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I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:

$F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar q)+...)$

${\it i.e.}$ only non-negative powers of $q$ and $\bar{q}$ arise, with possibly a logarithmic divergence (but no a pole divergence)

Does anyone know an example of such a function, or a theorem forbidding it?

Thank you very much!

Note added: Sorry for the non-standard notation. By $\bar{q}$ I mean $\bar{q}=\exp{(-2\pi I z^*)}$, the complex conjugated of $q$, and by real I mean invariant under complex conjugation. (they are non-holomorphic, the non-holomorphic Eisenstein series is an example of this, sorry for the language, it comes from Physics literature)

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For any holomorphic cuspform $f$ vanishing to order exactly $1$ at $i\infty$, $F(x+iy)=\log|f(x+iy)|$ is of the form you want.

The extreme simple case of this construction is the (in)famous $f(z)=q\prod (1-q^n)^{24}$, the log of whose absolute value appears in the Laurent expansion of the Eisenstein series at $s=1$ (Kronecker's limit formula).

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  • $\begingroup$ Thank you so much Paul! That answers my question indeed, sorry for not seeing that. Do you (or anyone!) know if a non-vanishing function exists with the properties above but which also vanishes at $q=\bar{q}$? Than you very much! $\endgroup$
    – fernando
    Aug 27, 2014 at 18:05
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    $\begingroup$ @fernando, this construction wouldn't be able to give you a function vanishing along $q=\overline{q}$, anyway, because then the corresponding $f$ would be of absolute value $1$ on the imaginary axis, so couldn't go to $0$ at $i\infty$. But on the compactified modular curve $\isom \mathbb P^1$, giving up the Riemannian structure inherited from the upper half-plane, the $q$ is a local coordinate at the image of $i\infty$, and the fundamental solution for the (spherical) Laplacian on that $\mathbb P^1$ will have a log-like singularity, yes. Is this the sort of thing you want? $\endgroup$ Aug 27, 2014 at 18:34

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