2.1 Basic definitions

Let K be an arbitrary number field in general and $O_K$ the ring of integers in K. We denote by $P_K$ the set of nonzero prime ideals of $O_K$ , and by $I_K$ the monoid of nonzero integral ideals of $O_K$ . For each prime $\mathfrak {p} \in P_K$ , $k_{\mathfrak {p}}$ denotes the residue field $k_{\mathfrak {p}} := O_K/ \mathfrak {p}$ ; and, for each $\mathfrak {a} \in I_K$ , its absolute norm is denoted $N\mathfrak {a}$ .

The major object of this work concerns the ring $W_{O_K}(A)$ of (generalized) Witt vectors over $O_K$ with components in an $O_K$ -algebra A introduced by Borger [Reference Borger1, Reference Borger2]; for our purpose, we only need to recall the definition of $W_{O_K}(A)$ for a flat $O_K$ -algebra A, in particular. In what follows, we denote by $A^{I_K}$ the $I_K$ -times product of A; and by $\psi _{\mathfrak {c}}: A^{I_K} \rightarrow A^{I_K}$ for $\mathfrak {c} \in I_K$ the $\mathfrak {c}$ -shift operator defined by $\psi _{\mathfrak {c}} (\xi ) := (\xi _{\mathfrak {c} \mathfrak {a}})_{\mathfrak {a}}$ for each $\xi = (\xi _{\mathfrak {a}})_{\mathfrak {a}} \in A^{I_K}$ . Informally speaking, the ring $W_{O_K}(A)$ of Witt vectors is defined as a subring of $A^{I_K}$ consisting of those $\xi \in A^{I_K}$ that are smooth with respect to arithmetic derivatives (in the sense of Remark 2.2 below).

Definition 2.1 (the ring $W_{O_K}(A)$ of Witt vectors).

Let A be a flat $O_K$ -algebra. Then the ring $W_{O_K}(A)$ of Witt vectors over $O_K$ with components in A is defined as the intersection $\bigcap _n U_n(A)$ of the following subalgebras $U_n (A) \subseteq A^{I_K}$ defined by induction on $n \geq 0$ :

$$ \begin{align*} U_0(A) &:= A^{I_K}, \\ U_{n+1}(A) &:= \{ \xi \in U_n(A) \mid \text{for all } \mathfrak{p} \in P_K,\ \psi_{\mathfrak{p}} \xi - \xi^{N\mathfrak{p}} \in \mathfrak{p} U_n(A) \}. \end{align*} $$

In general the ring $W_{O_K}(A)$ of Witt vectors enjoys the following properties, which means that $W_{O_K}(A)$ is not just a ring but in fact has a structure of a (generalized) $\Lambda $ -ring [Reference Borger1].

Amongst (generalized) Witt vectors, our major concern is with the integral Witt vectors over $O_K$ with coefficients in the algebraic integers $O_{\bar {K}}$ , in particular; the algebraic Witt vectors over K are also of our central concern, being defined through integral Witt vectors as follows.

By definition, an (integral) Witt vector $\xi \in A_K$ is a vector $\xi = (\xi _{\mathfrak {a}})$ that consists of algebraic integers $\xi _{\mathfrak {a}} \in O_{\bar {K}}$ indexed by the nonzero integral ideals $\mathfrak {a} \in I_K$ such that the distribution of the values $\xi _{\mathfrak {a}}$ on $I_K$ is arithmetically smooth; when K is of class number one, in particular, this means that the rth arithmetic derivatives $\delta _{\mathfrak {p}_1} \cdots \delta _{\mathfrak {p}_r} \xi $ of $\xi $ must have integral components for any $r \geq 0$ and any $\mathfrak {p}_1, \ldots , \mathfrak {p}_r \in P_K$ . In view of the fact that the arithmetic derivative $\delta _{\mathfrak {p}} \xi = (\psi _{\mathfrak {p}} \xi - \xi ^{N\mathfrak {p}})/\pi _{\mathfrak {p}}$ contains the fractional operation $\pi _{\mathfrak {p}}^{-1}$ , the arithmetic smoothness of $\xi \in A_K$ (that the components of $\delta _{\mathfrak {p}_1} \cdots \delta _{\mathfrak {p}_r} \xi $ must be all integers) imposes a strong constraint on the distribution of the integral values $\xi _{\mathfrak {a}}$ . In fact, for instance, the components $\xi _{\mathfrak {a}}$ of $\xi \in A_K$ are all forced to be abelian over K, despite the fact that we only assume that the $\xi _{\mathfrak {a}}$ s are algebraic integers (hence, a priori not necessarily abelian over K); see [Reference Borger and de Smit4, Section 3] and [Reference Uramoto13, Section 4].

Yet another constraint imposed on integral Witt vectors $\xi \in A_K$ is that the orbit of $\xi $ under the action of the operators $\psi _{\mathfrak {p}}$ (or Frobenius lifts) is finite; in fact, this orbit-finiteness characterizes the integral Witt vectors $\xi \in A_K$ among general Witt vectors in $W_{O_K}(O_{\bar {K}})$ . In what follows, note that if an integral ideal $\mathfrak {a} \in I_K$ decomposes as $\mathfrak {a} = \mathfrak {p}_1 \cdots \mathfrak {p}_n$ , the operator $\psi _{\mathfrak {a}}$ is equal to the composition $\psi _{\mathfrak {p}_1} \circ \cdots \circ \psi _{\mathfrak {p}_n}$ of the Frobenius lifts $\psi _{\mathfrak {p}_i}$ .

Due to this strong constraint on the components $\xi _{\mathfrak {a}}$ , only a few examples of integral Witt vectors have been explicitly constructed so far; the few examples include those over $\mathbb {Z}$ [Reference Uramoto15, Corollary 3.3.1] and the one induced by the j-function [Reference Uramoto15, Example 4.1.10]. This fact motivates the current work; and the subject of this paper is to give at least an effective algorithm to enumerate (in principle all) examples of integral Witt vectors over imaginary quadratic fields. For this purpose, the modularity theorem for algebraic Witt vectors [Reference Uramoto15] gives us a useful representation of algebraic Witt vectors as we briefly recall below.

2.2 The modularity theorem

Unlike the case of integral Witt vectors, we have a relatively concise description of algebraic Witt vectors; in particular, in the case of imaginary quadratic fields K, the distribution of the components $\xi _{\mathfrak {a}}$ of algebraic Witt vectors $\xi \in E_K$ admits an analytic description in terms of special values of a deformation family of Fricke functions $f_a$ (indexed by $a=(a_1, a_2) \in \mathbb {Q}^2/\mathbb {Z}^2$ ), which are modular functions on the upper-half plane $\mathfrak {H} :=\{z \in \mathbb {C} \mid \mathrm {Im}(\tau )> 0\}$ . Here we recall the definition of the Fricke functions $f_a$ (see [Reference Shimura11]), with which the modular vectors $\widehat {f}_a$ are constructed.

Definition 2.8 (Fricke functions $f_a$ ).

Let K be an imaginary quadratic field. The Fricke function $f_a$ indexed by $a = (a_1, a_2) \in \mathbb {Q}^2/\mathbb {Z}^2$ with $a \neq 0$ is defined as follows: for each $\tau \in \mathfrak {H}$ ,

$$ \begin{align*} f_a (\tau) & := \begin{cases} \dfrac{g_2(\tau) g_3(\tau)}{\Delta(\tau)} \cdot \wp (a_1 \tau + a_2; \tau) & \textrm{if }K \neq \mathbb{Q}(i), \mathbb{Q}(\sqrt{-3}), \\[6pt] \dfrac{g_2(\tau)^2}{\Delta(\tau)} \wp(a_1 \tau + a_2; \tau)^2 & \textrm{if }K = \mathbb{Q}(i), \\[8pt] \dfrac{g_3(\tau)}{\Delta(\tau)} \wp(a_1 \tau + a_2; \tau)^3 & \textrm{if }K = \mathbb{Q}(\sqrt{-3}); \end{cases} \end{align*} $$

for $a=0$ we define $f_0(\tau ) = j(\tau )$ with the j-function j; also, $\wp $ is the Weierstrass $\wp $ -function.

As shown in [Reference Uramoto15] this family of modular functions $f_a$ induces algebraic Witt vectors $\widehat {f}_a$ in a certain manner; technically, this construction is based on a canonical isomorphism between the $\Lambda $ -ring $E_K$ of algebraic Witt vectors and the $\Lambda $ -ring $\mathrm {Hom}_{G_K}(DR_K, K^{ab})$ of locally constant, $K^{ab}$ -valued, $G_K$ -equivariant functions on a profinite monoid $DR_K$ called the Deligne–Ribet monoid [Reference Uramoto15, Corollary 3.1.6]; this profinite monoid $DR_K$ is defined as the inverse limit of the following finite Deligne–Ribet monoids (or the ray class monoids).

Definition 2.9 (ray class monoid $DR_{\mathfrak {f}}$ , [Reference Deligne and Ribet7]).

For each $\mathfrak {f} \in I_K$ , we define the finite Deligne–Ribet monoid $DR_{\mathfrak {f}}$ (or the ray class monoid of conductor $\mathfrak {f}$ ) as the finite quotient $DR_{\mathfrak {f}} := I_K/\sim _{\mathfrak {f}}$ of the monoid $I_K$ with respect to the following congruence $\sim _{\mathfrak {f}}$ of finite index: for $\mathfrak {a}, \mathfrak {b} \in I_K$ ,

$$ \begin{align*} \mathfrak{a} \sim_{\mathfrak{f}} \mathfrak{b} & \Leftrightarrow \text{there exists } t \in K^+ \cap (1 + \mathfrak{f} \mathfrak{b}^{-1}), \quad \mathfrak{a} \mathfrak{b}^{-1} = (t), \end{align*} $$

where $K^+$ denotes the totally positive elements of K.

Since $DR_K$ has an action of $G_K$ , we may consider locally constant $K^{ab}$ -valued $G_K$ -equivariant functions on $DR_K$ , the set of which is denoted by $\mathrm {Hom}_{G_K}(DR_K, K^{ab})$ and forms a K-algebra with respect to the point-wise operations. Moreover, since $I_K$ embeds into $DR_K$ , we also have mutually commuting operators $\psi _{\mathfrak {p}}: \mathrm {Hom}_{G_K}(DR_K, K^{ab}) \rightarrow \mathrm {Hom}_{G_K}(DR_K, K^{ab})$ defined by $\psi _{\mathfrak {p}} \xi (x) := \xi (\mathfrak {p} x)$ for each $\xi \in \mathrm {Hom}_{G_K}(DR_K, K^{ab})$ and $x \in DR_K$ . With these structures, $\mathrm {Hom}_{G_K}(DR_K, K^{ab})$ forms a $\Lambda $ -ring, which is in fact isomorphic to $E_K$ .

Theorem 2.11 [Reference Uramoto15, Corollary 3.1.6].

We have a canonical isomorphism of $\Lambda $ -rings:

$$ \begin{align*} E_K &\simeq \mathrm{Hom}_{G_K}(DR_K, K^{ab}). \end{align*} $$

This theorem says that, to construct algebraic Witt vectors, it is sufficient to construct locally constant $K^{ab}$ -valued $G_K$ -equivariant functions on $DR_K$ ; our construction of modular vectors $\widehat {f}_a$ is based on this fact, together with the presentation of $DR_K$ due to Yalkinoglu [Reference Yalkinoglu17]; in this paper, we need this presentation for the case of imaginary quadratic fields, which is given as follows.

Proposition 2.12 (Yalkinoglu, [Reference Yalkinoglu17, Proposition 8.2]).

Let K be an imaginary quadratic field. Then we have a canonical isomorphism of profinite monoids:

$$ \begin{align*} DR_K &\simeq \widehat{O}_K \times_{\widehat{O}_K^\times} (\mathbb{A}_{K, f}^\times /K^\times), \end{align*} $$

where the monoid $\widehat {O}_K \times _{\widehat {O}_K^\times } (\mathbb {A}_{K, f}^\times /K^\times )$ is the quotient of the multiplicative monoid ${\widehat {O}_K \times (\mathbb {A}_{K, f}^\times /K^\times )}$ by the action of the group $\widehat {O}_K^\times $ given by $[\rho , s] \mapsto [\rho u, u^{-1} s]$ for each $u \in \widehat {O}_K^\times $ .

With these preparations, we now recall the construction of modular vectors $\widehat {f}_a$ over imaginary quadratic fields K (see [Reference Uramoto15, Section 4.1]). Below, throughout this paper, K denotes an imaginary quadratic field. The following definition first gives a general construction of $\widehat {f}_a$ ; for more detail the reader is referred to [Reference Uramoto15]; but we then recall a more explicit description of this construction given by [Reference Uramoto15, Lemma 4.1.9], which would be more useful for actual computations.

Definition 2.13 (modular vector $\widehat {f}_a$ ).

Let K be an imaginary quadratic field; for each $a \in \mathbb {Q}^2/\mathbb {Z}^2$ , the modular vector $\widehat {f}_a: DR_K \rightarrow K^{ab}$ is defined as follows:

$$ \begin{align*} \widehat{f}_a : DR_K &\rightarrow K^{ab} \\ \hspace{0cm} [\rho, s] &\longmapsto f_{a \cdot \rho \cdot u} (\tau_s), \end{align*} $$

where $\rho \in \widehat {O}_K$ acts on $a \in \mathbb {Q}^2/\mathbb {Z}^2$ via the natural isomorphisms $\mathbb {Q}^2/\mathbb {Z}^2 \simeq K/O_K \simeq \mathbb {A}_{K, f}/ \widehat {O}_K$ , the first one being determined by a basis of the $\mathbb {Z}$ -module $O_K = \mathbb {Z} \tau _K + \mathbb {Z}, \tau _K \in \mathfrak {H}$ ; this basis also induces an adelic representation $q: \mathbb {A}_{K, f} \rightarrow \mathrm {GL}_2(\mathbb {A}_{\mathbb {Q}, f})$ , so that $q(s) \in \mathrm {GL}_2(\mathbb {A}_{\mathbb {Q}, f})$ admits a decomposition of the form $q(s) = u \cdot \alpha _s$ , where $u \in \mathrm {GL}_2(\widehat {\mathbb {Z}})$ and $\alpha _s \in \mathrm {GL}_2(\mathbb {Q})$ . The former $u \in \mathrm {GL}_2(\widehat {\mathbb {Z}})$ naturally acts on $\mathbb {Q}^2/\mathbb {Z}^2 \simeq \mathbb {A}_{\mathbb {Q}, f}^2/ \widehat {\mathbb {Z}}^2$ ; the latter $\alpha _s$ determines $\tau _s \in \mathfrak {H}$ as the usual fractional transformation $\tau _s := \alpha _s(\tau _K)$ .

Theorem 2.15 (modularity theorem, [Reference Uramoto15, Theorem 4.3.1]).

For each $a \in \mathbb {Q}^2/\mathbb {Z}^2$ , the modular vector $\widehat {f}_a$ defines a locally constant $K^{ab}$ -valued $G_K$ -equivariant function on $DR_K$ , hence gives an algebraic Witt vector $\widehat {f}_a \in E_K$ ; moreover, these modular vectors generate $E_K$ over K. That is,

$$ \begin{align*} E_K &= K[\widehat{f}_a \mid a \in \mathbb{Q}^2/\mathbb{Z}^2]. \end{align*} $$

This theorem means that every algebraic Witt vector $\xi $ can be represented as $\xi = F(\widehat {f}_a)$ with a K-coefficient polynomial $F(z_a) \in K[z_a \mid a \in \mathbb {Q}^2/\mathbb {Z}^2]$ . Since the coefficients of $F(z_a)$ are all in $K = \mathbb {Q} \tau _K + \mathbb {Q}$ , each polynomial $F(z_a)$ can be identified with the finitely many rational numbers appearing in its coefficients, which enables us to manipulate all algebraic Witt vectors $\xi = F(\widehat {f}_a)$ symbolically. With this representation, it makes sense to ask the decidability of whether a given algebraic Witt vector $\xi = F(\widehat {f}_a) \in E_K$ belongs to $A_K$ ; and the main result of this paper is to prove this decidability.

As mentioned above, the function $\widehat {f}_a: DR_K \rightarrow K^{ab}$ induces an algebraic Witt vector actually by restricting this function onto its submonoid $I_K \hookrightarrow DR_K$ . In the current case, K being imaginary quadratic, this restriction can be calculated by [Reference Uramoto15, Lemma 4.1.9] more explicitly as follows.

Proposition 2.17 (see [Reference Uramoto15, Lemma 4.1.9]).

Let $K=\mathbb {Q}(\sqrt {-d})$ with a square-free natural number d; represent each nonzero integral ideal $\mathfrak {a} \in I_K$ as $\mathfrak {a} = \mathbb {Z} (a \tau _K + b) + \mathbb {Z} c$ with integers $a, b, c \in \mathbb {Z}$ . Then the $\mathfrak {a}$ th component of $\widehat {f}_a \in E_K$ is given as follows:

$$ \begin{align*} \widehat{f}_a(\mathfrak{a}) &= f_{a \cdot \beta^{-1}} (\tau_{\mathfrak{a}^{-1}}), \end{align*} $$

where $\beta \in \mathrm {GL}_2(\mathbb {Z})$ is determined by $\mathfrak {a}^{-1} = \mathbb {Z}^2 \cdot \beta \cdot (\tau _K, 1)^t = \mathbb {Z} w_1 + \mathbb {Z} w_2$ so that ${\tau _{\mathfrak {a}^{-1}} = w_1/w_2 \in \mathfrak {H}}$ . In particular,

(2-1) $$ \begin{align} \beta^{-1} &= \begin{cases} \begin{pmatrix} c & -b \\ 0 & -a \end{pmatrix} & \textrm{if }-d \equiv 2, 3 \ \pmod {\ 4}, \\ \begin{pmatrix} c & -a - b \\ 0 & -a \end{pmatrix} & \textrm{if} -d \equiv 1 \ \pmod {\ 4}. \end{cases} \end{align} $$

Proof. By [Reference Uramoto15, Lemma 4.1.9], we have $\widehat {f}_a (\mathfrak {a}) = f_{a \cdot \beta ^{-1}} (\tau _{\mathfrak {a}^{-1}})$ , where $\mathfrak {a}^{-1} = \mathbb {Z}^2 \cdot \beta \cdot (\tau _K, 1)^t = \mathbb {Z} w_1 + \mathbb {Z} w_2$ for $\beta \in M_2(\mathbb {Q})$ and $\tau _{\mathfrak {a}^{-1}} = w_1 / w_2$ . Here, the matrix $\beta $ can be computed more explicitly using the equalities $\mathfrak {a}^{-1} = (N\mathfrak {a})^{-1} \bar {\mathfrak {a}}$ and $N\mathfrak {a} = ac$ , where $\bar {\mathfrak {a}}$ is the Galois conjugate of $\mathfrak {a}$ ; thus,

$$ \begin{align*} \beta & = \begin{cases} (ac)^{-1} \cdot \begin{pmatrix} -a & b \\ 0 & c \end{pmatrix} & \textrm{if } -d \equiv 2, 3 \ \pmod {\ 4}, \\ (ac)^{-1} \begin{pmatrix} -a & a + b \\ 0 & c \end{pmatrix} & \textrm{if } -d \equiv 1 \ \pmod {\ 4} .\end{cases} \end{align*} $$

From this, the inverse $\beta ^{-1}$ can be computed as Equation (2-1), hence the claim.

Now we proceed to the problem of judging the integrality of algebraic Witt vectors. As outlined in Section 1, the major technical challenge is two-fold. First, we need to check whether each component $\xi _{\mathfrak {a}}$ of an algebraic Witt vector $\xi = F(\widehat {f}_a) \in E_K$ is an algebraic integer, which is the subject of Section 3; for this task, we employ the idea of division polynomials [Reference Silverman12] from the theory of elliptic curves. As remarked in Section 1 too, nevertheless, in order that $\xi \in E_K$ is an integral Witt vector (that is, $\xi \in A_K$ ), it is not sufficient that every component $\xi _{\mathfrak {a}}$ of $\xi $ is an algebraic integer: the distribution of the integers $\xi _{\mathfrak {a}}$ must be smooth with respect to arithmetic derivations, namely, infinitely differentiable by any $\mathfrak {p} \in P_K$ . The subject of Section 4 is thus to develop an effective procedure of checking this arithmetic smoothness, based on an elementary module-theoretic idea.

3.1 Outline of algorithm

The Fricke functions $f_a$ , used in the construction of the above modular vectors $\widehat {f}_a \in E_K$ , are defined in terms of Eisenstein series (see [Reference Shimura11]); hence, we can compute the components $\widehat {f}_a(\mathfrak {a})$ of the modular vectors $\widehat {f}_a \in E_K$ (= special values of $f_a$ s), and therefore those of all $\xi \in E_K$ as well, approximately with an arbitrary accuracy. The problem here is to judge whether the components $\xi _{\mathfrak {a}}$ of $\xi = F(\widehat {f}_a)$ are integers, using only this approximate numerical value.

Concerning this issue, note first that, for any $\xi = F(\widehat {f}_a) \in E_K$ , the indices $a \in \mathbb {Q}^2/\mathbb {Z}^2$ appearing in the polynomial $F(z_a) \in K[z_a \mid a\in \mathbb {Q}^2/\mathbb {Z}^2]$ are in $N^{-1} \mathbb {Z}^2/\mathbb {Z}^2$ for some natural number $N \geq 1$ , whence the components $\xi _{\mathfrak {a}}$ of $\xi \in E_K$ all belong in the ray class field $K_{NO_K}$ with the conductor $NO_K \in I_K$ ; in fact, this result can be stated more specifically as follows.

Therefore, the integrality of $\xi _{\mathfrak {a}}$ s can be judged by checking the integrality of the symmetric functions $s(\xi _{\mathfrak {a}}^{\sigma } \mid \sigma \in \mathrm {Gal}(K_{NO_K}/K)) \in K$ of the conjugates of $\xi _{\mathfrak {a}} \in K_{NO_K}$ over K, at least whose values are computable by Proposition 2.17. Of course, as mentioned above too, we can know only approximate values of ${s = s(\xi _{\mathfrak {a}}^{\sigma } \mid \sigma \in \mathrm { Gal}(K_{NO_K}/K)) \in K}$ ; but if we could estimate upper bounds of their denominators, we can judge the integrality of such values s since we know that the (for example, decimal) expansions of $s_1, s_2$ where $s = s_1 + s_2 \tau _K$ with $O_K = \mathbb {Z} + \mathbb {Z} \tau _K$ are periodic (because $s_1, s_2 \in \mathbb {Q}$ ), and that the periods can be estimated by the size of the denominator of s; also we can compute their decimal expansions in any accuracy.

Moreover, since all $\xi \in E_K$ are expressed as K-polynomials of $\widehat {f}_a$ s, we can estimate the upper bound of the denominator of the above s for all $\xi \in E_K$ if we could do this just for the special values of $f_a$ s at imaginary quadratic $\tau \in \mathfrak {H}$ . As seen in Definition 2.8, these special values are constant multiples of x-coordinates of torsion points in an elliptic curve for $\tau $ ; hence, the problem is reduced to an estimate of the denominators of coordinates of torsion points on elliptic curves, for which, however, some more information on arithmetic of elliptic curves is needed. Indeed we then solve this problem by employing the idea of division polynomials from the theory of elliptic curves (see [Reference Silverman12]); in fact, division polynomials are explicitly computable and tell us polynomials whose roots are the special values of $f_a$ s, hence are useful to give upper bounds of their denominators. Therefore, using this, we can judge the integrality of the components of the modular vectors $\widehat {f}_a$ ( $a \in \mathbb {Q}^2/\mathbb {Z}^2$ ), hence the integrality of the components $\xi _{\mathfrak {a}}$ of any $\xi = F(\widehat {f}_a) \in E_K$ as well.

3.2 Estimate of denominators

Before stating, we first recall the definition of division polynomials from [Reference Silverman12, Exercise III.3.7]. Let $E \colon y^2 = x^3 - Ax - B$ be an elliptic curve over a number field k. For $n \ge 0$ , the division polynomial $\psi _{A, B, n} = \psi _{A, B, n}(x, y) \in \mathbb {Z}[A, B, x, y]$ is defined by the following recurrence equation for $m \ge 2$

$$ \begin{align*} \begin{aligned} \psi_{A, B, 2m+1} &= \psi_{A, B, m+2}^3\psi_{A, B, m} - \psi_{A, B, m-1}\psi_{A, B, m+1}^3, \\ \psi_{A, B, 2m+2} &= \dfrac{\psi_{A, B, m+1}}{2y} ( \psi_{A, B, m+3}^3\psi_{A, B, m} - \psi_{A, B, m-1}\psi_{A, B, m+2}^3) \end{aligned} \end{align*} $$

with initial conditions

$$ \begin{align*} \begin{aligned} \psi_{A, B, 0}(x,y) &= 0, \\ \psi_{A, B, 1}(x,y) &= 1, \\ \psi_{A, B, 2}(x,y) &= 2y, \\ \psi_{A, B, 3}(x,y) &= 3x^4 + 6Ax^2 + 12Bx - A^2, \\ \psi_{A, B, 4}(x,y) &= 4y(x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - 8B^2 - A^3). \end{aligned} \end{align*} $$

Putting $\deg x = 2, \deg y = 3, \deg A = 4$ and $\deg B = 6$ , we can derive the following properties of these division polynomials $\psi_{A, B, N} (x, y)$ .

Moreover, we find the following description.

In our case, we can take

$$ \begin{align*} A = -\frac{g_2(\tau)}{4}, \quad B = -\frac{g_3(\tau)}{4}. \end{align*} $$

We write $E_{\tau } = E_{-g_2(\tau )/4, -g_3(\tau )/4}$ .

The $x, y$ -coordinates of $P \in E_\tau $ corresponding to $a = (a_1, a_2) \in \mathbb {R}^2 / \mathbb {Z}^2 \cong \mathbb {C} / (\mathbb {Z} \tau + \mathbb {Z})$ are described by

$$ \begin{align*} \bigg( \wp(a_1 \tau + a_2; \tau), \frac{\wp'(a_1 \tau + a_2; \tau)}{2} \bigg), \end{align*} $$

where $\wp '(z; \tau )$ is the derivation of $\wp (z; \tau )$ .

Let $N = 2m$ be even. The odd case is similar. Since $a \in ((N^{-1}\mathbb {Z})^2/\mathbb {Z}^2)$ corresponds to an N-torsion point of $E_\tau $ , taking products over $a = (a_1, a_2) \in ((N^{-1}\mathbb {Z})^2/\mathbb {Z}^2) / \pm $ with $2a \neq (0,0)$ ,

$$ \begin{align*} \prod_{\substack{a = (a_1, a_2) \in ((N^{-1}\mathbb{Z})^2/\mathbb{Z}^2) / \pm \\ 2a \neq (0,0)}} (x - \wp(a_1 \tau + a_2; \tau)) = \frac{\psi_{A, B, 2m+2}(x, y)}{2(m+1) y}. \end{align*} $$

We would like to modify this formula to one for the Fricke functions $f_a(\tau )$ .

Let us assume that $K \neq \mathbb {Q}(i), \mathbb {Q}(\sqrt {-3})$ ; in this case, the Fricke function is ${f_a(\tau ) = g_2g_3/\Delta \cdot \wp (a_1 \tau + a_2; \tau )}$ with

$$ \begin{align*} \Delta(\tau) = g_2(\tau)^3 - 27 g_3(\tau)^2 = -16(4A^3 + 27B^2). \end{align*} $$

Changing the coordinates

$$ \begin{align*} \begin{aligned} x' &= \frac{g_2(\tau)g_3(\tau)}{\Delta(\tau)} x, \\ A' &= \bigg(\frac{g_2(\tau)g_3(\tau)}{\Delta(\tau)}\bigg)^2 A = -\frac{g_2^3g_3^2}{4\Delta^2}, \\ B' &= \bigg(\frac{g_2(\tau)g_3(\tau)}{\Delta(\tau)}\bigg)^3 B = -\frac{g_2^3g_3^4}{4\Delta^3}, \end{aligned} \end{align*} $$

the left-hand side is changed to

$$ \begin{align*} \begin{aligned} \bigg( \dfrac{\Delta(\tau)}{g_2(\tau)g_3(\tau)} \bigg)^{2m^2 + 4m} \prod_{\substack{a = (a_1, a_2) \in ((N^{-1}\mathbb{Z})^2/\mathbb{Z}^2) / \pm \\ 2a \neq (0,0)}} (x' - f_a(\tau)). \end{aligned} \end{align*} $$

Since the right-hand side is homogeneous with respect to $(\deg x, \deg y, \deg A, \deg B) = (2, 3, 4, 6)$ and the variable y does not appear essentially by Lemma 3.2, we can modify it as

$$ \begin{align*} \begin{aligned} \frac{\psi_{A, B, 2m+2}(x, y)}{2(m+1) y} &= \bigg( \dfrac{\Delta(\tau)}{g_2(\tau)g_3(\tau)} \bigg)^{2m^2 + 4m} \frac{\psi_{A', B', 2m+2}(x', y)}{2(m+1) y}. \end{aligned} \end{align*} $$

Thus,

$$ \begin{align*} (m+1)\prod_{\substack{a = (a_1, a_2) \in ((N^{-1}\mathbb{Z})^2/\mathbb{Z}^2) / \pm \\ 2a \neq (0,0)}} (x' - f_a(\tau)) = \frac{\psi_{A', B', 2m+2}(x', y)}{2y}, \end{align*} $$

and it is a polynomial in $\mathbb {Z}[x', A', B']$ by Lemma 3.3.

Next we consider the case $K = \mathbb {Q}(i)$ . In this case, we have $g_3 = B = 0$ and $\Delta = g_2^3$ , and then we change the coordinates as

$$ \begin{align*} \begin{aligned} x' &= \sqrt{\frac{g_2(\tau)^2}{\Delta(\tau)}} x = \frac{x}{\sqrt{g_2(\tau)}}, & A' &= \left(\!\sqrt{\frac{g_2(\tau)^2}{\Delta(\tau)}}\right)^2 A = -\dfrac{1}{4}, & B' &= \left(\!\sqrt{\frac{g_2(\tau)^2}{\Delta(\tau)}}\right)^3 B = 0. \end{aligned} \end{align*} $$

Note that in this case, we have $B = 0$ , and if $x'$ is a coordinate of a N-torsion point P, then $-x'$ is also a coordinate of $[i]P$ . Hence,

$$ \begin{align*} (m+1)\prod_{\substack{a = (a_{1}, a_{2}) \in ((N^{-1}\mathbb{Z})^{2}/\mathbb{Z}^{2}) / \{\pm1, \pm i\} \\2a \neq (0,0)}} (x'^{2} - f_{a}(\tau)) = \frac{\psi_{-1/4,0,2m+2}(x', y)}{2y}, \end{align*} $$

and the polynomial is in $\mathbb {Z}[x', A', B']$ by Lemma 3.3. We can handle the case ${K = \mathbb {Q}(\sqrt {-3})}$ with the complex multiplication $[\omega ]P$ .

As a summary, we have the following proposition.

Proposition 3.4. Let us define

$$ \begin{align*} \begin{aligned} A' &= -\frac{g_2^3g_3^2}{4\Delta^2} = -\frac{j(j+64)}{4 \cdot 48^3}, & B' &= \frac{g_2^3g_3^4}{4\Delta^3} = -\frac{j^2(j+64)}{4 \cdot 576^3}. \end{aligned} \end{align*} $$

We set the degree of $x, A', B'$ as $2, 4, 6$ . Then we have the following:

1. (1) For $N = 2m+1$ with $m \ge 1$ , we define

   
   Then, this is a polynomial in $\mathbb {Z} [x, A', B' ]$ , homogeneous of degree $N^2-1$ . Moreover, as a polynomial of x, the leading coefficient is $N = 2m+1$ .

2. (2) For $n = 2m+2$ with $m \ge 1$ ,

   
   Then, this is a polynomial in $\mathbb {Z} [x, A', B' ]$ , homogeneous of degree $N^2-4$ . Moreover, as a polynomial in x, the leading coefficient is $N/2 = m$ .

Using this proposition, we estimate the denominator of the minimal polynomial of $f_a(\tau )$ over $\mathbb {Q}(j(\tau ))$ .

4.1 Outline of algorithm

4.1.2 The general case

The basic idea for the general-case algorithm is almost the same as the special case. But in the general case, the operators $\delta _{\mathfrak {p}}$ are no longer (globally) available; so we need some additional treatment for the general case.

As briefly predicted in [Reference Ishitsuka and Uramoto9] too, the key is the good behavior of the Witt vector functor [Reference Borger1] with respect to the localization operations, from which we can deduce the following natural isomorphism of rings (see Section 4.2 for more precise details):

$$ \begin{align*} W_{O_K}(O_{\bar{K}}) &\simeq \bigcap_{\mathfrak{p} \in P_K} W_{O_{K, \mathfrak{p}}, P_K^{(\mathfrak{p})}} (O_{K, \mathfrak{p}} \otimes O_{\bar{K}}). \end{align*} $$

From this isomorphism, the problem of judging whether $\xi \in W_{O_K}^a(O_{\bar {K}}) = A_K$ is reduced to the local problems of judging whether $\xi \in W_{O_{K, \mathfrak {p}}, P_K^{(\mathfrak {p})}} (O_{K, \mathfrak {p}} \otimes O_{\bar {K}})$ for all $\mathfrak {p}$ ; then we can use the operators $\delta _{\mathfrak {p}}$ locally at each $\mathfrak {p}$ .

As in the special case, we need only to consider finitely many $\mathfrak {p}$ such that $\mathfrak {p} \mid NO_K$ if $\xi \in X_{NO_K}$ ; therefore, the remaining task is to judge local smoothness of $\xi $ , namely, ${\xi \in W_{O_{K, \mathfrak {p}}, P_K^{(\mathfrak {p})}} (O_{K, \mathfrak {p}} \otimes O_{\bar {K}})}$ using the local (and thus single) derivative $\delta _{\mathfrak {p}}$ for each ${\mathfrak {p} \mid NO_K}$ . The first major difference from the special case is that, at each step of derivation, we need to judge whether the coefficients of the derivatives $\delta _{\mathfrak {p}}^r \xi $ are all in the localized ring $O_{K, \mathfrak {p}} \otimes O_{\bar {K}}$ rather than $O_{\bar {K}}$ ; hence, they do not need to be integers globally and may have nontrivial denominators in $O_K \setminus \mathfrak {p}$ , which makes our decision problem slightly more delicate.

To be specific, we need to check whether the coefficients $(\delta _{\mathfrak {p}}^r \xi )_{\mathfrak {a}} \in \bar {K}$ of $\delta _{\mathfrak {p}}^r \xi $ are all in $O_{K, \mathfrak {p}} \otimes O_{\bar {K}}$ , that is, whether $b \cdot (\delta _{\mathfrak {p}}^r \xi )_{\mathfrak {a}} \in O_{\bar {K}}$ for $\exists\ b \in O_K \setminus \mathfrak {p}$ ; the existential quantifier $\exists $ here is a new obstruction to our decision procedure. However, since we can now estimate the denominators of the coefficients of $\delta _{\mathfrak {p}}^r \xi $ thanks to Section 3, the possible denominators of the coefficients of $\delta _{\mathfrak {p}}^r \xi $ in $O_K \setminus \mathfrak {p}$ can be bounded, which implies that we in fact need only to consider finitely many possible denominators from $O_K \setminus \mathfrak {p}$ (see Section 4.2).

The second major difference from the special case is that, in the general case, we need to consider the $O_{K,\mathfrak {p}}$ -subalgebras (instead of $O_K$ -subalgebras) $D^{(\mathfrak {p})}_n$ of the K-algebra $X_\xi $ generated over $O_{K, \mathfrak {p}}$ by the derivatives $\delta _{\mathfrak {p}}^r \xi $ for $r \leq n$ . As in the special case, this gives rise to the ascending sequence $D_0^{(\mathfrak {p})} \subseteq D_1^{(\mathfrak {p})} \subseteq \cdots \subseteq D_n^{(\mathfrak {p})} \subseteq \cdots \subseteq X_\xi $ of $O_{K, \mathfrak {p}}$ -subalgebras of $X_\xi $ ; however, unlike the special case, the corresponding K-algebras $K \otimes D_0^{(\mathfrak {p})} \subseteq K \otimes D_1^{(\mathfrak {p})} \subseteq \cdots \subseteq K \otimes D_n^{(\mathfrak {p})} \subseteq \cdots \subseteq X_\xi $ do not necessarily converge to $X_\xi $ . Nevertheless, we can estimate when the K-algebra $K \otimes D_n^{(\mathfrak {p})}$ terminates, or in other words, when $D_n^{(\mathfrak {p})}$ becomes full rank in $\lim _n K \otimes D_n^{(\mathfrak {p})}$ . Denoting $X_{\xi }^{(\mathfrak {p})} := \lim _n K \otimes D_n^{(\mathfrak {p})} = K \otimes D_{n_1}$ for some $n_1 \geq 0$ , this K-algebra $X_{\xi }^{(\mathfrak {p})}$ is finite etale over K; thus the integral closure $B_\xi ^{(\mathfrak {p})}$ of $O_{K, \mathfrak {p}}$ within $X_\xi ^{(\mathfrak {p})}$ is finite over $O_{K, \mathfrak {p}}$ . Again, as in the special case, the quotient $O_{K, \mathfrak {p}}$ -module $B_\xi ^{(\mathfrak {p})} / D_{n_1}^{(\mathfrak {p})}$ is of finite cardinality, the size of which can be estimated from $\xi $ by elementary module-theoretic arguments; from this estimate, we can then bound $n_2 \geq n_1$ , whence $D_n^{(\mathfrak {p})}$ terminates. Based on these facts, it follows that $\xi $ is locally smooth (that is, $\xi \in W_{O_{K, \mathfrak {p}}, P_K^{(\mathfrak {p})}} (O_{K, \mathfrak {p}} \otimes O_{\bar {K}})$ ) if and only if it is locally differentiable at least $n_2$ -times by $\mathfrak {p}$ (thanks to $D_{n_2}^{(\mathfrak {p})} = D_{n_2 + 1}^{(\mathfrak {p})} = \cdots $ ). This can be checked within finite steps thanks to the above (first) argument concerning denominators in $O_K \setminus \mathfrak {p}$ ; on the whole, this is finitary and we are done.

4.2 Proof of the algorithm

Here we prove nontrivial claims appearing in the above outline of our algorithms. To this end, we first recall the structure of the K-algebra $X_\xi = K [\psi _{\mathfrak {a}} \xi \mid \mathfrak {a} \in I_K]$ generated over K by the orbit $I_K \xi $ of $\xi $ under the action of $\psi _{\mathfrak {p}}$ s; in particular, it is essential for our algorithm that this K-algebra $X_\xi $ is finite etale over K [Reference Uramoto13, Proposition 4]. To be more specific, the K-algebra $X_\xi $ has the following presentation given in terms of components $\xi _{\mathfrak {a}}$ of $\xi $ .

Proposition 4.1 (see [Reference Uramoto15, Proposition 3.2.5]).

For $\xi \in E_K$ , we have the following isomorphism as a finite etale K-algebra:

$$ \begin{align*} X_\xi &\simeq \prod_{[\mathfrak{a}] \in M_\xi / \mathcal{J}} K(\xi_{\mathfrak{a} \mathfrak{b}} \mid \mathfrak{b} \in I_K). \end{align*} $$

Here $M_\xi $ is the finite monoid defined as the quotient monoid $I_K / \equiv _\xi $ , where $\mathfrak {a} \equiv _\xi \mathfrak {b}$ for $\mathfrak {a}, \mathfrak {b} \in I_K$ if and only if $\psi _{\mathfrak {a}} \xi = \psi _{\mathfrak {b}} \xi $ ; and $\mathcal {J}$ is Green’s $\mathcal {J}$ -equivalence relation on the finite monoid $M_\xi $ , that is, $[\mathfrak {a}] \mathcal {J} [\mathfrak {b}]$ for $[\mathfrak {a}], [\mathfrak {b}] \in M_\xi $ if and only if $[\mathfrak {a} \mathfrak {c}] = [\mathfrak {b}]$ and $[\mathfrak {b} \mathfrak {d}] = [\mathfrak {a}]$ for some $[\mathfrak {c}], [\mathfrak {d}] \in M_\xi $ ; in other words, this is precisely when $[\mathfrak {a}]$ and $[\mathfrak {b}]$ generate the same two-sided ideal in $M_\xi $ . We abusively denote by $[\mathfrak {a}]$ (or by $[\mathfrak {a}]_{\mathcal {J}}$ ) the $\mathcal {J}$ -equivalence class for $[\mathfrak {a}] \in M_\xi $ . It is shown in [Reference Uramoto15, Lemma 3.2.4] that $[\mathfrak {a}] \in M_\xi $ and $[\mathfrak {b}] \in M_\xi $ are $\mathcal {J}$ -equivalent if and only if $[\mathfrak {a}] \cdot M_\xi ^\times = [\mathfrak {b}] \cdot M_\xi ^\times $ , where $M_\xi ^\times $ denotes the unit group of $M_\xi $ . Choosing complete representatives $[\mathfrak {a}_1]_{\mathcal {J}}, \ldots , [\mathfrak {a}_n]_{\mathcal {J}}$ of $M_\xi / \mathcal {J}$ , the above isomorphism of $X_\xi $ is given as follows (see [Reference Uramoto15, Proposition 3.2.5]):

$$ \begin{align*} X_\xi \rightarrow& \prod_{i=1}^n K(\xi_{\mathfrak{a}_i \mathfrak{b}} \mid \mathfrak{b} \in I_K) \\ \eta \longmapsto& (\eta_{\mathfrak{a}_1}, \ldots, \eta_{\mathfrak{a}_n}). \end{align*} $$

Here it is useful to recognize a more concise presentation of each component ${K(\xi _{\mathfrak {a}_i \mathfrak {b}} \mid \mathfrak {b} \in I_K)}$ of the right-hand side. For this, in relation to the above $\mathcal {J}$ -equivalence on $M_\xi $ , we define a quasiordering $\leq _{\mathcal {J}}$ on $M_\xi $ : that is, $[\mathfrak {a}] \leq _{\mathcal {J}} [\mathfrak {b}]$ for $[\mathfrak {a}], [\mathfrak {b}] \in M_\xi $ if and only if $[\mathfrak {a}] = [\mathfrak {b} \mathfrak {c}]$ for some $[\mathfrak {c}] \in M_\xi $ ; thus, by definition, $[\mathfrak {a}] \mathcal {J} [\mathfrak {b}]$ if and only if $[\mathfrak {a}] \leq _{\mathcal {J}} [\mathfrak {b}]$ and $[\mathfrak {a}] \geq _{\mathcal {J}} [\mathfrak {b}]$ . With this notation, notice first that if $[\mathfrak {a}] = [\mathfrak {b}]$ we have $\psi _{\mathfrak {a}} \xi = \psi _{\mathfrak {b}} \xi $ by definition, hence, in particular, $\xi _{\mathfrak {a}} = \xi _{\mathfrak {b}}$ . Therefore, it makes sense to denote $\xi _{[\mathfrak {a}]} := \xi _{\mathfrak {a}}$ ; the same is true for every $\eta \in X_\xi $ . With the above representatives $[\mathfrak {a}_1]_{\mathcal {J}}, \ldots , [\mathfrak {a}_n]_{\mathcal {J}}$ of $M_\xi / \mathcal {J}$ , the component $K(\xi _{\mathfrak {a}_i \mathfrak {b}} \mid \mathfrak {b} \in I_K)$ is then equal to $K(\xi _{[\mathfrak {b}]} \mid [\mathfrak {b}] \leq _{\mathcal {J}} [\mathfrak {a}_i])$ , hence,

$$ \begin{align*} X_\xi & \simeq \prod_{i=1}^n K(\xi_{[\mathfrak{b}]} \mid [\mathfrak{b}] \leq_{\mathcal{J}} [\mathfrak{a}_i]) \\ \eta & \longmapsto (\eta_{[\mathfrak{a}_1]}, \ldots, \eta_{[\mathfrak{a}_n]}). \end{align*} $$

Furthermore, the K-algebra $X_\xi $ is equipped with the $\Lambda $ -structure $\psi _{\mathfrak {p}}: X_\xi \rightarrow X_\xi $ for each $\mathfrak {p} \in I_K$ , which admits a simple description. That is, for each $\eta \in X_\xi $ and $\mathfrak {p} \in I_K$ , $\psi _{\mathfrak {p}} \eta \in X_\xi $ corresponds to $(\eta _{[\mathfrak {p} \mathfrak {a}_1]}, \ldots , \eta _{[\mathfrak {p} \mathfrak {a}_n]})$ under the above isomorphism.

From a computational standpoint, we need to further concsider how to compute (the multiplication table of) the finite monoid $M_\xi $ and how to effectively choose representatives of $M_\xi $ and $M_\xi / \mathcal {J}$ ; in fact, the finite monoid $M_\xi $ is defined as the quotient monoid $I_K / \equiv _\xi $ , but to judge the equivalence relation $\mathfrak {a} \equiv _\xi \mathfrak {b}$ , we need to judge the equality $\psi _{\mathfrak {a}} \xi = \psi _{\mathfrak {b}} \xi $ , namely $\xi _{\mathfrak {a} \mathfrak {c}} = \xi _{\mathfrak {b} \mathfrak {c}}$ for all $\mathfrak {c} \in I_K$ . This seemingly infinitary problem, however, can be reduced to a finitary problem, thanks to the fact that $M_\xi $ is a quotient of $DR_{NO_K}$ if the indices $a \in \mathbb {Q}^2 / \mathbb {Z}^2$ appearing in $\xi = F(\widehat {f}_a)$ are all in $N^{-1} \mathbb {Z}^2 / \mathbb {Z}^2$ , which gives a kind of upper bound for our search.

Definition 4.3 (conductor of $\xi $ ).

For $\xi \in E_K$ , the conductor $\mathfrak {f}_\xi \in I_K$ of $\xi $ is defined so that, for each $\mathfrak {p} \in P_K$ :

$$ \begin{align*} v_{\mathfrak{p}} (\mathfrak{f}_\xi) &:= \min \{ n \geq 0 \mid \psi_{\mathfrak{p}}^n X_\xi = \psi_{\mathfrak{p}}^{n+1} X_\xi\}. \end{align*} $$

We start from the special case; for a while, assume that K is of class number one, whence $O_K$ is a PID. The following says that, to judge the arithmetic smoothness of $\xi $ , we actually need to consider arithmetic derivatives $\delta _{\mathfrak {p}}$ for only finitely many $\mathfrak {p}$ s.

If $D_{n_1}$ already contains a nonintegral Witt vector, then we can conclude that $\xi $ is not integral; otherwise, we need to continue to check whether the arithmetic derivatives $\delta _{\mathfrak {p}_1} \cdots \delta _{\mathfrak {p}_n} \xi $ of $\xi $ have integral components for all $n \geq n_1$ . This task itself is an infinitely problem, especially when $\xi $ is an integral Witt vector; but by the following key estimate, we can actually stop this procedure at some finitely many times.

We now consider the general case; a key for this is the following.

Lemma 4.10. For $\xi \in E_K$ , we have the following equivalence:

$$ \begin{align*} \xi \in W_{O_K}^a(O_{\bar{K}}) &\Leftrightarrow \xi \in W_{O_{K, \mathfrak{p}}, P_K^{(\mathfrak{p})}} (O_{K, \mathfrak{p}} \otimes O_{\bar{K}}) \hspace{0.2cm} \textrm{for all}\ \mathfrak{p} \mid \mathfrak{f}_\xi. \end{align*} $$

Proof. To see this we note the following isomorphism:

$$ \begin{align*} W_{O_K}(O_{\bar{K}}) &\simeq \bigcap_{\mathfrak{p} \in P_K} W_{O_{K, \mathfrak{p}}, P_K^{(\mathfrak{p})}} (O_{K, \mathfrak{p}} \otimes O_{\bar{K}}). \end{align*} $$

To be precise, we first have the identity $W_{O_K}(O_{\bar {K}}) = \bigcap _{\mathfrak {p}} O_{K, \mathfrak {p}} \otimes W_{O_K} (O_{\bar {K}})$ ; then by Proposition 6.1 and by (2.4.2) [Reference Borger1], we have a canonical isomorphism $O_{K, \mathfrak {p}} \otimes W_{O_K}(O_{\bar {K}}) \simeq W_{O_{K, \mathfrak {p}}, P_K^{(\mathfrak {p})}} (O_{K, \mathfrak {p}} \otimes O_{\bar {K}})$ . Here $P_K^{(\mathfrak {p})}$ denotes the set of ideals $\mathfrak {m}_{\mathfrak {q}} \subseteq O_{K, \mathfrak {p}}$ indexed by $\mathfrak {q} \in P_K$ such that $\mathfrak {m}_{\mathfrak {p}} = \mathfrak {p} O_{K, \mathfrak {p}}$ and $\mathfrak {m}_{\mathfrak {q}} = O_{K, \mathfrak {p}}$ for $\mathfrak {q} \neq \mathfrak {p}$ , hence the claim follows.

Lemma 4.11. Let $\pi _{\mathfrak {p}} \in \mathfrak {p}$ be a uniformizer of $\mathfrak {p}$ ; and define $\delta _{\mathfrak {p}} \eta := \pi _{\mathfrak {p}}^{-1} (\psi _{\mathfrak {p}} \eta - \eta ^{N\mathfrak {p}})$ . With this notation, we have the following equivalence for $\xi \in E_K$ :

$$ \begin{align*} \xi \in W_{O_{K, \mathfrak{p}}, P_K^{(\mathfrak{p})}} (O_{K, \mathfrak{p}} \otimes O_{\bar{K}}) &\Leftrightarrow \delta_{\mathfrak{p}}^r \xi \in ( O_{K, \mathfrak{p}} \otimes O_{\bar{K}})^{I_K}\quad \textrm{for all }r \geq 0. \end{align*} $$

Thanks to the above lemmas, our target decision problem $\xi \in W_{O_K}^a(O_{\bar {K}})$ is now reduced to the local problems $\xi \in W_{O_{K, \mathfrak {p}}, P_K^{(\mathfrak {p})}} (O_{K, \mathfrak {p}} \otimes O_{\bar {K}})$ at $\mathfrak {p} \mid \mathfrak {f}_\xi $ ; but instead, we then need to be able to judge whether the coefficients of the derivatives $\delta _{\mathfrak {p}}^r \xi $ of $\xi $ is in the localized ring $O_{K, \mathfrak {p}} \otimes O_{\bar {K}}$ rather than $O_{\bar {K}}$ . The following lemma provides what we need for this task.

The next task is to estimate when the ascending sequence $K \otimes D_0^{(\mathfrak {p})} \subseteq K \otimes D_1^{(\mathfrak {p})} \subseteq \cdots \subseteq X_\xi $ of K-subalgebras of $X_\xi $ terminates; we denote by $X_\xi ^{(\mathfrak {p})} = \lim _n K \otimes D_n^{(\mathfrak {p})}$ the limit K-algebra.