Models and integral differentials of hyperelliptic curves

1.1. Overview

To describe the arithmetic of curves over global fields, for example in the study of the Birch & Swinnerton-Dyer conjecture, it is essential to understand regular models and integral differentials over all primes, including those with very bad reduction. Constructing regular models of curves over discrete valuation rings is not an easy problem, even in the hyperelliptic curve case. In fact, there is no practical algorithm able to determine a model, unless the genus of the curve is $1$ or we have some tameness or nondegeneracy hypothesis.

One possible approach to tackle this problem is giving a full classification of possible regular models in a fixed genus, as done by the Kodaira–Néron [Reference Kodaira7, Reference Néron19] and Namikawa–Ueno [Reference Liu10, Reference Namikawa and Ueno18] classifications for curves of genera $1$ and $2$ , respectively. However, this strategy seems impractical in general, since the number of models grows fast with the genus. Recently, new approaches based on clusters [Reference Dokchitser, Dokchitser, Maistret and Morgan14], Newton polytopes [Reference Dokchitser1], and MacLane valuations [Reference Obus and Wewers21], have been developed (see Section 1.5 for more detail).

On one side, clusters define nice and clear invariants from which one can extract information on the local arithmetic of hyperelliptic curves. Such invariants turn out to be particularly useful from a Galois theoretical point of view. However, for describing regular models, restrictions on the reduction type of the curve and on the residue characteristic of its base field [Reference Faraggi and Nowell5, Reference Dokchitser, Dokchitser, Maistret and Morgan14] need to be imposed. On the other side, Newton polytopes and MacLane valuations have a potential to solve the problem in general, but the respective constructions are more algorithmic and so do not give the result in closed form. Furthermore, they often depend on the chosen equation rather than on the curve itself.

In this paper, we present a new approach that preserves both positive aspects from the above and provides a link between the two sides. We describe a model from simple invariants defined from what we call rational cluster picture (Definition 1.10). This object modifies the theory in [Reference Dokchitser, Dokchitser, Maistret and Morgan14] and appears to be more suitable for our purpose (see Section 1.3). In fact, the rational cluster picture also carries intrinsic connections with the other presented approaches, as it is closely related to Newton polygons and to degree $1$ MacLane valuations (see [Reference Fernández, Guárdia, Montes and Nart3]). When these valuations are enough to describe a regular model we say that the curve has an almost rational cluster picture (Definition 1.1; see also Corollary 3.29, Proposition 3.31). It turns out that the approach even works in residue characteristic 2, under an extra assumption that the curve is $y$ -regular (Definition 1.4). Our main result is:

Let $K$ be a complete Footnote 1 discretely valued field with $\textrm{char}(K)\neq 2$ , and let $K^{nr}$ be its maximal unramified extension. Let $C/K$ be a hyperelliptic curve, having an almost rational cluster picture over $K^{nr}$ . If the residue characteristic of $K$ is $2$ , assume that $C_{K^{nr}}$ is $y$ -regular. Then via the rational cluster picture we determine:

1. (i) the minimal regular model with normal crossings $\mathcal{C}^{\textrm{min}}$ ,

2. (ii) a basis of integral differentials of $C$ .

This result applies to a wide class of curves, covering wild cases and base fields with even residue characteristic. For example, if $g=2$ , then $107$ out of $120$ Namikawa-Ueno types [Reference Namikawa and Ueno18] arise from hyperelliptic curves satisfying the conditions of our theorem. In addition, the author believes it has a potential to solve the problem in general. Heuristically speaking, the rational clusters invariants are expected to extend to general MacLane valuations. This approach could eventually lead to a full characterisation of minimal models with normal crossings of hyperelliptic curves (over any discretely valued field).

1.2. Main results

We will now present (a simplified version of) the main results of this paper. We will then illustrate them with an explicit example in Section 1.4.

Let $K$ be a complete discretely valued field of residue characteristic $p$ , with normalised discrete valuation $v$ and ring of integers $O_K$ . We require $\textrm{char}(K)$ to be not $2$ , but we allow $p=2$ and $p=0$ . In this subsection we will assume for simplicity that $K=K^{nr}$ . Extend the valuation $v$ to an algebraic closure $\bar{K}$ of $K$ . Let $C/K$ be a hyperelliptic curve, that is a geometrically connected smooth projective curve, double cover of ${\mathbb{P}}^1_K$ . Let $g$ be the genus of $C$ . Assume $g\geq 1$ . Fix a Weierstrass equation

\begin{align*} C\;:\;y^2=f(x). \end{align*}

Let $\mathfrak{R}$ be the set of roots of $f$ in $\bar{K}$ . Thus

\begin{align*} f(x)=c_f\prod _{r\in \mathfrak{R}}(x-r). \end{align*}

For any $r,r'\in \mathfrak{R}$ , with $r\neq r'$ , denote by $\mathcal{D}_{r,r^{\prime}}$ the smallest $v$ -adic disc containing $r$ and $r'$ .

In the following definition we introduce most of the notation and quantities, associated with rational clusters, needed in order to state our main theorems.

Definition 1.3. For any $\mathfrak{s}\in \Sigma _K$ we say:

$\mathfrak{s}$ proper,	$|\mathfrak{s}|\gt 1$
$\mathfrak{s}^{\prime}$ is a child of $\mathfrak{s}$ ,	$\mathfrak{s}^{\prime}\in \Sigma _K$ and $\mathfrak{s}^{\prime}\subsetneq \mathfrak{s}$ is a maximal subcluster
$\mathfrak{s}$ minimal,	$\mathfrak{s}$ has no proper children
$\mathfrak{s}$ übereven,	$\mathfrak{s}=\bigcup _{\mathfrak{s}^{\prime}\text{ child of }\mathfrak{s}}\mathfrak{s}^{\prime}$ and $|\mathfrak{s}^{\prime}|$ even for all children $\mathfrak{s}^{\prime}$ of $\mathfrak{s}$

Moreover, we write $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ , or $\mathfrak{s}=P(\mathfrak{s}^{\prime})$ , for a child $\mathfrak{s}^{\prime}\in \Sigma _K$ of $\mathfrak{s}$ , and $r\wedge \mathfrak{s}$ for the smallest rational cluster containing the root $r\in \mathfrak{R}$ and $\mathfrak{s}$ .

Let $\Sigma _K$ be the set of proper rational clusters. For any $\mathfrak{s}\in \Sigma _K$ , define its radius

\begin{align*} \rho _{\mathfrak{s}}=\max _{w\in K}\min _{r\in \mathfrak{s}} v(r-w) \end{align*}

and the following quantities:

$b_{\mathfrak{s}}$	denominator of $\rho_{\mathfrak{s}}$
$\epsilon _{\mathfrak{s}}$	$v(c_f) + \sum _{r\in \mathfrak{R}} \rho _{r\wedge \mathfrak{s}}$
$D_{\mathfrak{s}}$	$1$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ odd, $2$ if $b_{\mathfrak{s}}\epsilon _{\mathfrak{s}}$ even
$m_{\mathfrak{s}}$	$(3-D_{\mathfrak{s}})b_{\mathfrak{s}}$
$p_{\mathfrak{s}}$	$1$ if $|\mathfrak{s}|$ is odd, $2$ if $|\mathfrak{s}|$ is even
$s_{\mathfrak{s}}$	$\frac 12(|\mathfrak{s}|\rho _{\mathfrak{s}}+p_{\mathfrak{s}}\rho _{\mathfrak{s}}-\epsilon _{\mathfrak{s}})$
$\gamma _{\mathfrak{s}}$	$2$ if $|\mathfrak{s}|$ is even and $\epsilon _{\mathfrak{s}}\!-\!|\mathfrak{s}|\rho _{\mathfrak{s}}$ is odd, $1$ otherwise
$p_{\mathfrak{s}}^0$	$1$ if $\mathfrak{s}$ is minimal and $\mathfrak{s}\cap K\neq \varnothing$ , $2$ otherwise
$s_{\mathfrak{s}}^0$	$-\epsilon _{\mathfrak{s}}/2+\rho _{\mathfrak{s}}$
$\gamma _{\mathfrak{s}}^0$	2 if $p_{\mathfrak{s}}^0=2$ and $\epsilon _{\mathfrak{s}}$ is odd, 1 otherwise

Definition 1.5. Let $\mathfrak{s}\in \Sigma _K$ and let $c\in \{0,\dots,b_{\mathfrak{s}}-1\}$ such that $c\rho _{\mathfrak{s}}-\frac{1}{b_{\mathfrak{s}}}\in \mathbb{Z}$ . Define

\begin{align*} \tilde{\mathfrak{s}}=\{\mathfrak{s}^{\prime}\in \Sigma _K\cup \{\varnothing \}\mid \mathfrak{s}^{\prime}\lt \mathfrak{s}\text{ and }\tfrac{|\mathfrak{s}^{\prime}|}{b_{\mathfrak{s}}}-c\epsilon _{\mathfrak{s}}\notin 2\mathbb{Z}\}, \end{align*}

where $\varnothing \lt \mathfrak{s}$ if $\mathfrak{s}$ is minimal and $p_{\mathfrak{s}}^0=2$ .

The genus $g(\mathfrak{s})$ of a rational cluster $\mathfrak{s}\in \Sigma _K$ is defined as follows:

• If $D_{\mathfrak{s}}=1$ , then $g(\mathfrak{s})=0$ .

• If $D_{\mathfrak{s}}=2$ , then $2g(\mathfrak{s})+1$ or $2g(\mathfrak{s})+2$ equals $\dfrac{|\mathfrak{s}|-\sum _{\mathfrak{s}^{\prime}\lt \mathfrak{s}}|\mathfrak{s}^{\prime}|}{b_{\mathfrak{s}}}+|\tilde{\mathfrak{s}}|$ .

Notation 1.6. Let $\alpha \in \mathbb{Z}_+$ , $a,b\in \mathbb{Q}$ , with $a\gt b$ , and fix $\frac{n_i}{d_i}\in \mathbb{Q}$ so that

\begin{align*} \alpha a=\frac{n_0}{d_0}\gt \frac{n_1}{d_1}\gt \ldots \gt \frac{n_r}{d_r}\gt \frac{n_{r+1}}{d_{r+1}}=\alpha b,\quad \text{with}\quad \begin{vmatrix}n_i & &n_{i+1}\\ d_i & &d_{i+1}\\\end{vmatrix}=1, \end{align*}

and $r$ minimal. We write ${\mathbb{P}}^1(\alpha,a,b)$ for a chain of ${\mathbb{P}}^1$ s (Notation 4.16) of length $r$ and multiplicities $\alpha d_i,\dots,\alpha d_r$ . Denote by ${\mathbb{P}}^1(\alpha,a)$ the chain ${\mathbb{P}}^1(\alpha,a,\lfloor \alpha a-1\rfloor/\alpha )$ .

The following theorem describes the special fibre of a regular model of $C$ with strict normal crossings.Footnote 2 It follows from a more general result constructing a proper flat model of $C$ unconditionally (Theorem 4.18). For the special fibre $\mathcal{C}^{\textrm{min}}_s$ of the minimal regular model with normal crossings, the reader can refer to Theorem 4.23, where we also describe a defining equation for all components of $\mathcal{C}_s^{\textrm{min}}$ and discuss the Galois action (for general $K$ ). Finally, note that all these models are constructed in Section 5 by giving an explicit open affine cover (see Sections 5.1–5.3).

Theorem 1.7 (Regular SNC model). Suppose $C$ is $y$ -regular and has almost rational cluster picture. Then we can explicitly construct a regular model with strict normal crossings $\mathcal{C}/O_{K}$ of $C$ (Sections 5.1–5.3 ). Its special fibre $\mathcal{C}_s/k$ is given as follows.

1. (1) Every $\mathfrak{s}\in \Sigma _K$ gives a $1$ -dimensional closed subscheme $\Gamma _{\mathfrak{s}}$ of multiplicity $m_{\mathfrak{s}}$ . If $\mathfrak{s}$ is übereven and $\epsilon _{\mathfrak{s}}$ is even, then $\Gamma _{\mathfrak{s}}$ is the disjoint union of $\Gamma _{\mathfrak{s}}^{-}\simeq{\mathbb{P}}^1$ and $\Gamma _{\mathfrak{s}}^{+}\simeq{\mathbb{P}}^1$ , otherwise $\Gamma _{\mathfrak{s}}$ is a smooth geometrically integral curve of genus $g(\mathfrak{s})$ (write $\Gamma _{\mathfrak{s}}^{-}=\Gamma _{\mathfrak{s}}^{+}=\Gamma _{\mathfrak{s}}$ in this case).

2. (2) Every $\mathfrak{s}\in \Sigma _K$ with $D_{\mathfrak{s}}=1$ gives $(|\mathfrak{s}|-\sum _{\mathfrak{s}^{\prime}\in \Sigma _K,\,\mathfrak{s}^{\prime}\lt \mathfrak{s}}|\mathfrak{s}^{\prime}|+p_{\mathfrak{s}}^0-2)/b_{\mathfrak{s}}$ open-ended ${\mathbb{P}}^1$ s of multiplicity $b_{\mathfrak{s}}$ from $\Gamma _{\mathfrak{s}}$ .

3. (3) Finally, for any $\mathfrak{s}\in \Sigma _K$ draw the following chains of ${\mathbb{P}}^1$ s:

Conditions	Chain	From	To
$\mathfrak{s}$ minimal	${\mathbb{P}}^1(\gamma _{\mathfrak{s}}^0,-s_{\mathfrak{s}}^0)$	$\Gamma _{\mathfrak{s}}^-$	open-ended
$\mathfrak{s}$ minimal, $p_{\mathfrak{s}}^0/\gamma _{\mathfrak{s}}^0=2$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}}^0,-s_{\mathfrak{s}}^0)$	$\Gamma _{\mathfrak{s}}^+$	open-ended
$\mathfrak{s}\neq \mathfrak{R}$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$	$\Gamma _{\mathfrak{s}}^-$	$\Gamma _{P(\mathfrak{s})}^-$
$\mathfrak{s}\neq \mathfrak{R}$ , $p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}=2$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}},s_{\mathfrak{s}}-p_{\mathfrak{s}}\cdot \frac{\rho _{\mathfrak{s}}-\rho _{P(\mathfrak{s})}}{2})$	$\Gamma _{\mathfrak{s}}^+$	$\Gamma _{P(\mathfrak{s})}^+$
$\mathfrak{s}=\mathfrak{R}$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}})$	$\Gamma _{\mathfrak{s}}^-$	open-ended
$\mathfrak{s}=\mathfrak{R}$ , $p_{\mathfrak{s}}/\gamma _{\mathfrak{s}}=2$	${\mathbb{P}}^1(\gamma _{\mathfrak{s}},s_{\mathfrak{s}})$	$\Gamma _{\mathfrak{s}}^+$	open-ended

If $\mathfrak{s}^{\prime}\lt \mathfrak{s}$ and $w_{\mathfrak{s}^{\prime}}$ is a rational centre of $\mathfrak{s}^{\prime}$ , then $w_{\mathfrak{s}^{\prime}}$ is also a rational centre of $\mathfrak{s}$ . For any minimal rational cluster $\mathfrak{s}^{\prime}$ fix a rational centre $w_{\mathfrak{s}^{\prime}}$ . For any $\mathfrak{s}\in \Sigma _K$ fix $w_{\mathfrak{s}}=w_{\mathfrak{s}^{\prime}}$ for some minimal rational cluster $\mathfrak{s}^{\prime}\subseteq \mathfrak{s}$ .

The following result gives a basis of integral differentials when $K=K^{nr}$ . In Theorem 6.4 we extend it to the case $K\neq K^{nr}$ .

Theorem 1.9 (Theorem 6.3). Suppose $C$ is $y$ -regular and has almost rational cluster picture. For $i=0,\dots,g-1$ , inductively

1. (i) define $e_i\;:\!=\;\displaystyle \max _{\mathfrak{t}\in \Sigma _K}\bigg \{\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{\mathfrak{s}_j\wedge \mathfrak{t}}\bigg \}$ ;

2. (ii) let $\Sigma _i=\displaystyle \bigg \{\mathfrak{t}\in \Sigma _K\mid \,e_i=\frac{\epsilon _{\mathfrak{t}}}{2}-\rho _{\mathfrak{t}}-\sum _{j=0}^{i-1}\rho _{\mathfrak{s}_j\wedge \mathfrak{t}}\bigg \}$ ;

3. (iii) choose a maximal element $\mathfrak{s}_i$ of $\Sigma _i$ freely.

Then a basis of integral differentials is given by

\begin{align*} \mu _i=\pi ^{\lfloor e_i\rfloor }\prod _{j=0}^{i-1}(x-w_{\mathfrak{s}_j})\frac{dx}{2y},\qquad i=0,\dots,g-1. \end{align*}

Note that given $e_i$ as in the previous theorem, the sum $\sum _{i=0}^{g-1}\lfloor e_i \rfloor$ is the quantity, often denoted by $v({\omega ^\circ }/{\omega })$ , appearing in the period in the Birch and Swinnerton-Dyer conjecture (for more details see [Reference Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell4], [Reference van Bommel25, §1.3]).

1.3. Rational cluster picture

In this subsection we define the rational cluster picture and compare it with the classical cluster picture defined in [Reference Dokchitser, Dokchitser, Maistret and Morgan14]. We will show, via a simple example, in which sense the new object we introduce appears to be more suitable for the study of regular models.

Example 1.11. Let $p$ be any prime number and set $K=\mathbb{Q}_p^{nr}$ . Let $E_p/\mathbb{Q}_p^{nr}$ given by $y^2=x^3-p$ . Then $E_p$ is an elliptic curve with Kodaira-Néron reduction type II. Therefore, the minimal regular model (with normal crossings) of $E_p$ does not depend on $p$ . This is in accordance with the fact that the rational cluster picture of $E_p$ is the same for all $p$ . Indeed, the set of roots of the polynomial $x^3-p$ is $\mathfrak{R}=\{\sqrt [3]{p}, \zeta _3\sqrt [3]{p}, \zeta _3^2\sqrt [3]{p}\}$ , where $\zeta _3$ is a primitive $3$ rd of unity. Hence the rational cluster picture of $E_p$ is

where we denoted with bullet points the roots in $\mathfrak{R}$ , with a surrounding oval the only rational cluster $\mathfrak{R}$ , and with the subscript the radius $\rho _{\mathfrak{R}}$ of $\mathfrak{R}$ .

A different behaviour is observed when we consider the cluster picture [Reference Dokchitser, Dokchitser, Maistret and Morgan14 , Definition 1.26] of $E_p$ , collection of its clusters together with their depths. The cluster picture of $E_p$ is

where the subscripts represent the depth of the cluster $\mathfrak{R}$ . It does depend on $p$ and differs from the rational cluster picture when $p=3$ . Thus, although the cluster picture is particularly useful for Galois theoretical problems, the rational cluster picture appears to be a more suitable object for the study of regular models of the curve.

Finally, note that $E_p$ has an almost rational cluster picture. For any two distinct roots $r,r'\in \mathfrak{R}$ , the smallest $v$ -adic disc $D_{r,r^{\prime}}$ containing them also contains the whole $\mathfrak{R}$ . The element $0\in \mathbb{Q}_p^{nr}$ belongs to $D_{r,r^{\prime}}$ when $p\neq 3$ , while $|D_{r,r^{\prime}}\cap \mathfrak{R}|=3=|v(r)|_p$ , if $p=3$ .

The advantages of the rational cluster picture discussed in this subsection can also be observed in the following example where we study a more complex family of hyperelliptic curves having almost rational cluster picture.

1.4. Example

In this subsection we are going to present an example of a family of hyperelliptic curves $C_p$ satisfying the hypothesis of Theorems 1.7 and 1.9. Via those results we will then describe the special fibre of the minimal regular model and a basis of integral differentials of $C_p$ . All the computations involved are explained in detail in Examples 3.32, 4.25 and 6.5.

For any prime number $p$ , let $a\in \mathbb{Z}_p$ , $b\in \mathbb{Z}_p^\times$ such that the polynomial $x^2+ax+b$ is not a square modulo $p$ . Let $C_p/\mathbb{Q}_p$ be the hyperelliptic curve of genus $4$ given by $y^2=f(x)$ , where $f(x)=(x^6+ap^4x^3+bp^8)((x-p)^3-p^{11})$ . The curve $C_p/\mathbb{Q}_p^{nr}$ has an almost rational cluster picture and is $y$ -regular when $p=2$ . Its rational cluster picture is

where $\rho _{\mathfrak{t}_3}=\frac{4}{3}$ , $\rho _{\mathfrak{t}_4}=\frac{11}{3}$ , and $\rho _{\mathfrak{R}}=1$ . From Theorem 1.7 we can construct a regular model with strict normal crossings of $C_p$ with special fibre

over $\bar{\mathbb{F}}_p$ . Computing the self-intersection of each irreducible component we easily see that this model coincides with the minimal regular model $\mathcal{C}^{\textrm{min}}$ . Theorem 4.23 also describes the action of the Galois group $\textrm{Gal}(\bar{\mathbb{F}}_p/{\mathbb{F}}_p)$ on the special fibre $\mathcal{C}_s^{\textrm{min}}$ of $\mathcal{C}^{\textrm{min}}$ . If the roots of $x^2+ax+b\mod p$ are in ${\mathbb{F}}_p$ then the absolute Galois group acts trivially on each component, otherwise it swaps the $2$ irreducible components of multiplicity $3$ intersecting $\Gamma _{\mathfrak{t}_3}$ .

From Theorem 1.9 it follows that, for any $p$ , a basis of integral differentials of $C_p/\mathbb{Q}_p^{nr}$ is given by

\begin{align*} \mu _0=p^4\cdot \frac{dx}{2y},\quad \mu _1=p^3(x-p)\cdot \frac{dx}{2y},\quad \mu _2=p(x-p)x\cdot \frac{dx}{2y},\quad \mu _3=(x-p)x^2\cdot \frac{dx}{2y}. \end{align*}

In fact, this is also a basis of integral differentials of $C_p/\mathbb{Q}_p$ since they are all defined over $\mathbb{Q}_p$ (see Proposition B.2).

Below we will present related works of other authors concerning regular models and integral differentials of hyperelliptic curves. Note that the example presented here is not covered by [Reference Dokchitser, Dokchitser, Maistret and Morgan14] and [Reference Dokchitser1] since the curve $C_p$ is not semistable and not $\Delta _v$ -regular. In fact, if $p=3$ the curve $C_p$ does not even have tamely potential semistable reduction. The results in [Reference Faraggi and Nowell5] assume $p\gt 2$ and $C_p$ with tamely potential semistable reduction, hence they cannot be used when $p=2,3$ . Finally, there is no classification for genus $4$ curves.

1.5. Related works of other authors

Let $K$ be a discretely valued field with residue field $k$ of characteristic $p$ and let $C/K$ be a hyperelliptic curve of genus $g$ .

In genus $1$ , when $k$ is perfect, thanks to Tate’s algorithm, one can describe the minimal regular model and the space of integral differentials of an elliptic curve $C$ (see e.g., [Reference Silverman24, IV.8.2], [Reference Liu9, Theorem 9.4.35]).

If $K={\mathbb{C}}(t)$ and $C$ has genus $2$ , then Namikawa and Ueno [Reference Namikawa and Ueno18] and Liu [Reference Liu12] give a full classification of the possible configurations of the special fibre of the minimal regular model of $C$ .

If $p\neq 2$ , then Liu and Lorenzini show in [Reference Liu and Lorenzini13] that regular models of $C$ can be seen as double cover of well-chosen regular models of ${\mathbb{P}}^1_K$ . Since the latter can be found by using the MacLane valuations [Reference MacLane15] approach in [Reference Obus and Wewers21], this argument gives a way to describe any regular model of a hyperelliptic curve. At the moment there is no known closed form description of a regular model based on this approach and it has not been generalised to the $p=2$ case.

If $p\gt 2$ , $k$ finite, and $C$ is semistable, then in [Reference Dokchitser, Dokchitser, Maistret and Morgan14] the authors explicitly construct a minimal regular model in terms of the cluster picture of $C$ . Under the same assumptions, Kunzweiler [Reference Kunzweiler8] gives a basis of integral differentials rephrasing [Reference Kausz6, Proposition 5.5] in terms of the cluster invariants introduced in [Reference Dokchitser, Dokchitser, Maistret and Morgan14]. These results can be recovered from Theorem 4.23 (see Corollary 4.27) and Theorem 6.3.

If $p\gt 2$ and $C$ is semistable over some tamely ramified extension $L/K$ , then Faraggi and Nowell [Reference Faraggi and Nowell5] find the special fibre of the minimal regular model of $C$ with strict normal crossings taking the quotient of the stable model of $C_L$ and resolving the (tame) singularities. However, since they do not describe the charts of the model, their result does not immediately yield all arithmetic invariants, such as a basis of integral differentials.

The last work we want to recall represents an important ingredient of the strategy we will use in this paper (described more precisely in the next subsection). T. Dokchitser in [Reference Dokchitser1] shows that the toric resolution of $C$ gives a regular model in case of $\Delta _v$ -regularity [Reference Dokchitser1, Definition 3.9]. This result, used also in [Reference Faraggi and Nowell5], holds for general curves and in any residue characteristic. In his paper, Dokchitser also describes a basis of integral differentials since his model is given as open cover of affine schemes. In Corollary 3.25 and Theorem 6.1, we will rephrase his results for hyperelliptic curves by using rational cluster picture invariants from Section 3.

1.6. Strategy and outline of the paper

In [Reference Dokchitser1], Dokchitser not only describes a regular model of $C$ in case of $\Delta _v$ -regularity, but also constructs a proper flat model $\mathcal{C}_\Delta$ without any assumptions on $C$ . Assume $C$ is $y$ -regular and has an almost rational cluster picture over $K^{nr}$ with rational centres $w_1,\dots, w_m\in K^{nr}$ . Our approach to construct the minimal regular model with normal crossings of $C$ is composed by the following steps:

• Consider the $x$ -translated hyperelliptic curves $C^{w_h}/K^{nr}:y^2=f(x+w_h)$ , for $h=1,\dots,m$ . For each $h$ , [Reference Dokchitser1, Theorem 3.14] constructs a proper flat model $\mathcal{C}_\Delta ^{w_h}$ , possibly singular.

• We glue regular open subschemes of these models along common opens, and show that the result is a proper flat regular model $\mathcal{C}$ of $C_{K^{nr}}$ with strict normal crossings.

• We give a complete description of what components of the special fibre of $\mathcal{C}$ have to be blown down to obtain the minimal model with normal crossings $\mathcal{C}^{\textrm{min}}$ of $C_{K^{nr}}$ .

• Finally, we describe the action of the absolute Galois group $G_k$ of $k$ on the special fibre of $\mathcal{C}^{\textrm{min}}$ .

We will explicitly describe both the models $\mathcal{C}_\Delta ^{w_h}$ and $\mathcal{C}$ . This allows us to study the global sections of its relative dualising sheaf $\omega _{\mathcal{C}/O_K}(\mathcal{C})$ .

In Section 2, we present some results on Newton polygons used in the following sections. In Section 3, we recall the basic objects and notation of [Reference Dokchitser, Dokchitser, Maistret and Morgan14] and define the rational cluster picture. Moreover, we relate it with the notions given in Section 2. This comparison allows us to rephrase the objects in [Reference Dokchitser1] in terms of rational clusters invariants in Section 4. In the same section we also state the theorems which describe the special fibres of a proper flat model (Theorem 4.18) and of the minimal regular model with normal crossings (Theorem 4.23) of $C$ . The construction of these models, from which the two theorems above follow, is presented in Section 5. Finally, in Section 6, Theorems 6.3 and 6.4 describe a basis of integral differentials of $C$ , in terms of rational clusters invariants defined in Section 3.