This post is the first of a series dedicated to algebraic modular forms and -adic modular forms. I will be mostly following the foundational paper of Katz in the Antwerp proceedings.
A characteristic class with values in a cohomology theory is a rule which assigns to each bundle a cohomology class such that (i) depends only on the isomorphism class of ; (ii) commutes with base change: if is any morphism, then .
In order to illustrate, let us take the prototypical example of -bundles. If is a topological group, a -bundle on a topological space is a map of topological spaces equipped with a continuous action of , such that, locally over , is isomorphic to a product with its natural action. It can be thought of as a continously varying family of -homogeneous spaces.
Notice that -bundles can be base changed: if is a continuous map, the fibre product is naturally a -bundle.
Let denote the functor which takes . It takes a continuous map to the map induced by base change; it is therefore a contravariant functor. Let be a cohomology theory. For our purposes, this need only be a contravariant functor which is homotopy invariant. Then, by definition, a characteristic class with values in is simply a natural transformation .
Example of a characteristic class. In this example we take denote the first Cech cohomology pointed set with coefficients in the (non-abelian) sheaf of continuous functions . Then any -bundle determines a class in in the following manner: pick a trivializing cover of , with -bundle isomorphisms . Over the intersection , the map is an automorphism of the -bundle , or, what is the same, a continuous map , i.e. a section of the sheaf over . The collection is a Cech -cocycle, and therefore it determines a cohomology class . Different choices of trivializations give cocycles which differ by coboundaries, so the cohomology class is in fact well-defined.
The characteristic class constructed above actually determines the bundle up to isomorphism, so that the pointed cohomology set actually classifies isomorphism classes of -bundles on . There is, however, another way to classify -bundles, using classifying spaces. Given a topological group , one can construct a topological space which represents the functor in the homotopy category of topological spaces. This means means that there is a canonical isomorphism
where denotes the set of homotopy classes of maps .
Thanks to the homotopy invariance of cohomology, we have the following proposition:
Proposition 1 Given a cohomology theory , there exists a canonical bijection between the set of characteristic classes with values in , and the cohomology set of the classifying space .
Proof: Indeed, this is nothing but the Yoneda lemma:
This allows us to think of characteristic classes in two equivalent ways: either as rules which transform bundles into cohomology classes on the base, or as cohomology classes on a classifying space. The same thing will happen with modular forms.
An elliptic curve over a scheme (also called a relative elliptic curve over ) is defined as a smooth and proper morphism , equipped with a section , whose geometric fibres are smooth curves of genus one.
It can be shown that there is a unique structure of -group scheme on , for which is the identity, and which induces the group structure on each fibre (for which is the identity).
Elliptic curves can be based change: If is an elliptic curve and is a morphism, then the base change is an elliptic curve over . We think of as a family of elliptic curves, parametrized by the geometric points of .
Here are some examples:
- Let , where is an algebraically closed field. Then an elliptic curve over is an elliptic curve over in the classical sense.
- An elliptic curve over is an elliptic curve over having good reduction at .
- There are no elliptic curves over . Indeed, such a curve would provide an elliptic curve over having good reduction everywhere. However, by a result of Tate, this is impossible. (In fact, there are no abelian varieties over , of any dimension.)
In general, if is a ring which admits a morphism , then there is no elliptic curve over .
- Let and . Then an elliptic curve over is an elliptic curve over whose minimal discriminant divides .
- Weierstrass’ theory of elliptic functions shows that the equation
defines an elliptic curve over , where and are the formal power series
The Tate curve is therefore an elliptic curve over the formal punctured disc. It comes with the canonical differential , which is independent of . We write for the Tate curve and its canonical differential.
The Tate curve cannot be extended to the whole disc because the ring admits a morphism to , given by .
Given an elliptic curve , the sheaf of algebraic -forms is free of rank on . Since is proper, is free on . For example, if then can be identified with , the vector space of global differentials on (a one-dimensional vector space over ).
Now we define the notion of an algebraic modular form.
Definition 2 An algebraic modular form of weight is a rule which, to any elliptic curve over any scheme , assigns a section such that: (i) depends only on the isomorphism class of ; (ii) commutes with base change: if is any morphism, then .
Next time, I’ll expand on the definition and relate it with classical modular forms.