Namespaces
Variants
Actions

Difference between revisions of "Elliptic cohomology"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
A term first introduced in 1986 by P.S. Landweber, D.C. Ravenel and R.E. Stong (cf. [[#References|[a2]]] and [[#References|[a3]]]) to designate a [[Cohomology|cohomology]] theory obtained by tensoring oriented [[Cobordism|cobordism]] theory with a ring of modular forms in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100601.png" />. In the 1990{}s, numerous publications devoted to elliptic cohomology have appeared. On the one hand, they connect elliptic cohomology to other generalized cohomology theories, most notably to Morava <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100603.png" />-theories (cf. also [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100604.png" />-theory]]), on the other hand they attempt to give a geometric interpretation of elliptic cohomology. Despite a multitude of very interesting results, there seems to be no agreement on what exactly is elliptic cohomology. The situation is, however, quite well-understood outside the prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100605.png" />.
+
<!--
 +
e1100601.png
 +
$#A+1 = 24 n = 0
 +
$#C+1 = 24 : ~/encyclopedia/old_files/data/E110/E.1100060 Elliptic cohomology
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
 +
A term first introduced in 1986 by P.S. Landweber, D.C. Ravenel and R.E. Stong (cf. [[#References|[a2]]] and [[#References|[a3]]]) to designate a [[Cohomology|cohomology]] theory obtained by tensoring oriented [[Cobordism|cobordism]] theory with a ring of modular forms in characteristic $  \neq 2 $.  
 +
In the 1990{}s, numerous publications devoted to elliptic cohomology have appeared. On the one hand, they connect elliptic cohomology to other generalized cohomology theories, most notably to Morava $  K $-
 +
theories (cf. also [[K-theory| $  K $-
 +
theory]]), on the other hand they attempt to give a geometric interpretation of elliptic cohomology. Despite a multitude of very interesting results, there seems to be no agreement on what exactly is elliptic cohomology. The situation is, however, quite well-understood outside the prime number $  2 $.
  
 
==Landweber–Ravenel–Stong theory.==
 
==Landweber–Ravenel–Stong theory.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100606.png" /> be oriented bordism theory with coefficient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100607.png" />, and let
+
Let $  \Omega _ {*} ^ { { \mathop{\rm SO} } } (  ) $
 +
be oriented bordism theory with coefficient ring $  \Omega _ {*} ^ { { \mathop{\rm SO} } } $,  
 +
and let
  
<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/e/e110/e110060/e1100608.png" /></td> </tr></table>
+
$$
 +
\varphi : {\Omega _ {*} ^ { { \mathop{\rm SO} } } } \rightarrow {\mathbf Z [ {1 / 2 } , \delta, \varepsilon ] }
 +
$$
  
be the level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e1100609.png" /> elliptic genus (cf. [[Elliptic genera|Elliptic genera]]) with logarithm
+
be the level- $  2 $
 +
elliptic genus (cf. [[Elliptic genera|Elliptic genera]]) with logarithm
  
<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/e/e110/e110060/e11006010.png" /></td> </tr></table>
+
$$
 +
g ( z ) = \int\limits _ { 0 } ^ { z }  { {
 +
\frac{dt }{\sqrt {1 - 2 \delta t  ^ {2} + \varepsilon t  ^ {4} } }
 +
} } .
 +
$$
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006011.png" /> is the ring of level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006012.png" /> modular forms in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006013.png" /> (cf. also [[Modular form|Modular form]]) and can be viewed, via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006014.png" />, as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006015.png" />-module. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006016.png" /> has a natural grading, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006018.png" />.
+
The ring $  {\mathcal M} _ {*} \equiv \mathbf Z [ {1 / 2 } , \delta, \varepsilon ] $
 +
is the ring of level- $  2 $
 +
modular forms in characteristic $  \neq 2 $(
 +
cf. also [[Modular form|Modular form]]) and can be viewed, via $  \varphi $,  
 +
as an $  \Omega _ {*} ^ { { \mathop{\rm SO} } } $-
 +
module. $  {\mathcal M} _ {*} $
 +
has a natural grading, for which $  { \mathop{\rm deg} } \delta = 2 $,  
 +
$  { \mathop{\rm deg} } \varepsilon = 4 $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006019.png" /> be any homogeneous element of positive degree. Then the functor
+
Let $  \pi \in {\mathcal M} _ {*} $
 +
be any homogeneous element of positive degree. Then the functor
  
<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/e/e110/e110060/e11006020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
X \mapsto \Omega _ {*} ^ { { \mathop{\rm SO} } } ( X ) \otimes _ {\Omega _ {*}  ^ { { \mathop{\rm SO} } } } {\mathcal M} _ {*} [ \pi ^ {- 1 } ]
 +
$$
  
is a periodic homology theory (elliptic homology) with coefficient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006021.png" />.
+
is a periodic homology theory (elliptic homology) with coefficient ring $  {\mathcal M} _ {*} [ \pi ^ {- 1 } ] $.
  
A similar construction using oriented cohomology leads to a multiplicative periodic cohomology theory (elliptic cohomology). The proof of this theorem was first given by Landweber, Ravenel and Stong under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006022.png" /> is one of the factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006023.png" />, and was based on the one hand on Landweber's exact functor theorem, and on the other hand on interesting congruences for [[Legendre polynomials|Legendre polynomials]]. The general form stated above is due to J. Franke [[#References|[a1]]], who showed that the exactness of the functor (a1) is a consequence of the Deuring–Eichler theorem, saying that the height of the [[Formal group|formal group]] of an [[Elliptic curve|elliptic curve]] in positive characteristic is always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006024.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110060/e11006025.png" />.
+
A similar construction using oriented cohomology leads to a multiplicative periodic cohomology theory (elliptic cohomology). The proof of this theorem was first given by Landweber, Ravenel and Stong under the assumption that $  \pi $
 +
is one of the factors of $  \Delta = \varepsilon ( \delta  ^ {2} - \varepsilon )  ^ {2} $,  
 +
and was based on the one hand on Landweber's exact functor theorem, and on the other hand on interesting congruences for [[Legendre polynomials|Legendre polynomials]]. The general form stated above is due to J. Franke [[#References|[a1]]], who showed that the exactness of the functor (a1) is a consequence of the Deuring–Eichler theorem, saying that the height of the [[Formal group|formal group]] of an [[Elliptic curve|elliptic curve]] in positive characteristic is always $  1 $
 +
or $  2 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Franke, "On the construction of elliptic cohomology" ''Math. Nachr.'' , '''158''' (1992) pp. 43–65 {{MR|1235295}} {{ZBL|0777.55003}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.S. Landweber, "Elliptic cohomology and modular forms" P.S. Landweber (ed.) , ''Elliptic Curves and Modular Forms in Algebraic Topology (Proc., Princeton 1986)'' , ''Lecture Notes in Mathematics'' , '''1326''' , Springer (1988) pp. 55–68 {{MR|0970281}} {{ZBL|0649.57022}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.S. Landweber, D.C. Ravenel, R.E. Stong, "Periodic cohomology theories defined by elliptic curves" , ''The Čech Centennial (Boston, 1993)'' , ''Contemp. Math.'' , '''181''' , Amer. Math. Soc. (1995) {{MR|1320998}} {{ZBL|0920.55005}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Franke, "On the construction of elliptic cohomology" ''Math. Nachr.'' , '''158''' (1992) pp. 43–65 {{MR|1235295}} {{ZBL|0777.55003}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P.S. Landweber, "Elliptic cohomology and modular forms" P.S. Landweber (ed.) , ''Elliptic Curves and Modular Forms in Algebraic Topology (Proc., Princeton 1986)'' , ''Lecture Notes in Mathematics'' , '''1326''' , Springer (1988) pp. 55–68 {{MR|0970281}} {{ZBL|0649.57022}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.S. Landweber, D.C. Ravenel, R.E. Stong, "Periodic cohomology theories defined by elliptic curves" , ''The Čech Centennial (Boston, 1993)'' , ''Contemp. Math.'' , '''181''' , Amer. Math. Soc. (1995) {{MR|1320998}} {{ZBL|0920.55005}} </TD></TR></table>

Latest revision as of 19:37, 5 June 2020


A term first introduced in 1986 by P.S. Landweber, D.C. Ravenel and R.E. Stong (cf. [a2] and [a3]) to designate a cohomology theory obtained by tensoring oriented cobordism theory with a ring of modular forms in characteristic $ \neq 2 $. In the 1990{}s, numerous publications devoted to elliptic cohomology have appeared. On the one hand, they connect elliptic cohomology to other generalized cohomology theories, most notably to Morava $ K $- theories (cf. also $ K $- theory), on the other hand they attempt to give a geometric interpretation of elliptic cohomology. Despite a multitude of very interesting results, there seems to be no agreement on what exactly is elliptic cohomology. The situation is, however, quite well-understood outside the prime number $ 2 $.

Landweber–Ravenel–Stong theory.

Let $ \Omega _ {*} ^ { { \mathop{\rm SO} } } ( ) $ be oriented bordism theory with coefficient ring $ \Omega _ {*} ^ { { \mathop{\rm SO} } } $, and let

$$ \varphi : {\Omega _ {*} ^ { { \mathop{\rm SO} } } } \rightarrow {\mathbf Z [ {1 / 2 } , \delta, \varepsilon ] } $$

be the level- $ 2 $ elliptic genus (cf. Elliptic genera) with logarithm

$$ g ( z ) = \int\limits _ { 0 } ^ { z } { { \frac{dt }{\sqrt {1 - 2 \delta t ^ {2} + \varepsilon t ^ {4} } } } } . $$

The ring $ {\mathcal M} _ {*} \equiv \mathbf Z [ {1 / 2 } , \delta, \varepsilon ] $ is the ring of level- $ 2 $ modular forms in characteristic $ \neq 2 $( cf. also Modular form) and can be viewed, via $ \varphi $, as an $ \Omega _ {*} ^ { { \mathop{\rm SO} } } $- module. $ {\mathcal M} _ {*} $ has a natural grading, for which $ { \mathop{\rm deg} } \delta = 2 $, $ { \mathop{\rm deg} } \varepsilon = 4 $.

Let $ \pi \in {\mathcal M} _ {*} $ be any homogeneous element of positive degree. Then the functor

$$ \tag{a1 } X \mapsto \Omega _ {*} ^ { { \mathop{\rm SO} } } ( X ) \otimes _ {\Omega _ {*} ^ { { \mathop{\rm SO} } } } {\mathcal M} _ {*} [ \pi ^ {- 1 } ] $$

is a periodic homology theory (elliptic homology) with coefficient ring $ {\mathcal M} _ {*} [ \pi ^ {- 1 } ] $.

A similar construction using oriented cohomology leads to a multiplicative periodic cohomology theory (elliptic cohomology). The proof of this theorem was first given by Landweber, Ravenel and Stong under the assumption that $ \pi $ is one of the factors of $ \Delta = \varepsilon ( \delta ^ {2} - \varepsilon ) ^ {2} $, and was based on the one hand on Landweber's exact functor theorem, and on the other hand on interesting congruences for Legendre polynomials. The general form stated above is due to J. Franke [a1], who showed that the exactness of the functor (a1) is a consequence of the Deuring–Eichler theorem, saying that the height of the formal group of an elliptic curve in positive characteristic is always $ 1 $ or $ 2 $.

References

[a1] J. Franke, "On the construction of elliptic cohomology" Math. Nachr. , 158 (1992) pp. 43–65 MR1235295 Zbl 0777.55003
[a2] P.S. Landweber, "Elliptic cohomology and modular forms" P.S. Landweber (ed.) , Elliptic Curves and Modular Forms in Algebraic Topology (Proc., Princeton 1986) , Lecture Notes in Mathematics , 1326 , Springer (1988) pp. 55–68 MR0970281 Zbl 0649.57022
[a3] P.S. Landweber, D.C. Ravenel, R.E. Stong, "Periodic cohomology theories defined by elliptic curves" , The Čech Centennial (Boston, 1993) , Contemp. Math. , 181 , Amer. Math. Soc. (1995) MR1320998 Zbl 0920.55005
How to Cite This Entry:
Elliptic cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_cohomology&oldid=46808
This article was adapted from an original article by S. Ochanine (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article