# Elliptic genera

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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- genera in characteristic — 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 be any perfect field of characteristic and fix an algebraic closure of (cf. Algebraically closed field). Consider a triple consisting of:

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

ii) an invariant -rational differential ;

iii) a -rational primitive -division point . Following J.I. Igusa [a6] (up to a point), one can associate to these data two functions, and , as follows.

The set of -division points on can be described as follows. There are four -division points ( is one of them), four primitive -division points such that , and eight primitive -division points such that . Consider the degree- divisor . Since in and since Galois symmetries transform into itself, Abel's theorem (cf., for example, [a11], III.3.5.1, or Abel theorem) implies that there is a function , uniquely defined up to a multiplicative constant, such that .

The function is odd, satisfies , and undergoes sign changes under the two other translations of exact order . Moreover, if satisfies , then translation by transforms into for some non-zero constant . This constant depends on the choice of but only up to sign. It follows that does not depend on the choice of . This constant is written as , i.e.

One also defines

(the summation is over the primitive -division points such that ). If is one of the values of , the other values are , each taken twice. It follows that

and

It is now easy to see that

Using once more Abel's theorem, one sees that there is a unique such that , and . Since , one has .

The differential has four double poles . Also, it is easy to see that is a double zero of , hence a simple zero of . One concludes that

and that is an invariant differential on .

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

Accordingly, one defines the Euler formal group law by

Notice that since , is defined over .

## The elliptic genus.

At this point, one normalizes over by requiring that (the given invariant differential). All the objects , and are now completely determined by the initial data. Replacing by () yields:

 (a1)

As any formal group law, is classified by a unique ring homomorphism

from the complex cobordism ring. Since , it is easy to see that uniquely factors through a ring homomorphism

from the oriented cobordism ring. By definition, is the level- elliptic genus. Suppose now that . Define a local parameter near so that and . Then can be expanded into a formal power series which clearly satisfies and . In this case, the elliptic genus can be defined as the Hirzebruch genus (cf. [a4] or [a5]) corresponding to the series . Since , the logarithm of this elliptic genus is given by the elliptic integral

 (a2)

which gives the original definition in [a9].

## Modularity.

For any closed oriented manifold of dimension , is a function of the triple . As easily follows from (a1), multiplying by results in multiplying by . Also, depends only on the isomorphism class of the triple and commutes with arbitrary extensions of the scalar field . In the terminology of Katz ([a7]; adapted here to modular forms over fields), is a modular form of level and weight . Let be the graded ring of all such modular forms. Then , , . Moreover, one can prove that . If one identifies these two isomorphic rings, the elliptic genus becomes the Hirzebruch genus

with logarithm given by the formal integral (a2).

## Integrality.

Consider

i.e., the composition of with the forgetful homomorphism . As is shown in [a2],

The ring agrees with the ring of modular forms over . Thus: If is a -manifold of dimension , then .

## Example: the Tate curve.

Let be a local field, complete with respect to a discrete valuation , and let be any element satisfying . Consider . It is well-known (cf. [a11], § C.14) that can be identified with the elliptic curve (known as the Tate curve)

where

can be treated as an elliptic curve over with . Fix the invariant differential () on ( corresponds to the differential on the Tate curve). has three -rational primitive -division points: , and . To describe the corresponding Jacobi function , consider the theta-function

This is a "holomorphic" function on with simple zeros at points of (cf. [a10] for a justification of this terminology), satisfying

Consider the case where . Let be any square root of , and let

 (a3)

is a meromorphic function on satisfying and

i.e., is a multiple of the Jacobi function of .

Notice now that the normalization condition can be written as , where is the derivative with respect to . Since , one has . Differentiating (a3), one obtains

and

Finally, if , the function satisfies . It follows that the generating series is given by

The cases where or are treated similarly, with

and

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 be a principal -bundle (cf. also Principal -object) over an oriented manifold , where is a compact connected Lie group, and suppose acts on a compact -manifold . Then

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 [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 [a4] F. Hirzebruch, "Topological methods in algebraic geometry" , Grundlehren math. Wiss. , Springer (1966) (Edition: Third) [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) [a6] J.-I. Igusa, "On the transformation theory of elliptic functions" Amer. J. Math. , 81 (1959) pp. 436–452 [a7] N.M. Katz, "-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 [a8] 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 [a9] S. Ochanine, "Sur les genres multiplicatifs définis par des intégrales elliptiques" Topology , 26 (1987) pp. 143–151 [a10] P. Roquette, "Analytic theory of elliptic functions over local fields" , Hamburger Math. Einzelschrift. , 1 , Vandenhoeck and Ruprecht (1970) [a11] J.H. Silverman, "The arithmetic of elliptic curves" , GTM , 106 , Springer (1986) [a12] C. Taubes, " actions and elliptic genera" Comm. Math. Phys. , 122 (1989) pp. 455–526
How to Cite This Entry:
Elliptic genera. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_genera&oldid=12287
This article was adapted from an original article by S. Ochanine (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article