Genus 2 reduction types

Throughout, $R$ is a complete discrete valuation ring with fraction field $K$, uniformiser $\pi$, and algebraically closed residue field $k$ of characteristic $p\geq 0$. The curve $C/K$ is smooth, projective, geometrically connected, and of genus $2$ [ACDL, §1.6].

MRM reduction type

The reduction type of the minimal regular model (MRM) of $C$. It records the irreducible components of the special fibre, their multiplicities and geometric genera, their intersection pairing, and its singular points [ACDL, Def. 2.1]. The labels are those of [Namikawa–Ueno].

MRNC reduction type

The reduction type of the minimal regular normal crossings model (MRNC). The labels are those in [Dok25, Table G2]. For genus $2$, the MRM and MRNC reduction types determine one another [ACDL, Thm. 2.3], [ACDL, Cor. 2.7].

Potential stable type

The type of the special fibre of the stable model after passing to a finite extension over which $C$ is semistable. The notation $\I$–$\VII$ is due to [Liu]. The values in this column only apply when $p\neq 2$ and $C$ has tame reduction [ACDL, Thm. 3.2(2)], [ACDL, Table 7.8].

MRM special fibre

A schematic picture of the special fibre of the minimal regular model, using the conventions of Namikawa–Ueno [Namikawa–Ueno]. Following the notation of Ogg, for a component $\Gamma$ of the special fibre, the symbols have the following meanings (where $K$ is a canonical divisor):

Symbol	Arithmetic genus	$\Gamma^2$	$\Gamma\mathbin{\cdot}K$
$A$	$1$	$-1$	$1$
$B$	$0$	$-3$	$1$
$C$	$1$	$-2$	$2$
$D$	$0$	$-4$	$2$
none	$0$	$-2$	$0$

The number at each component denotes its multiplicity. In the case of multiplicity $1$, the number is omitted. The labels “elliptic” and “regular” indicate components of geometric genus $1$ and $2$, respectively, while “cusp” indicates a cuspidal singularity of the special fibre.

MRNC special fibre

Special fibre of the minimal regular normal crossings model, which are due to Tim Dokchitser [Dok25, Table G2]. The drawn curves represent the irreducible components of the special fibre, and the picture shows how these components intersect. A component marked by $m$ has multiplicity $m$. If it has positive geometric genus, this is also indicated: for example, a component marked $m\,\mathrm{g1}$ has multiplicity $m$ and geometric genus $1$. Components of geometric genus $0$ have no genus marking. Some rational components occur in chains. To keep the pictures compact, part of such a chain is replaced by dots. The expression beside the dots indicates how the length of the chain, which varies with the parameters of the reduction family. A chain of length zero (or a dashed blue line) between two principal components corresponds to the principal components intersecting transversely. After the parameters are specialised, the corresponding complete special fibre can be viewed on Tim Dokchitser's webpage [redlib].

# Components (MRM) / (MRNC)

The number of irreducible components in the special fibre of the minimal regular model and the minimal regular normal crossings model respectively.

Néron component group

The component group $\Phi_C$ of the Néron model of $\Jac(C)$. The following abbreviations are used:

$$ X_r = \begin{cases} \mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z, & r \text{ even},\\ \mathbb Z/4\mathbb Z, & r \text{ odd}, \end{cases} \qquad Y_r = \begin{cases} \mathbb Z/3\mathbb Z \times \mathbb Z/3\mathbb Z, & 3\mid r,\\ \mathbb Z/9\mathbb Z, & 3\nmid r, \end{cases} \qquad Z_{r,s} = \begin{cases} \mathbb Z/4s\mathbb Z, & rs \text{ even},\\ \mathbb Z/2\mathbb Z \times \mathbb Z/2s\mathbb Z, & rs \text{ odd}. \end{cases} $$

Here $(0)$ denotes the trivial group. The component groups are computed in [ACDL, Section 4] and also by Liu in [Liu94].

Tame conductor exponent

The tame part of the conductor exponent of $\Jac(C)$. If $C$ has tame reduction, this is its full conductor exponent [ACDL, Lem. 5.1 and Cor. 5.2].

Tame and wild reduction

The curve has tame reduction if it becomes semistable over a tamely ramified finite extension of $K$; otherwise its reduction is wild. The “Wild reduction iff $p=$” column lists the residue characteristics in which the given reduction-type family is wild. Whether the reduction is tame is determined by the MRNC special fibre [ACDL, Thm. 1.1(4)], [ACDL, Rem. 2.2].

$v(\Delta_{\min})$

The displayed entry is the valuation of the discriminant of a minimal Weierstrass model. A Weierstrass model $y^2=f(x)$ is minimal if $f(x)\in R[x]$ and $v(\operatorname{disc}(f))$ is minimal among all integral Weierstrass models of $C$. The values in this column only apply when $p\neq 2$ and $C$ has tame reduction [ACDL, Def. 5.6], [ACDL, Thm. 5.19].

$v(\omega_{\min}/\omega^0)$

The displayed entry is the valuation $v(\lambda)$, where $\omega_{\min}=\lambda\omega^0$. Here $\omega_{\min}=\frac{dx}{y}\wedge\frac{x\,dx}{y}$ is the exterior form associated to a minimal Weierstrass model, and $\omega^0$ is a Néron exterior form. The values in this column only apply when $p\neq 2$ [ACDL, Defs. 5.3–5.6], [ACDL, Cor. 5.20].

Curve example

An explicit genus $2$ curve realising the given reduction type, adapted from the examples of [Namikawa–Ueno].

Monodromy matrix

The monodromy matrix $M$ associated to the reduction type, as classified by [Namikawa–Ueno]. For a smooth proper family over a punctured complex disc, the monodromy action agrees, up to conjugation, with the action of a generator of tame inertia on $T_\ell(\operatorname{Jac}(C))$; see [ACDL, Cor. 4.2].

Cluster pictures

This column lists all possible degree-$6$ cluster pictures for the reduction type, up to an integral shift of all depths. The entries only apply when $p\neq 2$ and $C$ has tame reduction. For the definition of cluster pictures and the conventions used here, see [ACDL, §1.4].

For a Weierstrass equation $C:y^2=f(x)$, the roots of $f$ are drawn as red dots and clusters of size greater than $1$ as ovals around the corresponding roots. Each proper cluster other than the cluster of all roots is labelled by its relative depth; the cluster of all roots is labelled by its absolute depth. The cluster-picture data also include $v_c=v(c_f)$, where $c_f$ is the leading coefficient of $f$. Only the parity of $v_c$ is used in these pictures: the value of $v_c\bmod 2$ is displayed in the upper-right corner of each cluster picture.

The depth of the cluster containing all the roots is displayed modulo $1$. The integer parameters $a,b,j,s$ must make every relative depth positive, and $a$ and $b$ must satisfy $t=a+b$. Some cluster pictures impose additional parity restrictions on their parameters; these restrictions are recorded in brackets on the right-hand side of the picture. Quintic cluster pictures are not displayed separately: they are obtained from a displayed picture having a cluster of size $5$ by removing the root outside that cluster, while retaining the depths and $v_c$. See [ACDL, §7].