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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 $\mathbb{H}\backslash SL(2, \mathbb{Z})$.

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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
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1 Answer 1

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).

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To supplement Scott, see Gunning's Lectures on Modular Forms, Chapter 3. This "sum over cosets" technique is discussed there. books.google.com/… –  SandeepJ Jul 2 '10 at 12:41
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