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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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100701.png" /> genera in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100702.png" /> — 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]]].
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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.==
 
==Jacobi functions.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100703.png" /> be any [[Perfect field|perfect field]] of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100704.png" /> and fix an algebraic closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100705.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100706.png" /> (cf. [[Algebraically closed field|Algebraically closed field]]). Consider a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100707.png" /> consisting of:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100708.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e1100709.png" />, i.e. a smooth curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007010.png" /> with a specified <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007011.png" />-rational base-point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007012.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007013.png" />-rational differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007014.png" />;
+
ii) an invariant $  K $-
 +
rational differential $  \omega $;
  
iii) a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007015.png" />-rational primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007016.png" />-division point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007017.png" />. Following J.I. Igusa [[#References|[a6]]] (up to a point), one can associate to these data two functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007019.png" />, as follows.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007021.png" />-division points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007022.png" /> can be described as follows. There are four <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007023.png" />-division points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007024.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007025.png" /> is one of them), four primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007026.png" />-division points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007027.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007028.png" />, and eight primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007029.png" />-division points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007031.png" />. Consider the degree-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007032.png" /> divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007033.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007035.png" /> and since Galois symmetries transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007036.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007037.png" />, uniquely defined up to a multiplicative constant, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007038.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007039.png" /> is odd, satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007040.png" />, and undergoes sign changes under the two other translations of exact order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007041.png" />. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007042.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007043.png" />, then translation by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007044.png" /> transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007045.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007046.png" /> for some non-zero constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007047.png" />. This constant depends on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007048.png" /> but only up to sign. It follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007049.png" /> does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007050.png" />. This constant is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007051.png" />, i.e.
+
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.
  
<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/e110070/e11007052.png" /></td> </tr></table>
+
$$
 +
\varepsilon \equiv x ^ {- 2 } ( u + r ) x ^ {- 2 } ( u ) .
 +
$$
  
 
One also defines
 
One also defines
  
<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/e110070/e11007053.png" /></td> </tr></table>
+
$$
 +
\delta = {
 +
\frac{1}{8}
 +
} \sum x ^ {- 2 } ( s )
 +
$$
  
(the summation is over the primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007054.png" />-division points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007056.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007057.png" /> is one of the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007058.png" />, the other values are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007059.png" />, each taken twice. It follows that
+
(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
  
<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/e110070/e11007060.png" /></td> </tr></table>
+
$$
 +
\delta = {
 +
\frac{1}{2}
 +
} ( a ^ {- 2 } + \varepsilon a  ^ {2} )
 +
$$
  
 
and
 
and
  
<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/e110070/e11007061.png" /></td> </tr></table>
+
$$
 +
\prod ( X - x ( s ) ) = \varepsilon ^ {- 2 } ( 1 - 2 \delta X  ^ {2} + \varepsilon X  ^ {4} )  ^ {2} = \varepsilon ^ {- 2 } R ( X )  ^ {2} .
 +
$$
  
 
It is now easy to see that
 
It is now easy to see that
  
<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/e110070/e11007062.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm div} } ( R ( x ) ) = 2 \left ( \sum ( s ) - 2 \sum ( r ) \right ) .
 +
$$
  
Using once more Abel's theorem, one sees that there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007063.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007064.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007065.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007066.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007067.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007068.png" /> has four double poles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007069.png" />. Also, it is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007070.png" /> is a double zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007071.png" />, hence a simple zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007072.png" />. One concludes that
+
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
  
<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/e110070/e11007073.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm div} } ( dx ) = \sum ( s ) - 2 \sum ( r ) = { \mathop{\rm div} } ( y ) .
 +
$$
  
and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007074.png" /> is an invariant differential on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007075.png" />.
+
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
 
A slight modification of the argument given in [[#References|[a6]]] shows that the [[Jacobi elliptic functions|Jacobi elliptic functions]] satisfy the Euler addition formula
  
<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/e110070/e11007076.png" /></td> </tr></table>
+
$$
 +
x ( u + v ) ( 1 - \varepsilon x  ^ {2} ( u ) x  ^ {2} ( v ) ) = x ( u ) y ( v ) + x ( v ) y ( u ) .
 +
$$
  
Accordingly, one defines the Euler formal group law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007077.png" /> by
+
Accordingly, one defines the Euler formal group law $  F ( U,V ) \in K [ [ U,V ] ] $
 +
by
  
<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/e110070/e11007078.png" /></td> </tr></table>
+
$$
 +
F ( U,V ) = {
 +
\frac{U \sqrt {R ( V ) } + V \sqrt {R ( U ) } }{1 - \varepsilon U  ^ {2} V  ^ {2} }
 +
} .
 +
$$
  
Notice that since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007080.png" /> is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007081.png" />.
+
Notice that since $  { \mathop{\rm char} } K \neq 2 $,  
 +
$  F ( U,V ) $
 +
is defined over $  K $.
  
 
==The elliptic genus.==
 
==The elliptic genus.==
At this point, one normalizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007082.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007083.png" /> by requiring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007084.png" /> (the given invariant differential). All the objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007085.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007086.png" /> are now completely determined by the initial data. Replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007087.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007088.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007089.png" />) yields:
+
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:
  
<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/e110070/e11007090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
x \asR \lambda x, \quad y \asR y, \quad \delta \asR \lambda ^ {- 2 } \delta,
 +
$$
  
<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/e110070/e11007091.png" /></td> </tr></table>
+
$$
 +
\varepsilon \asR \lambda ^ {- 4 } \varepsilon, \quad F ( U,V ) \asR \lambda F ( \lambda ^ {- 1 } U, \lambda ^ {- 1 } V ) .
 +
$$
  
As any [[Formal group|formal group]] law, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007092.png" /> is classified by a unique ring homomorphism
+
As any [[Formal group|formal group]] law, $  F ( U,V ) $
 +
is classified by a unique ring 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/e/e110/e110070/e11007093.png" /></td> </tr></table>
+
$$
 +
\psi : {\Omega _ {*} ^ { { \mathop{\rm U} } } } \rightarrow K
 +
$$
  
from the complex cobordism ring. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007094.png" />, it is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007095.png" /> uniquely factors through a ring homomorphism
+
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
  
<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/e110070/e11007096.png" /></td> </tr></table>
+
$$
 +
\varphi : {\Omega _ {*} ^ { { \mathop{\rm SO} } } } \rightarrow K
 +
$$
  
from the oriented cobordism ring. By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007097.png" /> is the level-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e11007099.png" /> elliptic genus. Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070100.png" />. Define a local parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070101.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070102.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070104.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070105.png" /> can be expanded into a formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070106.png" /> which clearly satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070108.png" />. In this case, the elliptic genus can be defined as the Hirzebruch genus (cf. [[#References|[a4]]] or [[#References|[a5]]]) corresponding to the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070109.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070110.png" />, the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070111.png" /> of this elliptic genus is given by the [[Elliptic integral|elliptic integral]]
+
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]]
  
<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/e110070/e110070112.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
g ( z ) = \int\limits _ { 0 } ^ { z }  {
 +
\frac{dt }{\sqrt {1 - 2 \delta t  ^ {2} + \varepsilon t  ^ {4} } }
 +
} ,
 +
$$
  
 
which gives the original definition in [[#References|[a9]]].
 
which gives the original definition in [[#References|[a9]]].
  
 
==Modularity.==
 
==Modularity.==
For any closed oriented [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070113.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070115.png" /> is a function of the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070116.png" />. As easily follows from (a1), multiplying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070117.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070118.png" /> results in multiplying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070119.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070120.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070121.png" /> depends only on the isomorphism class of the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070122.png" /> and commutes with arbitrary extensions of the scalar field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070123.png" />. In the terminology of Katz ([[#References|[a7]]]; adapted here to modular forms over fields), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070124.png" /> is a [[Modular form|modular form]] of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070125.png" /> and weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070126.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070127.png" /> be the graded ring of all such modular forms. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070129.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070130.png" />. Moreover, one can prove that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070131.png" />. If one identifies these two isomorphic rings, the elliptic genus becomes the Hirzebruch genus
+
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
  
<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/e110070/e110070132.png" /></td> </tr></table>
+
$$
 +
\varphi : {\Omega _ {*} ^ { { \mathop{\rm SO} } } } \rightarrow {\mathbf Z [ {1 / 2 } , \delta, \varepsilon ] }
 +
$$
  
 
with logarithm given by the formal integral (a2).
 
with logarithm given by the formal integral (a2).
Line 81: Line 240:
 
Consider
 
Consider
  
<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/e110070/e110070133.png" /></td> </tr></table>
+
$$
 +
{ {\widetilde \varphi  } } : {\Omega _ {*} ^ { { \mathop{\rm Spin} } } } \rightarrow { {\mathcal M} _ {*} } ,
 +
$$
  
i.e., the composition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070134.png" /> with the forgetful homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070135.png" />. As is shown in [[#References|[a2]]],
+
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]]],
  
<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/e110070/e110070136.png" /></td> </tr></table>
+
$$
 +
{\widetilde \varphi  } ( \Omega _ {*} ^ { { \mathop{\rm Spin} } } ) = \mathbf Z [ 8 \delta, \varepsilon ] .
 +
$$
  
The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070137.png" /> agrees with the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070138.png" /> of modular forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070139.png" />. Thus: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070140.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070141.png" />-manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070142.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070143.png" />.
+
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.==
 
==Example: the Tate curve.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070144.png" /> be a [[Local field|local field]], complete with respect to a discrete [[Valuation|valuation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070145.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070146.png" /> be any element satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070147.png" />. Consider <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070148.png" />. It is well-known (cf. [[#References|[a11]]], § C.14) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070149.png" /> can be identified with the elliptic curve (known as 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)
  
<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/e110070/e110070150.png" /></td> </tr></table>
+
$$
 +
E _ {q  ^ {2}  } : Y  ^ {2} + XY = X  ^ {3} + a _ {4} X + a _ {6} ,
 +
$$
  
 
where
 
where
  
<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/e110070/e110070151.png" /></td> </tr></table>
+
$$
 +
a _ {4} = \sum _ {m \geq  1 } ( - 5m  ^ {3} ) {
 +
\frac{q ^ {2m } }{1 - q ^ {2m } }
 +
} ,
 +
$$
  
<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/e110070/e110070152.png" /></td> </tr></table>
+
$$
 +
a _ {6} = \sum _ {m \geq  1 } \left ( - {
 +
\frac{5m  ^ {3} + 7m  ^ {5} }{12 }
 +
} \right ) {
 +
\frac{q ^ {2m } }{1 - q ^ {2m } }
 +
} .
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070153.png" /> can be treated as an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070154.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070155.png" />. Fix the invariant differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070156.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070157.png" />) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070158.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070159.png" /> corresponds to the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070160.png" /> on the Tate curve). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070161.png" /> has three <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070162.png" />-rational primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070163.png" />-division points: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070165.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070166.png" />. To describe the corresponding Jacobi function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070167.png" />, consider the [[Theta-function|theta-function]]
+
$  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]]
  
<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/e110070/e110070168.png" /></td> </tr></table>
+
$$
 +
\Theta ( u ) = ( 1 - u ^ {- 2 } ) \prod _ {n > 0 } ( 1 - q ^ {2n } u ^ {- 2 } ) ( 1 - q ^ {2n } u  ^ {2} ) .
 +
$$
  
This is a "holomorphic" function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070169.png" /> with simple zeros at points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070170.png" /> (cf. [[#References|[a10]]] for a justification of this terminology), satisfying
+
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
  
<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/e110070/e110070171.png" /></td> </tr></table>
+
$$
 +
\Theta ( - u ) = \Theta ( u ) , \quad \Theta ( q ^ {- 1 } u ) = - u  ^ {2} \Theta ( u ) .
 +
$$
  
Consider the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070172.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070173.png" /> be any square root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070174.png" />, and let
+
Consider the case where $  \alpha = - 1 $.  
 +
Let $  i \in {\overline{K}\; } $
 +
be any square root of $  - 1 $,  
 +
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/e110070/e110070175.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
f ( u ) = {
 +
\frac{\Theta ( u ) }{\Theta ( iu ) }
 +
} =
 +
$$
  
<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/e110070/e110070176.png" /></td> </tr></table>
+
$$
 +
=  
 +
{
 +
\frac{u  ^ {2} - 1 }{u  ^ {2} + 1 }
 +
} \prod _ {n > 0 } {
 +
\frac{( 1 - q ^ {2n } u ^ {- 2 } ) ( 1 - q ^ {2n } u  ^ {2} ) }{( 1 + q ^ {2n } u ^ {- 2 } ) ( 1 + q ^ {2n } u  ^ {2} ) }
 +
} .
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070177.png" /> is a [[Meromorphic function|meromorphic function]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070178.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070179.png" /> and
+
$  f $
 +
is a [[Meromorphic function|meromorphic function]] on $  E $
 +
satisfying $  f ( iu ) = {1 / {f ( u ) } } $
 +
and
  
<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/e110070/e110070180.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm div} } ( f ) = ( 1 ) + ( - 1 ) + ( q ) + ( - q ) +
 +
$$
  
<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/e110070/e110070181.png" /></td> </tr></table>
+
$$
 +
- ( i ) - ( - i ) - ( iq ) - ( - iq ) ,
 +
$$
  
i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070182.png" /> is a multiple of the Jacobi function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070183.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070184.png" />.
+
i.e., $  f $
 +
is a multiple of the Jacobi function $  x $
 +
of $  ( E, \omega, - 1 ) $.
  
Notice now that the normalization condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070185.png" /> can be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070186.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070187.png" /> is the derivative with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070188.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070189.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070190.png" />. Differentiating (a3), one obtains
+
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
  
<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/e110070/e110070191.png" /></td> </tr></table>
+
$$
 +
f  ^  \prime  ( 1 ) = \prod _ {n > 0 } \left ( {
 +
\frac{1 - q ^ {2n } }{1 + q ^ {2n } }
 +
} \right )  ^ {2} ,
 +
$$
  
<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/e110070/e110070192.png" /></td> </tr></table>
+
$$
 +
x ( u ) = {
 +
\frac{u  ^ {2} - 1 }{u  ^ {2} + 1 }
 +
} \prod _ {n > 0 } {
 +
\frac{( 1 - q ^ {2n } u ^ {- 2 } ) ( 1 - q ^ {2n } u  ^ {2} ) ( 1 + q ^ {2n } )  ^ {2} }{( 1 + q ^ {2n } u ^ {- 2 } ) ( 1 + q ^ {2n } u  ^ {2} ) ( 1 - q ^ {2n } )  ^ {2} }
 +
} ,
 +
$$
  
 
and
 
and
  
<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/e110070/e110070193.png" /></td> </tr></table>
+
$$
 +
\varepsilon = \prod _ {n > 0 } \left ( {
 +
\frac{1 - q ^ {2n } }{1 + q ^ {2n } }
 +
} \right )  ^ {8} .
 +
$$
  
Finally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070194.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070195.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070196.png" />. It follows that the generating series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070197.png" /> is given by
+
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
  
<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/e110070/e110070198.png" /></td> </tr></table>
+
$$
 +
P ( z ) =
 +
$$
  
<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/e110070/e110070199.png" /></td> </tr></table>
+
$$
 +
=  
 +
{
 +
\frac{z}{ { \mathop{\rm tanh} } z }
 +
} \prod _ {n > 0 } {
 +
\frac{( 1 + q ^ {2n } e ^ {- 2z } ) ( 1 + q ^ {2n } e ^ {2z } ) ( 1 - q ^ {2n } )  ^ {2} }{( 1 - q ^ {2n } e ^ {- 2z } ) ( 1 - q ^ {2n } e ^ {2z } ) ( 1 + q ^ {2n } )  ^ {2} }
 +
} .
 +
$$
  
The cases where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070200.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070201.png" /> are treated similarly, with
+
The cases where $  \alpha = q $
 +
or $  \alpha = - q $
 +
are treated similarly, with
  
<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/e110070/e110070202.png" /></td> </tr></table>
+
$$
 +
f ( u ) = {
 +
\frac{u \Theta ( u ) }{\Theta ( q ^ {- 1/2 } u ) }
 +
}
 +
$$
  
 
and
 
and
  
<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/e110070/e110070203.png" /></td> </tr></table>
+
$$
 +
f ( u ) = {
 +
\frac{u \Theta ( u ) }{\Theta ( iq ^ {- 1/2 } u ) }
 +
} ,
 +
$$
  
 
respectively.
 
respectively.
  
 
==Strict multiplicativity.==
 
==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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070204.png" /> be a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070206.png" />-bundle (cf. also [[Principal G-object|Principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070207.png" />-object]]) over an oriented manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070208.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070209.png" /> is a compact connected [[Lie group|Lie group]], and suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070210.png" /> acts on a compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070211.png" />-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110070/e110070212.png" />. Then
+
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
  
<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/e110070/e110070213.png" /></td> </tr></table>
+
$$
 +
\varphi ( P \times _ {G} M ) = \varphi ( B ) \varphi ( M ) .
 +
$$
  
 
For the history of this conjecture, cf. [[#References|[a8]]].
 
For the history of this conjecture, cf. [[#References|[a8]]].

Latest revision as of 19:37, 5 June 2020


The name elliptic genus has been given to various multiplicative 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. [a7]) and view a modular form as a function of elliptic curves with a chosen invariant differential (cf. also Elliptic curve). A similar approach to elliptic genera was used by J. Franke [a3].

Jacobi functions.

Let $ K $ be any perfect field of characteristic $ \neq 2 $ and fix an algebraic closure $ {\overline{K}\; } $ of $ K $( cf. 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 [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, [a11], III.3.5.1, or 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 [a6] shows that the 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 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. [a4] or [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

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

which gives the original definition in [a9].

Modularity.

For any closed oriented 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 ([a7]; adapted here to modular forms over fields), $ \varphi ( M ) $ is a 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 [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, complete with respect to a discrete 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. [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

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

This is a "holomorphic" function on $ K ^ \times $ with simple zeros at points of $ \pm q ^ {\mathbf Z} $( cf. [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 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 [a12], then by R. Bott and Taubes [a1]. Let $ P $ be a principal $ G $- bundle (cf. also Principal $ G $- object) over an oriented manifold $ B $, where $ G $ is a compact connected 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. [a8].

References

[a1] R. Bott, C. Taubes, "On the rigidity theorems of Witten" J. Amer. Math. Soc. , 2 (1989) pp. 137–186 MR0954493 Zbl 0667.57009
[a2] D.V. Chudnovsky, G.V. Chudnovsky, P.S. Landweber, S. Ochanine, R.E. Stong, "Integrality and divisibility of the elliptic genus" Preprint (1988)
[a3] J. Franke, "On the construction of elliptic cohomology" Math. Nachr. , 158 (1992) pp. 43–65 MR1235295 Zbl 0777.55003
[a4] F. Hirzebruch, "Topological methods in algebraic geometry" , Grundlehren math. Wiss. , Springer (1966) (Edition: Third) MR0202713 Zbl 0138.42001
[a5] 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) MR1189136 Zbl 0752.57013 Zbl 0767.57014
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How to Cite This Entry:
Elliptic genera. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_genera&oldid=23820
This article was adapted from an original article by S. Ochanine (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article