I'm trying to understand the relationship between various aspects of the concept of "Eisenstein series" (as discussed for example in Diamond & Shurman's "A First Course in Modular Forms"), in particular:

A lowest-level Eisenstein series can be seen as a sum over cusps; that is, there's a bijection given by matching each term in the summation to the cusp at which it blows up; and moreover, this way of indexing the terms in the summation has the conceptual advantage of automatically giving a good normalization of the series without artificial introduction of a normalizing factor.

Eisenstein serieses span the Petersson-orthogonal complement of the ideal of cusp forms (the "antagonism" that I referred to).

Thus I want to know: Is there a direct connection from the "sum over cusps" aspect to the "orthogonality to cusp forms" aspect here, that I can re-use in other situations where the role played by the "cusps" is taken over by some sort of "generalized cusps"?

(This seems to me more of an entry-level question than a research-level one but I didn't have luck with it on lower-level forums. I'm open to (hearing) suggestions about possible better places for discussion of such a question, if there are any.)

In repsonse to comments:

To try to clear up any miscommunication about what I mean by "cusp" here: a cusp is a point where two arcs meet at a zero-degree angle, so joining into a single "direction-reversing arc". The arcs in this case are the mirror arcs of the (2,3,infinity) Coxeter group, pictured via the Poincare upper half-plane or the Poincare disk; thus the cusps are essentially the extended rational numbers on the boundary. A modular form can be thought of as a function on the upper half-plane with a certain automorphicity property, and as a special case of this a lowest-level Eisenstein series is a sum over the cusps, with the term corresponding to a given cusp blowing up at that cusp; thus a term with cx+d in the denominator blows up at the extended rational number x = -d/c.

I'm using this (pretty reasonable and etymologically prior) meaning of "cusp" because it facilitates the description of the sort of conjectural pattern that I'm asking about:

There's a Lie group G with a discrete subgroup L, and a G-space X with a G-orbit A in X and an L-orbit C in the closure of A. L-automorphic forms (wrt some automorphy factor) on A are considered, and the ones vanishing at the C-points are called the "C-forms". And forms that are "complementary" to the C-forms are created by summing over the C-points with the term corresponding to a C-point blowing up at that C-point.

Maybe this isn't quite the right pattern, or maybe the idea of looking for such a pattern at all is wrongheaded, but in any case my question is, roughly, whether there's some interesting pattern like this, generalizing the example of lowest-level Eisenstein serieses, which are modular forms "complementary" to the cusp forms and created by summing over the cusps.

There's still the issue of exactly what "complementary" should mean here, as from Paul Garrett's comment it seems that my first stab at it was too naive; I may have garbled something that I read somewhere.

Anyway, I have lots more follow-up questions, especially relating to Paul Garrett's comment, but I'll try to post those in a separate comment or comments, as it's already taking me long enough to write this.

[I should perhaps mention that although I'm not very clear on this forum's posting guidelines, I'm clear enough on them to know that I don't like them; in particular I'm not at all sympathetic to the official "no discussion, best answer floats to the top" philosophy of the forum. I'd be interested to know of possible good places for actual discussion of this sort of topic, where "good place for discussion" doesn't just mean "discussion is officially permitted".]

overcusps, but areattached_tocusps (and this is a bit misleading, too). Also, since Eisenstein series are not $L^2$ in any possible sense, there is something wrong with saying "orthogonal", even though it conveys intent. But that's "a sign". But, still, yes, thereareEisenstein-like things (called pseudo-Eisenstein series, or incomplete theta series) thatarein $L^2$ andareorthogonal to cuspforms. I can say more after you reformulate, if you are interested. $\endgroup$ – paul garrett May 21 '16 at 22:16