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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)

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...

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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).

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  • $\begingroup$ Ah I see what you're saying. That is, that the local monodromy around singular fibers of $E/S$ somehow doesn't depend on the actual singular fiber, but rather on the singular fiber of the proper regular minimal model of $E/S$? That's a little mysterious to me, since the fiber of the minimal model is obtained by just blowing-up then normalizing until you get something nice right? It's unclear to me why the local monodromy depends on the end result of such a process which is only really described algorithmically. $\endgroup$ – Will Chen Mar 29 '15 at 2:39
  • $\begingroup$ The local monodromy is not about the fiber; it's about its punctured neighborhood. And, a posteriori, this neighborhood knows what the fiber itself should be after everything is resolved. I find it very natural. E.g., any I type fiber can be blown down to a single nodal component (and so it is in the Weierstrass model); this does not mean that they all should have the same monodromy! $\endgroup$ – Alex Degtyarev Mar 29 '15 at 9:19

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