number theory - 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?

Answer

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.