I have already asked this question on MathStackExchange ([here](https://math.stackexchange.com/questions/1810985/why-is-12-the-smallest-weight-for-which-a-cusp-forms-exists)) but did not receive an answer, therefore I am trying it here:


On wikipedia ([here](https://en.wikipedia.org/wiki/12_%28number%29 "here")) I have read the following:



> 
> Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at $−1$ i.e. $\zeta(-1) = -1/12$, the fact that the abelianization of $SL(2,\mathbb{Z})$ has twelve elements, and even the properties of lattice polygons.
> 
> 
> 


I know the dimension formula and the weight formula for modular forms as well as the proof of the weight formula via the residue theorem, but I do not understand how this relates to any of the three concepts mentioned in the quote above. Can anybody tell me more?



Why is 12 the smallest weight for which a cusp forms exists
These facts are related via the geometric interpretation of cusp forms. Essentially, identifying cusp forms as differential forms on modular curves allows to establish the relation to the ${\rm SL}\_2\mathbb{Z}$-action on the upper half-plane. The following is very sketchy, but I hope it properly outlines the main points.


The first step is to use the transformation behaviour to interpret cusp forms as certain holomorphic differential forms on a modular curve. Alternatively, cusp forms are viewed as global sections of a suitable line bundle on the compactification of ${\rm SL}\_2\mathbb{Z}\backslash\mathbb{H}$. 
Then one can apply Riemann-Roch to compute the dimension of the space of cusp forms in terms of the Euler characteristic of the modular curve. 
A formula for dimensions of spaces of cusp forms for congruence subgroups can be found as Theorem 4.9 in [Milne's lecture notes.](http://www.jmilne.org/math/CourseNotes/MF110.pdf) Formulas also applicable to ${\rm SL}\_2\mathbb{Z}$ are discussed in the following lecture notes [these lecture notes.](https://www.math.wisc.edu/~hast/Notes/Math847-modular-forms-notes.pdf)
For ${\rm SL}\_2\mathbb{Z}\backslash\mathbb{H}$, the formula for the Euler characteristic gives $-\frac{1}{12}$ because there is one point with isotropy $\mathbb{Z}/4\mathbb{Z}$ and one point with isotropy $\mathbb{Z}/6\mathbb{Z}$. (There is some subtlety here with the isotropy which leads to the Euler characteristic not being integral. One can also deal with this by considering congruence subgroups $\Gamma$ such that $\Gamma\backslash\mathbb{H}$ is a Riemann surface and apply some form of Riemann-Hurwitz to compare to ${\rm SL}\_2\mathbb{Z}\backslash\mathbb{H}$.)


Now the Euler characteristic can be compared to the order of the abelianization. The fundamental domain for the ${\rm SL}\_2\mathbb{Z}$-action retracts equivariantly onto one of its boundary segments, leading to an equivariant retraction of the upper half plane onto a tree. From the action of ${\rm SL}\_2\mathbb{Z}$ on the tree we get the well-known amalgam decomposition $\mathbb{Z}/4\mathbb{Z}\ast\_{\mathbb{Z}/2\mathbb{Z}}\mathbb{Z}/6\mathbb{Z}$, where the subgroups are the isotropy group from before. The natural map from this amalgamated product to $\mathbb{Z}/12\mathbb{Z}$ is in fact the abelianization -- hence we get a relation between the abelianization and the isotropy of the action on $\mathbb{H}$. 


Now that we have the Euler characteristic, we can use the Gauss-Bonnet theorem (in a version for finite-volume hyperbolic surfaces) to relate volume and Euler characteristic 
$$
-{\rm vol}(M)=2\pi\chi(M)
$$
This should rather be applied to ${\rm PSL}\_2\mathbb{Z}$ because of the ubiquitous complication that the center (of order 2) in ${\rm SL}\_2\mathbb{Z}$ fixes all of $\mathbb{H}$ -- the result is a factor 2. In the end, the volume for ${\rm SL}\_2\mathbb{Z}\backslash\mathbb{H}$ turns out to be $\pi/3$. 


Now we need to compare this with another computation of the volume related to the zeta values. There is an equality of volumes
$$
{\rm Vol}({\rm SL}\_2\mathbb{Z}\backslash\mathbb{H})\times {\rm Vol}({\rm SO}(2,\mathbb{R}))=2{\rm Vol}({\rm SL}(2,\mathbb{R})/{\rm SL}\_2\mathbb{Z}).
$$
The right-hand side is the Tamagawa volume which is $\zeta(2)$. The volume of ${\rm SO}(2,\mathbb{R})$ is $\pi$. The combination tells us that the volume of ${\rm SL}\_2\mathbb{Z}\backslash\mathbb{H}$ is $2\zeta(2)/\pi$. Together with the functional equation for $\zeta$, we see that $-\zeta(-1)$ is the Euler characteristic of ${\rm SL}\_2\mathbb{Z}\backslash\mathbb{H}$ which entered in the computation for the cusp forms. For the volume statements, see chapter 11 of MacLachlan-Reid: The arithmetic of hyperbolic 3-manifolds.
The appearance of $\zeta(2)$ can be seen directly in the volume computation.


