# Eisenstein series as sections of line bundles on moduli spaces

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!

• I wish an algebraic definition of Eisenstein series, as special section of line bundles , under some condition on cusps , hecke action, etc. Not need to mention cusp forms. – user4245 Aug 5 '10 at 14:58
• Well "in the space spanned by non-cuspidal eigenforms" would work, right? – Kevin Buzzard Aug 5 '10 at 16:06
• @Kevin This is exactly what I think. But I couldn't convince myself. Could you explain it with more detail? e.g. compare with the classical or representatio-theoric defintions? Thank you! – user4245 Aug 5 '10 at 16:11
• @unknown: use Hecke operators T_p only for p prime to the level. Then classical theory says that these are all simultaneously diagonalisable. The cusp forms are a Hecke-stable subspace, spanned by eigenforms, and the Eisenstein series are a Hecke-stable subspace, spanned by eigenforms. Any eigenform is either a cusp form, so it lives in the cuspidal subspace, or an Eisenstein eigenform, so it lives in the Eisenstein subspace. My comment is really a rather trivial consequence, and Emerton's comment has the same idea at its heart. I don't really see what there is to prove :-/ – Kevin Buzzard Aug 5 '10 at 19:10

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).
• Sction 3 of: G. Faltings, B. Jordan, Crystalline cohomology and ${\rm GL}(2,Q)$. Israel J. Math. 90 (1995), no. 1-3, 1--66. – Felipe Voloch Aug 5 '10 at 18:52
• I tried looking in Faltings-Jordan's paper. They indeed define Eisenstein series as eigenforms for which the Hecke action is determined by that on the coefficients. But still, this seems (to me) to be done in a very "analytic" way, by embedding suitable rings/fields in $\mathbb{C}$, ecc... May be I am wrong, and glad to be corrected, but I am still looking for a neat (detailed) construction in the spirit of Emerton's answer. – Filippo Alberto Edoardo Oct 22 '12 at 3:19