Let $N$ be an integer $\geq 3$ and $X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N]$ is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of $N$ for which the special fibre $X_{\mathbb{F}_p}(N)$ of $X(N)$ at some prime $p \nmid N$ is isomorphic to the projective $\mathbb{P}^{1}_{\mathbb{F}_p}$?

Another question : Is there some technics to compute the genus of $X_{\mathbb{F}_p}(N)$?


(1) The Riemann-Hurwitz applied to the covering $X(N)\to X(1)$ (over the complex numbers) furnishes a number theoretical formula for the genus of $X(N)$. (The degree of this covering is the cardinality of $\mathop{\rm SL}(2,\mathbf Z/N\mathbf Z)/\{\pm \mathrm I_2\}$, the ramification can be analysed.) Unless $N$ is small (ie, $N\leq 5$), the genus will be strictly positive. Formulas are to be found in probably all good textbooks about modular forms (but Wikipedia only gives the computation for prime number $N$).

(2) As you write, the scheme $X(N)$ is projective and smooth over $\mathop{\rm Spec}(\mathbf Z[1/N])$. In particular, the genus of all fibers are the same.

  • $\begingroup$ I find in EGA III that in our case the genus is locally constant, thence the genus is constant in the case where $X$ is geometrically irreducible. But i don't understand why the smoothness implies that the genus is constant. Can you give me a reference? $\endgroup$ – Adel BETINA Apr 18 '16 at 9:41
  • $\begingroup$ One needs to agree on the definition of the genus. For a proper irreducible curve $X$, you have arithmetic genus $1-\chi(X,\mathscr O_X)$, and the geometric genus (genus of normalization); for a reducible curve, I don't know how it is defined. In a proper flat family, the Euler characteristic of the fibers do not change (EGA III), and smooth implies flat. $\endgroup$ – ACL Apr 18 '16 at 14:18

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