My question considers the curve $E$ over the affine $j$-line $S$ given by
$$Y^2 - (j-1728)XY = X^3 - 36(j-1728)^3X - (j-1728)^5$$
This curve has the property that it's $j$-invariant is $j$ (see [Questions about the "universal elliptic curve" over the affine $j$-line punctured at 0 and 1728](https://mathoverflow.net/questions/187110/questions-about-the-universal-elliptic-curve-over-the-affine-j-line-puncture))


Standard computations show that at $j = 0,1728$, $E$ becomes a cuspidal cubic (in fact we only need that $E$ is cuspidal at $j = 1728$, which is obvious)


Let $j\_0\ne 0,1728\in S$. Then the fiber $E\_{j\_0}$ is an elliptic curve, and for any $\gamma\in\pi\_1(S\setminus\{0,1728\},j\_0)$ we have an action of $\gamma$ on $H^1(E\_{j\_0},\mathbb{Z})\cong\mathbb{Z}^2$. This action is ``orientation preserving'', and so $\gamma$ acts with determinant 1. Various sources compute this local monodromy around singular fibers.


For example, see chapter VI of <http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf> or <http://en.wikipedia.org/wiki/Elliptic_surface> where it's shown that the local monodromy around a cuspidal fiber (Kodaira type II) is a conjugate of $\begin{bmatrix}1 & 1\\-1 & 0\end{bmatrix}$ (of order 6).


This map $\pi\_1(S\setminus\{0,1728\},j\_0)\rightarrow SL\_2(\mathbb{Z})$ is called the homological invariant of $E/S$. 


On the other hand...


Let $\mathcal{H}^\circ$ be the upper half plane with the $PSL\_2(\mathbb{Z})$-orbits of $i$ and $e^{2\pi i/3}$ removed, then the map $\mathcal{H}^\circ\rightarrow S\setminus\{0,1728\}$ is galois with galois group $PSL\_2(\mathbb{Z})$, so we have a natural map $J\_\* : \pi\_1(S\setminus\{0,1728\},j\_0)\rightarrow PSL\_2(\mathbb{Z})$, which is surjective and takes a loop around $0\in S$ to an element of order 3, and a loop around 1728 to an element of order 2. It's well-known that "the homological invariant belongs to $J\_\*$", ie the map $J\_\*$ factors as:
$$\pi\_1(S\setminus\{0,1728\},j\_0)\longrightarrow SL\_2(\mathbb{Z})\longrightarrow PSL\_2(\mathbb{Z})$$
where the first map is the homological invariant, and the second map is the usual quotient map (this is in Miranda as well as Kodaira "On Compact Complex Analytic Surfaces II"). In particular, the monodromy around the fiber above 1728 should be an element of order 4, which seems to contradict the fact that the fiber above $j=1728$ is a cuspidal cubic.


Can anyone help resolve this apparent contradiction?


It's also interesting that the special fiber of the *minimal* model for $E/S$ at $j=1728$ is actually of type III\*, which has the right local monodromy action ($\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$), but I don't see how that's relevant here... 



homological invariant of the "universal elliptic curve" over the punctured $j$-line
This is an illusion that the fiber is of type II, even though it looks like a cuspidal cubic. Your Weierstrass model is a **singular** surface with a type III fiber, where one of the two components has been blown down (so the other one, tangent to the first, does get a cusp).


