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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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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