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The name elliptic genus has been given to various multiplicative [[Cobordism|cobordism]] invariants taking values in a ring of modular forms. The following is an attempt to present the simplest case — level- $  2 $
genera in characteristic  $  \neq 2 $— 
in a unified way. It is convenient to use N. Katz's approach to modular forms (cf. [[#References|[a7]]]) and view a [[Modular form|modular form]] as a function of elliptic curves with a chosen invariant differential (cf. also [[Elliptic curve|Elliptic curve]]). A similar approach to elliptic genera was used by J. Franke [[#References|[a3]]].

==Jacobi functions.==
Let  $  K $
be any [[Perfect field|perfect field]] of characteristic  $  \neq 2 $
and fix an algebraic closure  $  {\overline{K}\; } $
of  $  K $(
cf. [[Algebraically closed field|Algebraically closed field]]). Consider a triple  $  ( E, \omega, \alpha ) $
consisting of:

i) an elliptic curve  $  E $
over  $  K $, 
i.e. a smooth curve of genus  $  1 $
with a specified  $  K $-
rational base-point  $  O $;

ii) an invariant  $  K $-
rational differential  $  \omega $;

iii) a  $  K $-
rational primitive  $  2 $-
division point  $  \alpha $. 
Following J.I. Igusa [[#References|[a6]]] (up to a point), one can associate to these data two functions,  $  x $
and  $  y $, 
as follows.

The set  $  E _ {4} \subset  E ( {\overline{K}\; } ) $
of  $  4 $-
division points on  $  E $
can be described as follows. There are four  $  2 $-
division points  $  t $(
$  \alpha $
is one of them), four primitive  $  4 $-
division points  $  r $
such that  $  2r = \alpha $, 
and eight primitive  $  4 $-
division points  $  s $
such that  $  2s \neq \alpha $. 
Consider the degree- $  0 $
divisor  $  D = \sum ( t ) - \sum ( r ) $. 
Since  $  \sum t - \sum r = 0 $
in  $  E $
and since Galois symmetries transform  $  D $
into itself, Abel's theorem (cf., for example, [[#References|[a11]]], III.3.5.1, or [[Abel theorem|Abel theorem]]) implies that there is a function  $  x \in K ( E )  ^  \times  $, 
uniquely defined up to a multiplicative constant, such that  $  { \mathop{\rm div} } ( x ) = D $.

The function  $  x $
is odd, satisfies  $  x ( u + \alpha ) \equiv x ( u ) $, 
and undergoes sign changes under the two other translations of exact order  $  2 $. 
Moreover, if  $  r \in E _ {4} $
satisfies  $  2r = \alpha $, 
then translation by  $  r $
transforms  $  x $
into  $  Cx ^ {- 1 } $
for some non-zero constant  $  C $. 
This constant depends on the choice of  $  r $
but only up to sign. It follows that  $  x  ^ {2} ( u + r ) x  ^ {2} ( u ) $
does not depend on the choice of  $  r $. 
This constant is written as  $  \varepsilon ^ {- 1 } $, 
i.e.

$$ 
\varepsilon \equiv x ^ {- 2 } ( u + r ) x ^ {- 2 } ( u ) .
$$

One also defines

$$ 
\delta = {
\frac{1}{8}
 } \sum x ^ {- 2 } ( s )
$$

(the summation is over the primitive  $  4 $-
division points  $  s $
such that  $  2s \neq \alpha $). 
If  $  a $
is one of the values of  $  x ( s ) $, 
the other values are  $  \pm  a, \pm  \varepsilon ^ {- {1 / 2 } } a ^ {- 1 } $, 
each taken twice. It follows that

$$ 
\delta = {
\frac{1}{2}
 } ( a ^ {- 2 } + \varepsilon a  ^ {2} )
$$

and

$$ 
\prod ( X - x ( s ) ) = \varepsilon ^ {- 2 } ( 1 - 2 \delta X  ^ {2} + \varepsilon X  ^ {4} )  ^ {2} = \varepsilon ^ {- 2 } R ( X )  ^ {2} .
$$

It is now easy to see that

$$ 
{ \mathop{\rm div} } ( R ( x ) ) = 2 \left ( \sum ( s ) - 2 \sum ( r ) \right ) .
$$

Using once more Abel's theorem, one sees that there is a unique  $  y \in K ( E )  ^  \times  $
such that  $  { \mathop{\rm div} } ( y ) = \sum ( s ) - 2 \sum ( r ) $, 
and  $  y ( O ) = 1 $. 
Since  $  x ( O ) = 0 $, 
one has  $  y  ^ {2} = R ( x ) $.

The differential  $  dx $
has four double poles  $  r $. 
Also, it is easy to see that  $  s $
is a double zero of  $  x - x ( s ) $, 
hence a simple zero of  $  dx $. 
One concludes that

$$ 
{ \mathop{\rm div} } ( dx ) = \sum ( s ) - 2 \sum ( r ) = { \mathop{\rm div} } ( y ) .
$$

and that  $  { {dx } / y } $
is an invariant differential on  $  E $.

A slight modification of the argument given in [[#References|[a6]]] shows that the [[Jacobi elliptic functions|Jacobi elliptic functions]] satisfy the Euler addition formula

$$ 
x ( u + v ) ( 1 - \varepsilon x  ^ {2} ( u ) x  ^ {2} ( v ) ) = x ( u ) y ( v ) + x ( v ) y ( u ) .
$$

Accordingly, one defines the Euler formal group law  $  F ( U,V ) \in K [ [ U,V ] ] $
by

$$ 
F ( U,V ) = {
\frac{U \sqrt {R ( V ) } + V \sqrt {R ( U ) } }{1 - \varepsilon U  ^ {2} V  ^ {2} }
 } .
$$

Notice that since  $  { \mathop{\rm char} } K \neq 2 $, 
$  F ( U,V ) $
is defined over  $  K $.

==The elliptic genus.==
At this point, one normalizes  $  x $
over  $  K $
by requiring that  $  { {dx } / y } = \omega $(
the given invariant differential). All the objects  $  x, y, \delta, \varepsilon $, 
and  $  F ( U,V ) $
are now completely determined by the initial data. Replacing  $  \omega $
by  $  \lambda \omega $(
$  \lambda \in K  ^  \times  $) 
yields:

$$ \tag{a1 }
x \asR \lambda x, \quad y \asR y, \quad \delta \asR \lambda ^ {- 2 } \delta,
$$

$$ 
\varepsilon \asR \lambda ^ {- 4 } \varepsilon, \quad F ( U,V ) \asR \lambda F ( \lambda ^ {- 1 } U, \lambda ^ {- 1 } V ) .
$$

As any [[Formal group|formal group]] law,  $  F ( U,V ) $
is classified by a unique ring homomorphism

$$ 
\psi : {\Omega _ {*} ^ { { \mathop{\rm U} } } } \rightarrow K
$$

from the complex cobordism ring. Since  $  F ( - U, - V ) = - F ( U,V ) $, 
it is easy to see that  $  \psi $
uniquely factors through a ring homomorphism

$$ 
\varphi : {\Omega _ {*} ^ { { \mathop{\rm SO} } } } \rightarrow K
$$

from the oriented cobordism ring. By definition,  $  \varphi $
is the level- $  2 $
elliptic genus. Suppose now that  $  { \mathop{\rm char} } K = 0 $. 
Define a local parameter  $  z $
near  $  O $
so that  $  z ( O ) = 0 $
and  $  dz = \omega $. 
Then  $  x $
can be expanded into a formal power series  $  x ( z ) \in K [ [ z ] ] $
which clearly satisfies  $  x ( z ) = z + o ( z ) $
and  $  x ( - z ) = - x ( z ) $. 
In this case, the elliptic genus can be defined as the Hirzebruch genus (cf. [[#References|[a4]]] or [[#References|[a5]]]) corresponding to the series  $  P ( z ) = {z / {x ( z ) } } $. 
Since  $  { {d  x ( z ) } / {dz } } = y ( z ) $, 
the logarithm  $  g ( z ) $
of this elliptic genus is given by the [[Elliptic integral|elliptic integral]]

$$ \tag{a2 }
g ( z ) = \int\limits _ { 0 } ^ { z }  {
\frac{dt }{\sqrt {1 - 2 \delta t  ^ {2} + \varepsilon t  ^ {4} } }
 } ,
$$

which gives the original definition in [[#References|[a9]]].

==Modularity.==
For any closed oriented [[Manifold|manifold]]  $  M $
of dimension  $  4k $, 
$  \varphi ( M ) $
is a function of the triple  $  ( E, \omega, \alpha ) $. 
As easily follows from (a1), multiplying  $  \omega $
by  $  \lambda $
results in multiplying  $  \varphi ( M ) $
by  $  \lambda ^ {- 2k } $. 
Also,  $  \varphi ( M ) $
depends only on the isomorphism class of the triple  $  ( E, \omega, \alpha ) $
and commutes with arbitrary extensions of the scalar field  $  K $. 
In the terminology of Katz ([[#References|[a7]]]; adapted here to modular forms over fields),  $  \varphi ( M ) $
is a [[Modular form|modular form]] of level  $  2 $
and weight  $  2k $. 
Let  $  {\mathcal M} _ {*} $
be the graded ring of all such modular forms. Then  $  \varphi ( M ) \in {\mathcal M} _ {2k }  $, 
$  \delta \in {\mathcal M} _ {2} $, 
$  \varepsilon \in {\mathcal M} _ {4} $. 
Moreover, one can prove that  $  {\mathcal M} _ {*} \cong \mathbf Z [ {1 / 2 } , \delta, \varepsilon ] $. 
If one identifies these two isomorphic rings, the elliptic genus becomes the Hirzebruch genus

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

with logarithm given by the formal integral (a2).

==Integrality.==
Consider

$$ 
{ {\widetilde \varphi   } } : {\Omega _ {*} ^ { { \mathop{\rm Spin} } } } \rightarrow { {\mathcal M} _ {*} } ,
$$

i.e., the composition of  $  \varphi $
with the forgetful homomorphism  $  \Omega _ {*} ^ { { \mathop{\rm Spin} } } \rightarrow \Omega _ {*} ^ { { \mathop{\rm SO} } } $. 
As is shown in [[#References|[a2]]],

$$ 
{\widetilde \varphi   } ( \Omega _ {*} ^ { { \mathop{\rm Spin} } } ) = \mathbf Z [ 8 \delta, \varepsilon ] .
$$

The ring  $  \mathbf Z [ 8 \delta, \varepsilon ] $
agrees with the ring  $  {\mathcal M} _ {*} ( \mathbf Z ) $
of modular forms over  $  \mathbf Z $. 
Thus: If  $  M $
is a  $  { \mathop{\rm Spin} } $-
manifold of dimension  $  4k $, 
then  $  \varphi ( M ) \in {\mathcal M} _ {2k }  ( \mathbf Z ) $.

==Example: the Tate curve.==
Let  $  K $
be a [[Local field|local field]], complete with respect to a discrete [[Valuation|valuation]]  $  v $, 
and let  $  q \in K  ^  \times  $
be any element satisfying  $  v ( q ) < 0 $. 
Consider  $  E = K  ^  \times  /q ^ {2 \mathbf Z } $. 
It is well-known (cf. [[#References|[a11]]], § C.14) that  $  E $
can be identified with the elliptic curve (known as the Tate curve)

$$ 
E _ {q  ^ {2}  } : Y  ^ {2} + XY = X  ^ {3} + a _ {4} X + a _ {6} ,
$$

where

$$ 
a _ {4} = \sum _ {m \geq  1 } ( - 5m  ^ {3} ) {
\frac{q ^ {2m } }{1 - q ^ {2m } }
 } ,
$$

$$ 
a _ {6} = \sum _ {m \geq  1 } \left ( - {
\frac{5m  ^ {3} + 7m  ^ {5} }{12 }
 } \right ) {
\frac{q ^ {2m } }{1 - q ^ {2m } }
 } .
$$

$  E $
can be treated as an elliptic curve over  $  K $
with  $  O = 1 $. 
Fix the invariant differential  $  \omega = { {du } / u } $(
$  u \in K  ^  \times  $) 
on  $  E $(
$  \omega $
corresponds to the differential  $  \omega _ {\textrm{ can  }  } = { {dX } / {( 2Y + X ) } } $
on the Tate curve).  $  E $
has three  $  K $-
rational primitive  $  2 $-
division points:  $  - 1 $, 
$  q $
and  $  - q $. 
To describe the corresponding Jacobi function  $  x $, 
consider the [[Theta-function|theta-function]]

$$ 
\Theta ( u ) = ( 1 - u ^ {- 2 } ) \prod _ {n > 0 } ( 1 - q ^ {2n } u ^ {- 2 } ) ( 1 - q ^ {2n } u  ^ {2} ) .
$$

This is a "holomorphic" function on  $  K  ^  \times  $
with simple zeros at points of  $  \pm  q  ^ {\mathbf Z} $(
cf. [[#References|[a10]]] for a justification of this terminology), satisfying

$$ 
\Theta ( - u ) = \Theta ( u ) , \quad \Theta ( q ^ {- 1 } u ) = - u  ^ {2} \Theta ( u ) .
$$

Consider the case where  $  \alpha = - 1 $. 
Let  $  i \in {\overline{K}\; } $
be any square root of  $  - 1 $, 
and let

$$ \tag{a3 }
f ( u ) = {
\frac{\Theta ( u ) }{\Theta ( iu ) }
 } =
$$

$$ 
= 
{
\frac{u  ^ {2} - 1 }{u  ^ {2} + 1 }
 } \prod _ {n > 0 } {
\frac{( 1 - q ^ {2n } u ^ {- 2 } ) ( 1 - q ^ {2n } u  ^ {2} ) }{( 1 + q ^ {2n } u ^ {- 2 } ) ( 1 + q ^ {2n } u  ^ {2} ) }
 } .
$$

$  f $
is a [[Meromorphic function|meromorphic function]] on  $  E $
satisfying  $  f ( iu ) = {1 / {f ( u ) } } $
and

$$ 
{ \mathop{\rm div} } ( f ) = ( 1 ) + ( - 1 ) + ( q ) + ( - q ) +
$$

$$ 
- ( i ) - ( - i ) - ( iq ) - ( - iq ) ,
$$

i.e.,  $  f $
is a multiple of the Jacobi function  $  x $
of  $  ( E, \omega, - 1 ) $.

Notice now that the normalization condition  $  { {du } / u } = { {dx } / y } $
can be written as  $  y ( u ) = ux  ^  \prime  ( u ) $, 
where  $  x  ^  \prime  ( u ) $
is the derivative with respect to  $  u $. 
Since  $  y ( 1 ) = 0 $, 
one has  $  x  ^  \prime  ( 1 ) = 1 $. 
Differentiating (a3), one obtains

$$ 
f  ^  \prime  ( 1 ) = \prod _ {n > 0 } \left ( {
\frac{1 - q ^ {2n } }{1 + q ^ {2n } }
 } \right )  ^ {2} ,
$$

$$ 
x ( u ) = {
\frac{u  ^ {2} - 1 }{u  ^ {2} + 1 }
 } \prod _ {n > 0 } {
\frac{( 1 - q ^ {2n } u ^ {- 2 } ) ( 1 - q ^ {2n } u  ^ {2} ) ( 1 + q ^ {2n } )  ^ {2} }{( 1 + q ^ {2n } u ^ {- 2 } ) ( 1 + q ^ {2n } u  ^ {2} ) ( 1 - q ^ {2n } )  ^ {2} }
 } ,
$$

and

$$ 
\varepsilon = \prod _ {n > 0 } \left ( {
\frac{1 - q ^ {2n } }{1 + q ^ {2n } }
 } \right )  ^ {8} .
$$

Finally, if  $  { \mathop{\rm char} } K = 0 $, 
the function  $  z = { \mathop{\rm log} } u $
satisfies  $  dz = { {du } / u } $. 
It follows that the generating series  $  P ( z ) = {z / {x ( z ) } } $
is given by

$$ 
P ( z ) =
$$

$$ 
= 
{
\frac{z}{ { \mathop{\rm tanh} } z }
 } \prod _ {n > 0 } {
\frac{( 1 + q ^ {2n } e ^ {- 2z } ) ( 1 + q ^ {2n } e ^ {2z } ) ( 1 - q ^ {2n } )  ^ {2} }{( 1 - q ^ {2n } e ^ {- 2z } ) ( 1 - q ^ {2n } e ^ {2z } ) ( 1 + q ^ {2n } )  ^ {2} }
 } .
$$

The cases where  $  \alpha = q $
or  $  \alpha = - q $
are treated similarly, with

$$ 
f ( u ) = {
\frac{u \Theta ( u ) }{\Theta ( q ^ {- 1/2 } u ) }
 }
$$

and

$$ 
f ( u ) = {
\frac{u \Theta ( u ) }{\Theta ( iq ^ {- 1/2 } u ) }
 } ,
$$

respectively.

==Strict multiplicativity.==
The following theorem, also known (in an equivalent form) as the Witten conjecture, was proven first by C. Taubes [[#References|[a12]]], then by R. Bott and Taubes [[#References|[a1]]]. Let  $  P $
be a principal  $  G $-
bundle (cf. also [[Principal G-object|Principal  $  G $-
object]]) over an oriented manifold  $  B $, 
where  $  G $
is a compact connected [[Lie group|Lie group]], and suppose  $  G $
acts on a compact  $  { \mathop{\rm Spin} } $-
manifold  $  M $. 
Then

$$ 
\varphi ( P \times _ {G} M ) = \varphi ( B ) \varphi ( M ) .
$$

For the history of this conjecture, cf. [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Bott, C. Taubes, "On the rigidity theorems of Witten" ''J. Amer. Math. Soc.'' , '''2''' (1989) pp. 137–186 {{MR|0954493}} {{ZBL|0667.57009}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.V. Chudnovsky, G.V. Chudnovsky, P.S. Landweber, S. Ochanine, R.E. Stong, "Integrality and divisibility of the elliptic genus" ''Preprint'' (1988)</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top"> F. Hirzebruch, "Topological methods in algebraic geometry" , ''Grundlehren math. Wiss.'' , Springer (1966) (Edition: Third) {{MR|0202713}} {{ZBL|0138.42001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Hirzebruch, Th. Berger, R. Jung, "Manifolds and modular forms" , ''Aspects of Mathematics'' , '''E20''' , Vieweg (1992) (Appendices by Nils-Peter Skoruppa and by Paul Baum) {{MR|1189136}} {{ZBL|0752.57013}} {{ZBL|0767.57014}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.-I. Igusa, "On the transformation theory of elliptic functions" ''Amer. J. Math.'' , '''81''' (1959) pp. 436–452 {{MR|0104668}} {{ZBL|0131.28102}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> N.M. Katz, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070214.png" />-adic properties of modular schemes and modular forms" W. Kuyk (ed.) J.-P. Serre (ed.) , ''Modular Functions in One Variable III. Proc. Internat. Summer School, Univ. of Antwerp, RUCA, July 17--August 3, 1972'' , ''Lecture Notes in Mathematics'' , '''350''' (1973) pp. 69–190 {{MR|0447119}} {{ZBL|0271.10033}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> P.S. Landweber, "Elliptic genera: An introductory overview" P.S. Landweber (ed.) , ''Elliptic Curves and Modular Forms in Algebraic Topology (Proc., Princeton 1986)'' , ''Lecture Notes in Mathematics'' , '''1326''' , Springer (1988) pp. 1–10 {{MR|0970279}} {{ZBL|0649.57021}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> S. Ochanine, "Sur les genres multiplicatifs définis par des intégrales elliptiques" ''Topology'' , '''26''' (1987) pp. 143–151 {{MR|0895567}} {{ZBL|0626.57014}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Roquette, "Analytic theory of elliptic functions over local fields" , ''Hamburger Math. Einzelschrift.'' , '''1''' , Vandenhoeck and Ruprecht (1970) {{MR|0260753}} {{ZBL|0194.52002}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> J.H. Silverman, "The arithmetic of elliptic curves" , ''GTM'' , '''106''' , Springer (1986) {{MR|0817210}} {{ZBL|0585.14026}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> C. Taubes, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070215.png" /> actions and elliptic genera" ''Comm. Math. Phys.'' , '''122''' (1989) pp. 455–526 {{MR|0998662}} {{ZBL|0683.58043}} </TD></TR></table>