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This is a question in Serre's book "A Course in Arithmetic". He said "We can thus identify modular functions of weight $k$ with some lattice functions of weight $k$."

I am explain my question as follows. He definites "modular functions" to require that it must meromorphic at upper half plane $H$ and $\infty$, of course, satisfing $f\left(\frac{az+b}{cz+d}\right)=(cz+d)^{k}f(z).$ However, in his definition of lattice function, he hasn't mentioned "meromorphic" condition. enter image description here

Thus, $f$ may not a meromorphic function. What I mean is that we cannot regard a lattice functions of weight $k$ as a modular functions of weight $k$. Am I wrong...?

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    I don't see the problem. He says "**some** lattice functions", which means there is a particular *subset* of lattice functions of weight $k$ that can be identified with the modular functions of equal weight. – 2017-01-14
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    Yes, given a mordular funcition we can always get a lattice function. But we can't always regard a lattice functions as a modular functions. How can we get a "if and only if" condition? Or whether we can make clear what kind of subset of lattice function can be identified......@ Erick Wong – 2017-01-14
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    It is simply the lattice functions for which $\omega_2^{2k} F(\omega_1,\omega_2)$ is meromorphic as a function of $\omega_1/\omega_2$. The association is meant to be more tautological than deep. The point is just that we can then try to construct modular functions by starting from functions on lattices (which proves to be a very fruitful building block indeed). – 2017-01-14
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    Thanks for your answer! I got that $\omega_2^{2k}F(\omega_1,\omega_2)$ is meromorphic, but this can't be a property of $F$. Can we define a "meromorphic lattice function" to this correspondence. As you say this may be more tautological than deep, it may not necessary. Further, I am more wondering how we construct modular functions from lattice functions. This may be another question. Thank you again~~ :) – 2017-01-14
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    Probably one can define meromorphy for lattice functions, but I don't know a formal definition in the general case. But constructing lattice functions is easy: Eisenstein series are defined on the next page as $G_k(\Gamma) = \sum_{z \in \Gamma \setminus\{0\}} z^{-2k}$. This is by construction independent of the choice of lattice basis, so the $\mathbf{SL}_2$ invariance essentially comes for free. I think this justifies forming the connection between modularity and lattice functions. – 2017-01-14

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