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This is an arithmetic follow-up to my previous question http://mathoverflow.net/questions/64112/does-there-exist-a-non-trivial-semi-stable-curve-of-genus-1-with-only-4-singular

Let $k$ be an algebraically closed field and let $f:X\longrightarrow \mathbf{P}^1_{k}$ be a semi-stable curve. Let $s$ denote the number of singular fibres. If $X$ is non-isotrivial and of positive genus, we have that $s>2$ (Beauville and Szpiro). As you can see Angelo stated in my previous question, for genus >1 and $k=\mathbf{C}$, Sheng-Li Tan has shown that $s>4$.

Now, let $S=\textrm{Spec} \mathbf{Z}$ and let $X\longrightarrow S$ be a (regular) semi-stable arithmetic surface. Let $s$ be the number of singular fibres. Fontaine has shown that $s>0$ if $X$ is of positive genus.

Question. Let $g>0$ be an integer. Does there exist a semi-stable arithmetic surface $X\longrightarrow S$ of genus $g$ with precisely one singular fibre?

I expect the answer to be yes for $g=1$ but no for $g>1$.

Example. The modular curve $X_1(\ell)$ ($\ell$ big enough) has semi-stable reduction over Spec $\mathbf{Z}[\zeta_{l}]$. This model has precisely one singular fibre. Note that the modular curve $X_1(l)$ does not have semi-stable reduction over $\mathbf{Z}$.

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# The number of singular fibres of a semi-stable arithmetic surface over \Z

This is an arithmetic follow-up to my previous question http://mathoverflow.net/questions/64112/does-there-exist-a-non-trivial-semi-stable-curve-of-genus-1-with-only-4-singular

Let $k$ be an algebraically closed field and let $f:X\longrightarrow \mathbf{P}^1_{k}$ be a semi-stable curve. Let $s$ denote the number of singular fibres. If $X$ is non-isotrivial and of positive genus, we have that $s>2$ (Beauville and Szpiro). As you can see in my previous question, for genus >1 and $k=\mathbf{C}$, Sheng-Li Tan has shown that $s>4$.

Now, let $S=\textrm{Spec} \mathbf{Z}$ and let $X\longrightarrow S$ be a (regular) semi-stable arithmetic surface. Let $s$ be the number of singular fibres. Fontaine has shown that $s>0$ if $X$ is of positive genus.

Question. Let $g>0$ be an integer. Does there exist a semi-stable arithmetic surface $X\longrightarrow S$ of genus $g$ with precisely one singular fibre?

I expect the answer to be yes for $g=1$ but no for $g>1$.

Example. The modular curve $X_1(\ell)$ ($\ell$ big enough) has semi-stable reduction over Spec $\mathbf{Z}[\zeta_{l}]$. This model has precisely one singular fibre. Note that the modular curve $X_1(l)$ does not have semi-stable reduction over $\mathbf{Z}$.