# Elliptic genera

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

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How to Cite This Entry:
Elliptic genera. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_genera&oldid=46811
This article was adapted from an original article by S. Ochanine (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article