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?