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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105601.png" /> be the ring of integers of an algebraic number [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105602.png" /> (cf. also [[Algebraic number|Algebraic number]]). The Milnor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105604.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105605.png" />, which is also called the tame kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105606.png" />, is an [[Abelian group|Abelian group]] of finite order.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105607.png" /> denote the [[Dedekind zeta-function|Dedekind zeta-function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105608.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b1105609.png" /> is totally real, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056010.png" /> is a non-zero rational number, and the Birch–Tate conjecture is about a relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056011.png" /> and the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056012.png" />.
+
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{{TEX|done}}
  
Specifically, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056013.png" /> be the largest natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056014.png" /> such that the [[Galois group|Galois group]] of the cyclotomic extension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056015.png" /> obtained by adjoining the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056016.png" />th roots of unity to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056017.png" />, is an elementary Abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056018.png" />-group (cf. [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056019.png" />-group]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056020.png" /> is a rational integer, and the Birch–Tate conjecture states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056021.png" /> is a totally real number field, then
+
Let  $  {\mathcal O} _ {F} $
 +
be the ring of integers of an algebraic number [[Field|field]]  $  F $(
 +
cf. also [[Algebraic number|Algebraic number]]). The Milnor  $  K $-
 +
group  $  K _ {2} ( {\mathcal O} _ {F} ) $,
 +
which is also called the tame kernel of $  F $,  
 +
is an [[Abelian group|Abelian group]] of finite order.
  
<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/b/b110/b110560/b11056022.png" /></td> </tr></table>
+
Let  $  \zeta _ {F} $
 +
denote the [[Dedekind zeta-function|Dedekind zeta-function]] of  $  F $.  
 +
If  $  F $
 +
is totally real, then  $  \zeta _ {F} ( - 1 ) $
 +
is a non-zero rational number, and the Birch–Tate conjecture is about a relationship between  $  \zeta _ {F} ( - 1 ) $
 +
and the order of  $  K _ {2} ( {\mathcal O} _ {F} ) $.
  
A numerical example is as follows. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056023.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056025.png" />; so it is predicted by the conjecture that the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056026.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056027.png" />, which is correct.
+
Specifically, let  $  w _ {2} ( F ) $
 +
be the largest natural number  $  N $
 +
such that the [[Galois group|Galois group]] of the cyclotomic extension over  $  F $
 +
obtained by adjoining the  $  N $
 +
th roots of unity to  $  F $,
 +
is an elementary Abelian  $  2 $-
 +
group (cf. [[P-group| $  p $-
 +
group]]). Then  $  w _ {2} ( F ) \cdot \zeta _ {F} ( - 1 ) $
 +
is a rational integer, and the Birch–Tate conjecture states that if  $  F $
 +
is a totally real number field, then
  
What is known for totally real number fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056028.png" />?
+
$$
 +
\# K _ {2} ( {\mathcal O} _ {F} ) = \left | {w _ {2} ( F ) \cdot \zeta _ {F} ( - 1 ) } \right | .
 +
$$
  
By work on the main conjecture of Iwasawa theory [[#References|[a6]]], the Birch–Tate conjecture was confirmed up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056029.png" />-torsion for Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056031.png" />.
+
A numerical example is as follows. For  $  F = \mathbf Q $
 +
one has  $  w _ {2} ( \mathbf Q ) = 24 $,
 +
$  \zeta _ {\mathbf Q} ( - 1 ) = - {1 / {12 } } $;
 +
so it is predicted by the conjecture that the order of $  K _ {2} ( \mathbf Z ) $
 +
is  $  2 $,
 +
which is correct.
  
Subsequently, [[#References|[a7]]], the Birch–Tate conjecture was confirmed up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056032.png" />-torsion for arbitrary totally real number fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056033.png" />.
+
What is known for totally real number fields $  F $?
  
Moreover, [[#References|[a7]]] (see the footnote on page 499) together with [[#References|[a4]]], also the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056034.png" />-part of the Birch–Tate conjecture is confirmed for Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056036.png" />.
+
By work on the main conjecture of Iwasawa theory [[#References|[a6]]], the Birch–Tate conjecture was confirmed up to  $  2 $-
 +
torsion for Abelian extensions $  F $
 +
of $  \mathbf Q $.
  
By the above, all that is left to be considered is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056037.png" />-part of the Birch–Tate conjecture for non-Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056038.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056039.png" />. In this regard, for extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056041.png" /> for which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056042.png" />-primary subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056043.png" /> is elementary Abelian, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056044.png" />-part of the Birch–Tate conjecture has been confirmed [[#References|[a3]]].
+
Subsequently, [[#References|[a7]]], the Birch–Tate conjecture was confirmed up to  $  2 $-
 +
torsion for arbitrary totally real number fields  $  F $.
  
In addition, explicit examples of families of non-Abelian extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056046.png" /> for which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056047.png" />-part of the Birch–Tate conjecture holds, have been given in [[#References|[a1]]], [[#References|[a2]]].
+
Moreover, [[#References|[a7]]] (see the footnote on page 499) together with [[#References|[a4]]], also the  $  2 $-
 +
part of the Birch–Tate conjecture is confirmed for Abelian extensions  $  F $
 +
of  $  \mathbf Q $.
  
The Birch–Tate conjecture is related to the Lichtenbaum conjectures [[#References|[a5]]] for totally real number fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056048.png" />. For every odd natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056049.png" />, the Lichtenbaum conjectures express, up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056050.png" />-torsion, the ratio of the orders of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056052.png" /> in terms of the value of the zeta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056053.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056054.png" />.
+
By the above, all that is left to be considered is the  $  2 $-
 +
part of the Birch–Tate conjecture for non-Abelian extensions  $  F $
 +
of  $  \mathbf Q $.
 +
In this regard, for extensions  $  F $
 +
of  $  \mathbf Q $
 +
for which the  $  2 $-
 +
primary subgroup of  $  K _ {2} ( {\mathcal O} _ {F} ) $
 +
is elementary Abelian, the  $  2 $-
 +
part of the Birch–Tate conjecture has been confirmed [[#References|[a3]]].
 +
 
 +
In addition, explicit examples of families of non-Abelian extensions  $  F $
 +
of  $  \mathbf Q $
 +
for which the  $  2 $-
 +
part of the Birch–Tate conjecture holds, have been given in [[#References|[a1]]], [[#References|[a2]]].
 +
 
 +
The Birch–Tate conjecture is related to the Lichtenbaum conjectures [[#References|[a5]]] for totally real number fields $  F $.  
 +
For every odd natural number $  m $,  
 +
the Lichtenbaum conjectures express, up to $  2 $-
 +
torsion, the ratio of the orders of $  K _ {2m }  ( {\mathcal O} _ {F} ) $
 +
and $  K _ {2m+1 }  ( {\mathcal O} _ {F} ) $
 +
in terms of the value of the zeta-function $  \zeta _ {F} $
 +
at $  - m $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.E. Conner,  J. Hurrelbrink,  "Class number parity" , ''Pure Math.'' , '''8''' , World Sci.  (1988)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Hurrelbrink,  "Class numbers, units, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056055.png" />"  J.F. Jardine (ed.)  V. Snaith (ed.) , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056056.png" />-theory: Connection with Geometry and Topology'' , ''NATO ASI Ser. C'' , '''279''' , Kluwer Acad. Publ.  (1989)  pp. 87–102</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Kolster,  "The structure of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056057.png" />-Sylow subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056058.png" /> I"  ''Comment. Math. Helv.'' , '''61'''  (1986)  pp. 376–388</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Kolster,  "A relation between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056059.png" />-primary parts of the main conjecture and the Birch–Tate conjecture"  ''Canad. Math. Bull.'' , '''32''' :  2  (1989)  pp. 248–251</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Lichtenbaum,  "Values of zeta functions, étale cohomology, and algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056060.png" />-theory"  H. Bass (ed.) , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056061.png" />-theory II'' , ''Lecture Notes in Mathematics'' , '''342''' , Springer  (1973)  pp. 489–501</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Mazur,  A. Wiles,  "Class fields of abelian extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110560/b11056062.png" />"  ''Invent. Math.'' , '''76'''  (1984)  pp. 179–330</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Wiles,  "The Iwasawa conjecture for totally real fields"  ''Ann. of Math.'' , '''131'''  (1990)  pp. 493–540</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.E. Conner,  J. Hurrelbrink,  "Class number parity" , ''Pure Math.'' , '''8''' , World Sci.  (1988)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Hurrelbrink,  "Class numbers, units, and $K_2$"  J.F. Jardine (ed.)  V. Snaith (ed.) , ''Algebraic K-theory: Connection with Geometry and Topology'' , ''NATO ASI Ser. C'' , '''279''' , Kluwer Acad. Publ.  (1989)  pp. 87–102</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Kolster,  "The structure of the $2$-Sylow subgroup of $K_2(\mathcal{O})$ I"  ''Comment. Math. Helv.'' , '''61'''  (1986)  pp. 376–388</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Kolster,  "A relation between the $2$-primary parts of the main conjecture and the Birch–Tate conjecture"  ''Canad. Math. Bull.'' , '''32''' :  2  (1989)  pp. 248–251</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Lichtenbaum,  "Values of zeta functions, étale cohomology, and algebraic K-theory"  H. Bass (ed.) , ''Algebraic K-theory II'' , ''Lecture Notes in Mathematics'' , '''342''' , Springer  (1973)  pp. 489–501</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Mazur,  A. Wiles,  "Class fields of abelian extensions of $\QQ$"  ''Invent. Math.'' , '''76'''  (1984)  pp. 179–330</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Wiles,  "The Iwasawa conjecture for totally real fields"  ''Ann. of Math.'' , '''131'''  (1990)  pp. 493–540</TD></TR></table>

Latest revision as of 08:21, 26 March 2023


Let $ {\mathcal O} _ {F} $ be the ring of integers of an algebraic number field $ F $( cf. also Algebraic number). The Milnor $ K $- group $ K _ {2} ( {\mathcal O} _ {F} ) $, which is also called the tame kernel of $ F $, is an Abelian group of finite order.

Let $ \zeta _ {F} $ denote the Dedekind zeta-function of $ F $. If $ F $ is totally real, then $ \zeta _ {F} ( - 1 ) $ is a non-zero rational number, and the Birch–Tate conjecture is about a relationship between $ \zeta _ {F} ( - 1 ) $ and the order of $ K _ {2} ( {\mathcal O} _ {F} ) $.

Specifically, let $ w _ {2} ( F ) $ be the largest natural number $ N $ such that the Galois group of the cyclotomic extension over $ F $ obtained by adjoining the $ N $ th roots of unity to $ F $, is an elementary Abelian $ 2 $- group (cf. $ p $- group). Then $ w _ {2} ( F ) \cdot \zeta _ {F} ( - 1 ) $ is a rational integer, and the Birch–Tate conjecture states that if $ F $ is a totally real number field, then

$$ \# K _ {2} ( {\mathcal O} _ {F} ) = \left | {w _ {2} ( F ) \cdot \zeta _ {F} ( - 1 ) } \right | . $$

A numerical example is as follows. For $ F = \mathbf Q $ one has $ w _ {2} ( \mathbf Q ) = 24 $, $ \zeta _ {\mathbf Q} ( - 1 ) = - {1 / {12 } } $; so it is predicted by the conjecture that the order of $ K _ {2} ( \mathbf Z ) $ is $ 2 $, which is correct.

What is known for totally real number fields $ F $?

By work on the main conjecture of Iwasawa theory [a6], the Birch–Tate conjecture was confirmed up to $ 2 $- torsion for Abelian extensions $ F $ of $ \mathbf Q $.

Subsequently, [a7], the Birch–Tate conjecture was confirmed up to $ 2 $- torsion for arbitrary totally real number fields $ F $.

Moreover, [a7] (see the footnote on page 499) together with [a4], also the $ 2 $- part of the Birch–Tate conjecture is confirmed for Abelian extensions $ F $ of $ \mathbf Q $.

By the above, all that is left to be considered is the $ 2 $- part of the Birch–Tate conjecture for non-Abelian extensions $ F $ of $ \mathbf Q $. In this regard, for extensions $ F $ of $ \mathbf Q $ for which the $ 2 $- primary subgroup of $ K _ {2} ( {\mathcal O} _ {F} ) $ is elementary Abelian, the $ 2 $- part of the Birch–Tate conjecture has been confirmed [a3].

In addition, explicit examples of families of non-Abelian extensions $ F $ of $ \mathbf Q $ for which the $ 2 $- part of the Birch–Tate conjecture holds, have been given in [a1], [a2].

The Birch–Tate conjecture is related to the Lichtenbaum conjectures [a5] for totally real number fields $ F $. For every odd natural number $ m $, the Lichtenbaum conjectures express, up to $ 2 $- torsion, the ratio of the orders of $ K _ {2m } ( {\mathcal O} _ {F} ) $ and $ K _ {2m+1 } ( {\mathcal O} _ {F} ) $ in terms of the value of the zeta-function $ \zeta _ {F} $ at $ - m $.

References

[a1] P.E. Conner, J. Hurrelbrink, "Class number parity" , Pure Math. , 8 , World Sci. (1988)
[a2] J. Hurrelbrink, "Class numbers, units, and $K_2$" J.F. Jardine (ed.) V. Snaith (ed.) , Algebraic K-theory: Connection with Geometry and Topology , NATO ASI Ser. C , 279 , Kluwer Acad. Publ. (1989) pp. 87–102
[a3] M. Kolster, "The structure of the $2$-Sylow subgroup of $K_2(\mathcal{O})$ I" Comment. Math. Helv. , 61 (1986) pp. 376–388
[a4] M. Kolster, "A relation between the $2$-primary parts of the main conjecture and the Birch–Tate conjecture" Canad. Math. Bull. , 32 : 2 (1989) pp. 248–251
[a5] S. Lichtenbaum, "Values of zeta functions, étale cohomology, and algebraic K-theory" H. Bass (ed.) , Algebraic K-theory II , Lecture Notes in Mathematics , 342 , Springer (1973) pp. 489–501
[a6] B. Mazur, A. Wiles, "Class fields of abelian extensions of $\QQ$" Invent. Math. , 76 (1984) pp. 179–330
[a7] A. Wiles, "The Iwasawa conjecture for totally real fields" Ann. of Math. , 131 (1990) pp. 493–540
How to Cite This Entry:
Birch-Tate conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birch-Tate_conjecture&oldid=11217
This article was adapted from an original article by J. Hurrelbrink (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article