# Does viewing an Eisenstein series as a sum over cusps explain the antagonism between Eisenstein serieses and cusp forms?

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:

1. 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.

2. 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.)

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".]

• Perhaps you'll reformulate after some reconsiderations: if anything, it'd not be that Eisenstein series are sums over cusps, but are attached_to cusps (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, there are Eisenstein-like things (called pseudo-Eisenstein series, or incomplete theta series) that are in $L^2$ and are orthogonal to cuspforms. I can say more after you reformulate, if you are interested. – paul garrett May 21 '16 at 22:16
• I take it the reference to "lower-level forums" was to this: math.stackexchange.com/questions/1792635/… If you also posted it on some additional "lower-level forums", you should include a link, please. – Gerry Myerson May 21 '16 at 22:20
• I'm not sure if this is what the poster means, but one way to view Eisenstein series as sums over cusps is as follows: one can think of the cusps of the locally symmetric space associated to $\mathrm{GL}_2(F)$ $\textit{at infinite level}$ as being parametrized by $\mathbb{P}^1(F)$. That is, the compactified locally symmetric space associated to a congruence subgroup $\Gamma$ of $\mathrm{GL}_2(F)$ is $\Gamma \backslash \mathcal{H} \times \mathbb{P}^1(F)$. And of course, the Eisenstein series $\textit{is}$ a sum over $\mathbb{P}^1(F) = \mathrm{GL}_2(F) / B(F)$, for $B$ a Borel – user25514 May 21 '16 at 22:43
• @user25514 Probably that's what I meant but I'll need to read what you wrote more carefully (trying to make sure I understand the notation and terminology and ideas) before I can say more definitely. – JWilds May 21 '16 at 23:03
• @paul garrett I'm definitely interested but I probably need to think things over a lot more before I can usefully reformulate anything. – JWilds May 21 '16 at 23:06

I'd say the approximate answer to the question in the title is "not exactly, but partly". But I'd also claim that clarity here is made considerably more difficult by the context in which the question is posed, and in which, apparently, an answer is desired to be.

That is, although for elliptic modular forms we can draw attractive pictures of fundamental domains, and although there are compactifications (due to Satake, Bailey-Borel, Borel-Serre, et al) in various generality, and although the analogy with the elliptic modular case is attractive, for understanding decompositions of various spaces of automorphic forms it seems (by this year...) that this viewpoint is not generally the most efficient. Namely, a more group-theoretic description of automorphic forms seems (for many purposes) to be more direct, and to show the how/why of "orthogonality" [sic] of cuspforms and Eisenstein series (noting that this needs qualification).

Even at an elementary level, the holomorphic Poincare series, also written as "sums over cusps" (literally $\Gamma_\infty\backslash\Gamma$) construct elliptic modular forms that are easily shown to be cuspforms. So it's not just the indexing set $\{\hbox{cusps}\}\approx \Gamma_\infty\backslash\Gamma$ (given by $\gamma\cdot\infty\leftarrow \gamma$).

I would claim that the happy entry-level notion of (holomorphic) cuspform, while indeed conveniently straightforward for elliptic modular forms, gives several misleading impressions about how things work more generally, both outside the holomorphic class, and on larger groups, whether situations like $Sp_n(\mathbb Z)\subset Sp_n(\mathbb R)$ where there are holomorphic modular forms, or $SL_n(\mathbb Z)\backslash SL_n(\mathbb R)$ where (for $n>2$) there are "waveforms", nothing holomorphic, (although "cohomological" things exist...).

Specifically, somewhat in contrast to the holomorphic case, more generally a/the "constant term" is the zeroth Fourier component, and is a function, not just a number. The condition of holomorphy often (though not always) constrains that function completely. Without such a constraint, that function can be almost anything (invariant under the unipotent radical along which the Fourier expansion is taken...). In particular, the condition for a "cuspform" is somewhat more complicated, since it's no longer the vanishing of a number, but identical-vanishing of a function. (The Gelfand condition.) This vanishing is perhaps most sanely characterized distributionally as "vanishing when integrated against any (unipotent-radical-invariant) test function". When the latter are "automorphized" (summed over the proper analogue of $\Gamma_\infty\backslash\Gamma$) they produce pseudo-Eisenstein series, which are in $L^2$, and in fact are test functions on the quotients $\Gamma\backslash G$. That is, the condition of being a cuspform is literally orthogonality to pseudo-Eisenstein series (with test-function data).

(Then, generally, pseudo-Eisenstein series themselves are spectrally decomposed in terms of genuine Eisenstein series.)

Unsurprisingly, since test functions are never holomorphic, such a discussion falls outside of the world of holomorphic things.

In any case, I'd say that the "cuspform" condition is best generally understood in terms of the Gelfand condition, rather than "vanishing at a cusp", which becomes progressively incorrect in general.

• [trying to figure out how to cancel a mistake here] – JWilds May 24 '16 at 10:42
• "Clarity here is made considerably more difficult by the context in which the question is posed, and in which, apparently, an answer is desired to be." - Actually I'm happy to hear suggestions about steering the question in a somewhat different direction, as you're giving here; that's part of the kind of feedback I'm looking for. I still really need to think over everything you said, though (particularly this "holomorphic Poincare series" idea). – JWilds May 24 '16 at 10:51