It is shown by M. Moeller (M. Moeller, Shimura- and Teichmüller curves) that there are only 2 Shimura and Teichmüller curves in the moduli space of curves $M_g$, namely, the ones given by $y^4=x(x-1)(x-t)$ and $y^6=x(x-1)(x-t)$ in $M_3$ and $M_4$ respectively. I have difficulties in understanding this result, because we know that it is true that families of cyclic covers of $\mathbb{P}^{1}$ branched over $1$ point in $\mathbb{P}^{1}\setminus \{0,1,\infty\}$ (square-tiled surfaces) generate Teichmüller curves in $M_g$ (for example by A. Wright, SCHWARZ TRIANGLE MAPPINGS AND TEICHMüULLER CURVES:ABELIAN SQUARE-TILED SURFACES). On the other hand, the Shimura curves arising from cyclic covers of the projective line are already classified (By Moonen and other authors) and we know that beside the curves given above there are other examples of Teichmüller curves which, when considered in $A_g$ generate Shimura curves. I must have made a mistake somewhere in understanding this result, but can't figure out where! I would appreciate any help.
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1$\begingroup$ Perhaps the point is that $M_g$ is not closed in $A_g$. $\endgroup$– nafCommented Dec 20, 2015 at 7:11
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$\begingroup$ I don't understand what you mean. If we have $C\to M_g\xrightarrow{i} A_g$, then since the Teichmüller curves are not compact and $i$ is an immersion, we get an open curve in both $M_g$ and $A_g$. The above Shimura curves are also non-compact, i.e., they have cusps. So I don't see the contradiction! $\endgroup$– CyrusCommented Dec 20, 2015 at 9:11
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$\begingroup$ There could still be points on the (non-compact) Shimura curve that are not in $M_g$. (This is just a suggestion, I am not claiming that this is indeed the explanation.) $\endgroup$– nafCommented Dec 20, 2015 at 11:29
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I think the point is that, when you have a family $f:X\to C$, then the induced map $C\to M_g$ is not necessarily one to one. It is not even generically one to one but in reasonably good cases it is generically finite. Now by a ST-curve completely inside $M_g$ we mean that this latter map is injective and the image $C\to A_g$ is a Shimura curve. These two conditions together are very restrictive. The families you mentioned are Teichmüller in the sense that the image of $C\to M_g$ is Teichmüller, not that the family completely sits in a Teichmüller curve.
