{\rm Gal}  ( \widetilde{k}  /k)  \rightarrow \
 +
\mathop{\rm Aut} _ {\mathbf Q _ {l}  }  H _ {l}  ^ {2i}
 +
( V \otimes _ {k} \overline{k}\; ) ( i)
 +
$$
  
is the natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922507.png" />-adic representation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922508.png" />, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t0922509.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225010.png" />, the space of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225011.png" /> annihilated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225012.png" />, is generated by the homology classes of algebraic cycles of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225014.png" /> (cf. also [[Algebraic cycle|Algebraic cycle]]).
+
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|Algebraic cycle]]).
  
Conjecture 2. The rank of the group of classes of algebraic cycles of codimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225016.png" /> modulo homology equivalence coincides with the order of the pole of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225017.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225018.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225019.png" /> and on the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225020.png" />.
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.T. Tate, "Algebraic cycles and poles of zeta-functions" D.F.G. Schilling (ed.) , ''Arithmetical Algebraic geometry (Proc. Purdue Conf. 1963)'' , Harper &amp; Row (1965) pp. 93–110 {{MR|0225778}} {{ZBL|0213.22804}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.T. Tate, "Algebraic cycles and poles of zeta-functions" D.F.G. Schilling (ed.) , ''Arithmetical Algebraic geometry (Proc. Purdue Conf. 1963)'' , Harper &amp; Row (1965) pp. 93–110 {{MR|0225778}} {{ZBL|0213.22804}} </TD></TR></table>
  
 +
====Comments====
 +
In conjecture 2 above  $  L _ {i} ( V, s ) $
 +
is the  $  L $-
 +
series of  $  V $,
 +
defined by
  
 +
$$
 +
L _ {i} ( V, s)  =  \prod _ {\mathfrak p }
 +
\{ P _ {i} ( q  ^ {-s} ) \}  ^ {-1} ,
 +
$$
  
====Comments====
+
where the product is over all primes  $  \mathfrak p $
In conjecture 2 above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225021.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225023.png" />-series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225024.png" />, defined by
+
where  $  V $
 
+
has good reduction and where  $  P _ {i} ( q  ^ {-s} ) $
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225025.png" /></td> </tr></table>
+
is the $  i $-
 +
th polynomial factor appearing in the [[Zeta-function|zeta-function]] of the variety  $  V  \mathop{\rm mod}  \mathfrak p $
 +
over the residue field  $  \mathbf F _ {q} $
 +
of  $  k $
 +
at  $  \mathfrak p $,
  
where the product is over all primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225026.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225027.png" /> has good reduction and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225028.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225029.png" />-th polynomial factor appearing in the [[Zeta-function|zeta-function]] of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225030.png" /> over the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225032.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225033.png" />,
+
$$
 +
\zeta _ {V  \mathop{\rm mod}  \mathfrak p } ( s)  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225034.png" /></td> </tr></table>
+
\frac{P _ {1} ( q  ^ {-s} ) \dots P _ {2d-1}  ( q  ^ {-s} ) }{P _ {0} ( q  ^ {-s} ) \dots P _ {2d} ( q  ^ {-s} ) }
 +
.
 +
$$
  
In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225035.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225037.png" /> Abelian varieties, conjecture 1 takes for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225038.png" /> (i.e. for divisors) the following form: The natural homomorphism
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225039.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Hom} _ {k} ( A, B) \otimes \mathbf Z _ {l}  \rightarrow \
 +
\mathop{\rm Hom} _ { \mathop{\rm Gal}  ( \overline{k} / k ) }
 +
( T _ {l} ( A), T _ {l} ( B) )
 +
$$
  
is an isomorphism (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225040.png" /> is the [[Tate module|Tate module]] of the Abelian variety) (see [[#References|[1]]]). This case of the conjecture has been proved: i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225041.png" /> is a finite field by J. Tate [[#References|[a1]]]; ii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225042.png" /> is a function field over a finite field by J.G. Zarkin [[#References|[a2]]]; and iii) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225043.png" /> is a number field by G. Faltings [[#References|[a3]]].
+
is an isomorphism (where $  T _ {l} (-) $
 +
is the [[Tate module|Tate module]] of the Abelian variety) (see [[#References|[1]]]). This case of the conjecture has been proved: i) $  k $
 +
is a finite field by J. Tate [[#References|[a1]]]; ii) if $  k $
 +
is a function field over a finite field by J.G. Zarkin [[#References|[a2]]]; and iii) if $  k $
 +
is a number field by G. Faltings [[#References|[a3]]].
  
For examples of particular cases where the Tate conjecture has been proved see, e.g., [[#References|[a4]]] for ordinary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225044.png" />-surfaces over finite fields and [[#References|[a5]]] for Hilbert modular surfaces.
+
For examples of particular cases where the Tate conjecture has been proved see, e.g., [[#References|[a4]]] for ordinary $  K3 $-
 +
surfaces over finite fields and [[#References|[a5]]] for Hilbert modular surfaces.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Tate, "Endomorphisms of Abelian varieties over finite fields" ''Invent. Math.'' , '''2''' (1966) pp. 104–145 {{MR|0206004}} {{ZBL|0147.20303}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" ''Invent. Math.'' , '''73''' (1983) pp. 349–366 (Erratum: Invent. Math <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225045.png" /> (1984), 381) {{MR|0718935}} {{MR|0732554}} {{ZBL|0588.14026}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> N.O. Nygaard, "The Tate conjecture for ordinary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092250/t09225046.png" />-surfaces over finite fields" ''Invent. Math.'' , '''74''' (1983) pp. 213–237 {{MR|723215}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. van der Geer, "Hilbert modular surfaces" , Springer (1987) {{MR|}} {{ZBL|0634.14022}} {{ZBL|0511.14021}} {{ZBL|0483.14009}} {{ZBL|0418.14021}} {{ZBL|0349.14022}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Wüstholz (ed.) , ''Rational points'' , Vieweg (1984) {{MR|0766568}} {{ZBL|0588.14027}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Tate, "Endomorphisms of Abelian varieties over finite fields" ''Invent. Math.'' , '''2''' (1966) pp. 104–145 {{MR|0206004}} {{ZBL|0147.20303}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" ''Invent. Math.'' , '''73''' (1983) pp. 349–366 (Erratum: Invent. Math '''75''' (1984), 381) {{MR|0718935}} {{MR|0732554}} {{ZBL|0588.14026}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> N.O. Nygaard, "The Tate conjecture for ordinary $K3$-surfaces over finite fields" ''Invent. Math.'' , '''74''' (1983) pp. 213–237 {{MR|723215}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> G. van der Geer, "Hilbert modular surfaces" , Springer (1987) {{MR|}} {{ZBL|0634.14022}} {{ZBL|0511.14021}} {{ZBL|0483.14009}} {{ZBL|0418.14021}} {{ZBL|0349.14022}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G. Wüstholz (ed.) , ''Rational points'' , Vieweg (1984) {{MR|0766568}} {{ZBL|0588.14027}} </TD></TR></table>

Latest revision as of 16:54, 21 December 2020


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
How to Cite This Entry:
Tate conjectures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_conjectures&oldid=23992
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article