MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How does one write down $\mathbb{R}$-valued functions on the modular surface? I am considering taking an arbitrary function on the upper half plane $f:\mathbb{H} \to \mathbb{R}$ and averaging over the elements of $SL(2, \mathbb{Z})$. So, $$ f_1(z) = \sum_{g \in SL(2, \mathrm{Z})} f(gz) $$ there may have to be a decay condition on $f$ so the function will converge. I am not necessarily looking for holomorphic functions, just smooth and well defined on $\mathbb{H}\backslash SL(2, \mathbb{Z})$.

share|cite|improve this question
I must be missing something: what counts as "writing down a function"? There are going to be a plethora of non-zero smooth functions on this surface, so what other properties or identities are you looking for? – Yemon Choi Jul 2 '10 at 1:45
Take any smooth function of the j-invariant? – Qiaochu Yuan Jul 2 '10 at 1:59
In the absence of an application or additional context, it is hard to tell if you would prefer a universal answer as given by Yemon and Qiaochu, or something more structured. – S. Carnahan Jul 2 '10 at 2:08
It could be the word I am looking for is "modular function". Does the j-invvariant have a Fourier series? What are the critical values? – john mangual Jul 2 '10 at 19:32

You may have an easier time starting with a function that is periodic under translation by 1, then summing over cosets of translation in $SL_2(\mathbb{Z})$. If your initial function is well-behaved, your sum will converge (although often one introduces correction terms to get sections of a line bundle, i.e., modular forms of nonzero weight). This is a common method for constructing Poincaré series, Real-analytic Eisenstein series (where $f$ is given by a power of the imaginary part), and Rademacher sums (where $f$ is exponential).

share|cite|improve this answer
To supplement Scott, see Gunning's Lectures on Modular Forms, Chapter 3. This "sum over cosets" technique is discussed there.… – SandeepJ Jul 2 '10 at 12:41

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.