Question

It is well known that a modular form of weight k and level \Gamma is a global section of k-power of a Hodge line bundle over some modular curve. e.g. H^0(X,E^k).

My question is

How to characterize Eisenstein series among such sections using geometric datas?

For example, we know cusp forms are just sections of H^0(X,E^k(-cusps)).But how about Eisenstein series?

Actually in his Introduction to "Abelian Varieties" 1970, Mumford writes:

"It is interesting to ask whether further ties between the analytic and algebraic theories exist: e.g. an algebraic defintion of the Eisenstein series as a section of a line bundle on the moduli space. ..."

Could somebody explain the analytic-algebraic-representation aspects of Eisenstein series in some detail?

Thank you!

Answer 1

Here is one construction:

We have the exact sequence $$0 \to H^0(\omega^{\otimes k}(-\text{cusps})) \to H^0(\omega^{\otimes k}) \to H^0(\text{cusps}, \omega^{\otimes k}_{| \text{cusps}}).$$ (Here I am using $\omega$ for what you called $E$; this is the traditional notation for modular forms people.) It is easy to define a Hecke action on the third $H^0$ so that this exact sequence is Hecke equivariant.

The right hand map is surjective if $k > 2$, and its image has codimension one when $k = 2$. In any event, write $\mathcal I$ to denote the image, so that $$0 \to H^0(\omega^{\otimes k}(-\text{cusps})) \to H^0(\omega^{\otimes k}) \to \mathcal I \to 0$$ is short exact. One then shows that this short exact sequence has a unique Hecke equivariant splitting; i.e. there is a uniquely determined Hecke equivariant subspace $\mathcal E \subset H^0(\omega^{\otimes k})$ such that $\mathcal E$ projects isomorphically onto $\mathcal I$. This space $\mathcal E$ is the space of weight $k$ Eisenstein series (for whatever level we are working at).