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Daniele Tampieri
Daniele Tampieri
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Any meromorphic modular function of weight 0$0$ for SL2Z$\mathrm{SL}(2,\Bbb Z)$ is a rational function of j$j$, say P(j)$P(j)$. Since your function is holomorphic, P$P$ is a polynomial. Since your function has a simple pole at infinity, P$P$ has degree one. But P$P$ fixes 0$0$ and 1$1$, so it is the identity.

Any meromorphic modular function of weight 0 for SL2Z is a rational function of j, say P(j). Since your function is holomorphic, P is a polynomial. Since your function has a simple pole at infinity, P has degree one. But P fixes 0 and 1, so it is the identity.

Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a simple pole at infinity, $P$ has degree one. But $P$ fixes $0$ and $1$, so it is the identity.

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David Loeffler
David Loeffler
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Any meromorphic modular function of weight 0 for SL2Z is a rational function of j, say P(j). Since your function is holomorphic, P is a polynomial. Since your function has a simple pole at infinity, P has degree one. But P fixes 0 and 1, so it is the identity.