# Relation between zeta value and genus of modular curve

This question is sort of vague, so I don't mind a vague answer.

We have the special value formula

$\zeta(-1)=-B_2/2 = -1/12$,

where $\zeta$ is the Riemann zeta function. Also, the "genus" of the level 1 modular curve $X(1)$ is $1/12$, where genus is meant in the sense of orbifolds. Is this just a numerical coincidence, or is there a deeper underlying phenomenon?

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There does appear to be some relationship coming from the Dedekind eta function, see math.ucr.edu/home/baez/numbers/24.pdf . – Qiaochu Yuan Jun 8 '12 at 0:17

One way to interpret this result, I think, is as a Tamagawa number computation. More precisely, for the simply connected semisimple algebraic group $SL_2$, the Tamagawa number is famously equal to $1$. If you try to compute what this means in classical terms, you will find a relationship between the volume (and hence, by Gauss--Bonnet, the genus) of $X(1)$, and a $\zeta$-value, which will be the relationship you are asking about.