# 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. An Eisenstein series can be seen as a sum over cusps; moreover this seems to be the "correct" way to look at it, in that it automatically yields a nicer normalization than looking at it as a sum over nonzero lattice points.

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

Thus I ask the obvious question here: Is this more than just a coincidence? That is, is there a straightforward conceptual connection between the "sum over cusps" aspect and the "orthogonality to cusp forms" aspect of "Eisenstein series"?

The kind of answer that I'm hoping for is one that I can re-use in other situations, where the role played by the "cusps" is taken over by some sort of "generalized cusps".

Perhaps there's some obvious answer given somewhere but if so then I somehow missed it.

(I'm the original question-asker and I want to counter-reply to the reply asking me "how do you define the "sum over cusps"?", but since I'm a new user and the documentation on how to post such a counter-reply is so poor, I'll just go ahead and post my counter-reply here.)

It's a sum over cusps because there's a bijection between terms in the summation and cusps at which they blow up.

Thus look at the definition of "weight k Eisenstein series" on page 4 of Diamond & Shurman; it's a sum over certain pairs (c,d) of integers, where the (c,d)th term blows up just at the cusp corresponding to the rational number -d/c; but then at the top of page 6 they admit via Exercise 1.1.7(b) that it's more sensible to include only those terms where -d/c is in lowest terms, resulting in a better normalization of the series. This gives a normalization factor of zeta(k) because of the way that the unnormalized Eisenstein series is a sort of "2-dimensional analog of the Riemann zeta function", as Diamond & Shurman put it; and a further normalization factor of 2 accounts for how d/-c and -d/c are lowest-terms representations for the same cusp.

So I'm getting more convinced now that there's some good conceptual-geometric story here about how the blowing up of the terms at one cusp apiece connects to the orthogonality of their cumulative sum to the cusp forms, but I don't really understand it even in this case yet, let alone in more generality than that.

I'd thought that this was a pretty low-level question, given that you're studying automorphic forms at all; Is it too advanced for this forum? It seems too low-level for a "research-level" forum.

• there's a link to the book 148.206.53.84/tesiuami/S_pdfs/… . and you should read the chapter 2.4 "cusps page 57 May 21, 2016 at 7:50
• @user1952009 I don't see anything in section 2.4 especially responsive to my question. Section 2.4 seems to be mostly about cusp points on (compact) modular curves, whereas my question is phrased more in terms of the extended rational numbers interpreted as the cusp points on the boundary of the upper-half-plane. Are you asking me to explain what I mean by "interpreting the extended rational numbers as the cusp points on the boundary of the upper-half-plane"? I thought it was pretty straightforward. May 21, 2016 at 8:31
• @user1952009 Again, my question is about the orthogonality relationship between Eisenstein serieses and cusp forms, and how this relates to looking at an Eisenstein series as a sum over cusps. May 21, 2016 at 8:34
• @user1952009 I'm sorry but your second comment also didn't seem helpful or responsive to my question. I don't think that you understand what kind of answer I'm looking for. May 21, 2016 at 8:38
• @user1952009 I'm not sure why it should be difficult to understand what it means for a term to "blow up at a cusp" in this particular context. If you look at the terms in that summation on page 4, they have factors of cx+d in the denominator (by "x" I mean whatever input variable they're using) so each individual term blows up at that cusp corresponding to the rational number x = -d/c due to division by zero. Anyway, perhaps I'll try MathOverflow instead. May 21, 2016 at 16:26