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?