Diameter of reduction graph of a curve over a complete discrete valuation ring

2010-06-14T12:28:16Z

Let $R$ be a complete discrete valuation ring with field of fractions $K$ and algebraically closed residue field $k$, and let $X$ be a proper, smooth, geometrically connected curve over $K$. Take a finite extension $L/K$ and a regular proper model $\widetilde{X}$ of $X$ over the ring of integers of $L$ whose special fibre $\widetilde X_k$ is semi-stable. Let $e$ be the ramification index of $L/K$. The reduction graph of $X$ is a metrised graph obtained as follows. Take a set of intervals of lengh $1/e$ indexed by the singular points of $\widetilde X_k$. For every singular point $x$ of $\widetilde X_k$, label the endpoints of the corresponding edge by the two irreducible components (possibly the same) on which $x$ lies. For every irreducible component $C$ of $\widetilde X_k$, identify all the endpoints labelled $C$. The result (as a metric space) is independent of the choice of $L$.

Question: Suppose we have a regular proper model of $X$ over $R$ whose special fibre $X_k$ is reduced, but not necessarily semi-stable. Suppose furthermore that we know the irreducible components and singular points of $X_k$ and their intersection multiplicities in this model. What can be said about the reduction graph of $X$?

It seems reasonable to ask this question in this generality, but I am actually interested in the modular curves X₁(n) over W_p[ζ_p²], where p is a prime number dividing n exactly twice and W_p is the ring of Witt vectors of an algebraic closure of F_p. In this situation non-semi-stable models as above were found by Katz and Mazur. What I would like specifically is an upper bound on the diameter of the reduction graph for these curves.

Answer by Qing Liu for Diameter of reduction graph of a curve over a complete discrete valuation ring

2010-06-15T21:06:45Z

This is a sequel to the above comments.

Consider an elliptic curve $E$ over $K$ with additive reduction over $K$ and multiplicative reduction over some extension $L/K$. Then we can find a quadratic extension $L/K$, and the Kodaira symbole of $E$ over $K$ is $I^*_m$ for some $m$ (see below), and the group of components of $E_L$ is $Z/nZ$, where $n=[L:K]\nu_K(\Delta)$ and $\Delta$ is the minimal discriminant of $E$ over $K$.

Now how to compute $n$ from data over $K$ ? I mean from data that you can read from the special fiber of the minimal regular model over $K$ ? The link between $n$ and $m$ is $n/2=(2+\delta) + (m+4)-1$ (Ogg's formula), and $\delta$ is the Swan conductor of $E$. If the residue characteristic is different from $2$, then $\delta=0$ and $n=2(m+5)$. Perfect.

But if the residue characteristic is 2 (so wild ramification happens), then $\delta$ can be arbitrarily big (if the absolute ramification index of $K$ is big). So $n$ is not a function of data from the special fiber over $K$.

Diameter of reduction graph of a curve over a complete discrete valuation ring - MathOverflow

Diameter of reduction graph of a curve over a complete discrete valuation ring

Answer by Qing Liu for Diameter of reduction graph of a curve over a complete discrete valuation ring