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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.==
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|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|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|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====
<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>