Abstract
We prove that there exists a supersingular nonsingular curve of genus 4 in arbitrary characteristic \(p>0\). For \(p>3\) we shall prove that the desingularization of a certain fiber product over \(\mathbf{P }^1\) of two supersingular elliptic curves is supersingular.
Introduction
Let K be an algebraically closed field of positive characteristic. For a nonsingular algebraic curve C over K we call C supersingular (resp. superspecial) if its Jacobian J(C) is isogenous (resp. isomorphic) to a product of supersingular elliptic curves.
As to supersingular curves, the following is a basic problem (cf. [19, Question 2.2]).
For given g, does there exist a supersingular curve of genus g in any characteristic p?
For \(g\le 3\), this problem was solved affirmatively. The case of \(g=1\), i.e., elliptic curves is due to Deuring [4]. Ibukiyama, Katsura and Oort in [11, Proposition 3.1] proved the existence of superspecial curves of genus 2 for \(p>3\). For the existence of supersingular curves of genus 3 in any characteristic \(p>0\), see Oort [17, Theorem 5.12]. Also, as a proof for \(g=2\) with \(p>3\) and for \(g=3\) with \(p>2\), we refer to the stronger fact that there exists a maximal curve of genus g over \(\mathbb {F}_{p^{2e}}\) if \(g=2\) and \(p^{2e}\ne 4,9\) (cf. Serre [20, Théorème 3]) and if \(g=3\), \(p\ge 3\) and e is odd (cf. Ibukiyama [10, Theorem 1]), where we recall the general fact that any maximal curve over \(\mathbb {F}_{p^2}\) is superspecial (and therefore supersingular). Even for \((g,p)=(2,3)\), there exists a supersingular curve: for example \(y^2=x^5+1\) is supersingular (but is not superspecial), since its CartierManin matrix is nilpotent, see (9) and (10) in Sect. 2 for the Cartier–Manin matrix and a criterion for the supersingularity. For the case of \(p=2\), we refer to the celebrated paper [22] by van der Geer and van der Vlugt, where they proved that there exists a supersingular curve of an arbitrary genus in characteristic 2.
This paper focuses on the first open case, i.e., the case of \(g=4\) (cf. [19, Question 3.4]). Let us recall some recent works, restricting ourselves to the case of \(g=4\). According to [16, Remark 7.2] and [15, Theorem 7.1] by Li, Mantovan, Pries and Tang, there exists a supersingular curve of genus 4 if \(p \equiv 5 \bmod {6}\) or if \(p \equiv 2, 3, 4 \bmod {5}\). Among them, we review the existence for \((g,p)=(4,3)\) for the reader’s convenience. Indeed \(y^{10} = x(1x)\) is supersingular, as this is a quotient of the Hermitian curve \(X^{q+1}+Y^{q+1}=1\) with \(q=3^2\), where the quotient map is given by \(x=X^{q+1}\) and \(y=XY\). For odd \(p \equiv 2 \bmod {3}\), in [14] the first author showed that there exists a superspecial (and thus supersingular) nonsingular curve of genus 4.
This paper aims to remove any condition on p for the existence of supersingular curves of genus 4. For this, we use curves introduced by Howe in [9], where he studied a curve of genus 4 defined as the desingularization of a fiber product over \(\mathbf{P }^1\) of two elliptic curves. In this paper, we call such a curve a Howe curve, see Definition 2.1 for the precise definition of Howe curves. Our main theorem is:
Theorem 1.1
For any \(p > 3\), there exists a supersingular Howe curve in characteristic p. The number of isomorphism classes of supersingular Howe curves over an algebraically closed field is finite.
The next corollary is deduced from Theorem 1.1 together with the existence results for \(p=2\) and for \((g,p)=(4,3)\) explained above.
Corollary 1.2
There exists a supersingular nonsingular curve of genus 4 in arbitrary characteristic \(p>0\).
As any supersingular Howe curve has anumber \(\ge 3\) for odd p (cf. Sect. 2), Theorem 1.1 is a stronger assertion than the affirmative answer for \(p>3\) to the question by Pries [19, Question 3.6], which predicts that there exists a nonsingular curve of genus 4 with prank 0 and anumber at least 2.
Let us describe an outline of the proof of Theorem 1.1, with an overview of this paper. In Sect. 2, we review the definition of Howe curves and their properties, and show that the existence of a supersingular Howe curve of genus 4 is equivalent to that of two supersingular elliptic curves \(E_1: y^2 = f_1\) and \(E_2 : y^2 = f_2\) with coprime pair \((f_1, f_2)\) of separable polynomials of degree 3 such that the hyperelliptic curve \(C : y^2 = f:=f_1 f_2\) of genus 2 is also supersingular. Here \(E_1\) and \(E_2\) can be isomorphic. For the supersingularity of C, we use the fact that any curve of genus 2 is supersingular if and only if \(M M^{\sigma } = 0\) holds for its CartierManin matrix M, where \(\sigma \) denotes the Frobenius map. The moduli space \({\mathcal M}_2\) (resp. \(\overline{{\mathcal M}_2}\)) of nonsingular (resp. stable) curves of genus two is of dimension 3 and it is known that the supersingular locus on \(\overline{{\mathcal M}_2}\) (naturally extending that of \({\mathcal M}_2\)) is one. On the other hand, C’s constructed as above from supersingular \(E_1\) and \(E_2\) make a twodimensional family, say \(\mathcal P\), in \({\mathcal M}_2\). Then, the Zariski closure \(\overline{{\mathcal P}}\) of \({\mathcal P}\) in \(\overline{{\mathcal M}_2}\) must inersect the supersingular locus in \(\overline{{\mathcal M}_2}\). The difficulty lies in showing that not all of these intersection points lie outside \({\mathcal M}_2\). Almost all of the remaining sections are devoted to overcoming the difficulty. In the proof, instead of \(\overline{{\mathcal P}}\), we use a more concrete space, see Sect. 3 for our modulitheoretic framework. In Sect. 4, we prove two assertions of cubic polynomials (Corollaries 4.4 and 4.7) by reducing the problems to two propositions (Propositions 4.3 and 4.6) for the Legendre form \(y^2=g:=x(x1)(xt)\). In Sect. 5, based on the two assertions, we investigate properties of entries of \(M M^{\sigma }\) as polynomials, where we regard coefficients in \(f_1\) and \(f_2\) as variables. The properties show that we get a desired \((f_1, f_2)\) from a solution of a multivariate system obtained by removing trivial factors from \(M M^{\sigma } = 0\), where “desired” means that \(f_1\) and \(f_2\) are separable and are coprime (in other words, the corresponding point belongs to the interior \({\mathcal M}_2\)). Finally, we show the existence of such a solution by proving an analogous result of the quasiaffineness of EkedahlOort strata in the case of abelian varieties.
Howe curves
In this section, we recall the definition of Howe curves and properties of these curves, and study the supersingularity of them.
Definition 2.1
A Howe curve is a curve which is isomorphic to the desingularization of the fiber product \(E_1 \times _{\mathbf{P }^1} E_2\) of two double covers \(E_i\rightarrow \mathbf{P }^1\) ramified over \(S_i\), where \(S_i\) consists of 4 points and \(S_1\cap S_2=1\) holds.
To achieve our goal, for \(p>3\) we realize a Howe curve in the following way. Let K be an algebraically closed field of characteristic p. Let
be two (nonsingular) elliptic curves, where \(A_1,B_1,A_2,B_2\in K\). Here two elliptic curves (1) and (2) can be isomorphic. Let \(\lambda , \mu , \nu \) be elements of K and set
Consider two elliptic curves
with the double covers
Note that the isomorphism classes of \(E_1\) and \(E_2\) are independent of the choice of \((\lambda ,\mu ,\nu )\) provided \(\mu \ne 0\) and \(\nu \ne 0\). We say that \((\lambda ,\mu ,\nu )\) is of Howe type if

(i)
\(\mu \ne 0\) and \(\nu \ne 0\);

(ii)
\(f_1\) and \(f_2\) are coprime.
If \((\lambda ,\mu ,\nu )\) is of Howe type, then the desingularization of the fiber product \(E_1\times _{\mathbf{P }^1}E_2\) is a Howe curve, since \(E_i\rightarrow \mathbf{P }^1\) is ramified over the set consisting of 4 points, say \(S_i\), and \(S_1\cap S_2 = \{(1:0)\}\). Conversely, any Howe curve is realized in this way. Indeed, let \(E_i\) for \(i=1,2\) be two elliptic curves with double covers \(E_i\rightarrow \mathbf{P }^1\), defining a Howe curve. An automorphism of \(\mathbf{P }^1\) moves the ramified point of both of \(E_i\rightarrow \mathbf{P }^1\) (\(i=1,2\)) to (1 : 0). By a usual coordinatechange, we make \(E_1\) to be of the form (5) if \(p>3\). Note that any elliptic curve ramified at (1 : 0) is written as in (6).
Suppose that \((\lambda ,\mu ,\nu )\) is of Howe type. Put
and consider the hyperelliptic curve C of genus 2 defined by
It was proven by Howe [9, Theorem 2.1] that H is of genus 4 and there exist two isogenies
such that \(\varphi \circ \psi \) and \(\psi \circ \varphi \) are the multiplication by 2, see [13, Theorem C] for a more general result. Hence H is supersingular if and only if \(E_1\), \(E_2\) and C are supersingular. For an abelian variety Z, let Z[p] denote the kernel of the multiplicationbyp map on Z. If p is odd, then \(\psi \circ \varphi \) (resp. \(\varphi \circ \psi \)) is an automorphism of J(H)[p] (resp. \((E_1 \times E_2 \times J(C))[p]\)), whence J(H)[p] and \(E_1[p] \times E_2[p] \times J(C)[p]\) are isomorphic, see [5, Corollary 2] for a more general result. Hence the anumber of H is equal to the sum of the anumbers of \(E_1\), \(E_2\) and J(C), whence any supersingular Howe curve is of anumber \(\ge 3\).
Now we recall a criterion for the supersingularity of C. Let \(\gamma _i\) be the \(x^i\)coefficient of \(f(x)^{(p1)/2}\), i.e.,
Put
In Sect. 4, we shall use the fact that \(\gamma _i\) and therefore a, b, c and d are homogeneous when we regard them as polynomials in \(\lambda \), \(\mu \) and \(\nu \). Let M be the CartierManin matrix of C, that is a matrix representing the Cartier operator on \(H^0(C,\Omega ^1_C)\) (referred to as the modified Cartier operator in [23]). It is known (cf. [6, 4.1] and [23, §2]) that the Cartier–Manin matrix of C is given by
Recall the fact (cf. [11, Lemma 1.1, (i)]) that a curve D of genus 2 is supersingular if and only if the Cartier operator V on \(H^0(D,\Omega _D^1)\) satisfies \(V^2=0\). Hence C is supersingular if and only if
where \(M^\sigma = \begin{pmatrix}a^p &{}\quad b^p \\ c^p &{}\quad d^p\end{pmatrix}\). See [2, Step 2 of the proof of Theorem 4.8] and also [7, 4.0.3] for a sufficient condition for supersingularity, generalized in the higher genus case.
Remark 2.2
As Achter and Howe suggested in [1], we need to be careful when we use Cartier–Manin matrices. A notation different from (9) for Cartier–Manin matrices exists: the transpose A of M, see [23]. In the notation, the supersingularity for genus2 curves is equivalent to \(A^\sigma A=0\), as in [11, Lemma 1.1, (i)].
Proposition 2.3
Assume that \(E_1\) and \(E_2\) are supersingular. Then H is supersingular if and only if \(adbc=0\), \(ab^{p1}+d^p=0\) and \(a^p+c^{p1}d=0\).
Proof
Since \(E_1\) and \(E_2\) are supersingular, H is supersingular if and only if C is supersingular. As explained above, C is supersingular if and only if (10) holds. First remark that \(adbc=0\) implies
To prove the “if”part, suppose that \(adbc=0\), \(ab^{p1}+d^p=0\) and \(a^p+c^{p1}d=0\). It follows from \(adbc=0\) that (11)–(14) hold. This means that every entry of the matrix in (10) is a multiple of \(ab^{p1}+d^p\) or \(a^p+c^{p1}d\). Thus the “if”part is true.
Conversely, suppose that (10) holds. Since \({\text {det}}(M M^\sigma ) = {\text {det}}(M)^{p+1}\), we have \({\text {det}}(M)=adbc=0\), which implies
Also by (11)–(14) we have \(a^p+c^{p1}d=0\) unless \(a=c=0\) and \(ab^{p1}+d^p=0\) unless \(b=d=0\). If \(a=c=0\), by (15) and (16) we have \(a^p+c^{p1}d=0\) unless \(b=d=0\). Similarly, if \(b=d=0\), by (15) and (16) we have \(ab^{p1}+d^p=0\) unless \(a=c=0\). Obviously \((a,b,c,d)=(0,0,0,0)\) satisfies \(adbc=0\), \(ab^{p1}+d^p=0\) and \(a^p+c^{p1}d=0\). Thus the “only if”part is true. \(\square \)
For later use, we review how the Cartier–Manin matrix is converted by a linear change of variables.
Lemma 2.4
Let X be a new variable and consider substituting \(uX+v\) for x, where \(u,v\in K\) with \(u\ne 0\). Let \(\gamma '_i\) be the \(X^i\)coefficient of \(f(uX+v)^{(p1)/2}\) and set
Then we have
with
In particular, we have \(\det (M') = \det (M)\) if \(u=1\).
Proof
We have
as
Then we have
by calculating, in characteristic p, the binomials
for \(j\le 3p3\). These are obtained by applying a general formula
for \(0\le \ell ,\ell '\le p1\) to the case of \(k'=0,1\) and \(\ell '=p1,p2\) with \(\left( {\begin{array}{c}\ell \\ \ell '\end{array}}\right) =0\) for \(\ell '>\ell \). The formula (17) follows from
in characteristic p. \(\square \)
A modulitheoretic observation and ingredients of the proof
From the next section onward, we go into the details of the proof of the main theorem. Before that, let us give an outline of the proof with a modulitheoretic observation. We use the same notation as in Sect. 2. We choose \(A_1\), \(B_1\), \(A_2\) and \(B_2\) so that the elliptic curves (1) and (2) are supersingular. Let \({\tilde{C}}\) be the desingularizaion only at the infinity of C defined in (7). Note that \({\tilde{C}}\) is nonsingular if and only if the corresponding point \((\lambda : \mu : \nu )\) is of Howe type. Considering \(\lambda \), \(\mu \) and \(\nu \) as indeterminates, we have a family of curves \({\tilde{C}}\) over \(\mathbf{P }^2={\text {Proj}}K[\lambda , \mu , \nu ]\). To prove the existence of a supersingular Howe curve, it suffices to find a nonsingular supersingular fiber from this family. The possible Newton polygons of nonsingular \({\tilde{C}}\)’s are supersingular, prankone and ordinary. Note that a nonsingular \({\tilde{C}}\) has prank \(\le 1\) if and only if its Cartier–Manin matrix \(M=\begin{pmatrix}a&{}\quad b\\ c&{}\quad d\end{pmatrix}\) has determinant zero. Since the determinant \(adbc\) turns out to be a multiple of a power of \(\mu \nu \) and every fiber \({\tilde{C}}\) over the locus \(\mu \nu =0\) is singular, it would be better to divide \(adbc\) by \(\mu \nu \) as much as possible. We shall prove in Lemma 5.3 that the order of \(adbc\) with respect to \(\mu \nu \) is exactly \((p+1)/2\). The exactness is important and its proof requires an assertion (Corollary 4.4) on cubic polynomials from the next section.
Put
We consider
as the Newton polygon stratification for this family over \(\mathbf{P }^2\), where \(V(h_0,h_1,h_2)\), \(V(h_0)\smallsetminus V(h_0,h_1,h_2)\) and \(\mathbf{P }^2\smallsetminus V(h_0)\) are called the supersingular locus, the prankone locus and the ordinary locus respectively. The expected dimensions of the loci would be 0, 1 and 2 respectively. Indeed, this is proved in Proposition 5.7. (A priori, on the open subscheme where \({\tilde{C}}\) is nonsingular, the dimension of each locus is greater than or equal to the expected one by the purity theorem [3, 4.1] due to de Jong and Oort.) If the prankone locus is proved not to be projective, then \(V(h_0,h_1,h_2)\ne \emptyset \) follows from it (cf. Proposition 5.7). Our proof is along this line. However, an obstruction is that \(V(h_0,h_1,h_2)\) may be contained in the locus consisting of points which are not of Howe type (equivalently, over which \({\tilde{C}}\) is singular). The main part of our proof is to show that there is no obstruction, i.e., the condition \(h_i=0\) for \(i=0,1,2\) implies that \({\tilde{C}}\) is nonsingular. This is proven in Proposition 5.6, where the proof needs another assertion (Corollary 4.7) on cubic polynomials from the next section.
Two assertions on cubic polynomials
In this section, we prove two assertions on cubic polynomials (Corollaries 4.4 and 4.7), which play important roles in the proof of the main theorem. The proof is done by reducing the problems to those of the Legendre form (Propositions 4.3 and 4.6).
Assume \(p\ge 3\). Let \(g(x)=x(x1)(xt)\) and \(e=(p1)/2\), where we regard t as an indeterminate. We define a polynomial \(H_p (t)\) by
Let \(\delta _{p1}(t)\) be the \(x^{p1}\)coefficient of \(g(x)^e\). It follows from
that \(y^2 = x(x1)(xt_0)\) is a supersingular elliptic curve for \(t_0 \in \overline{\mathbb {F}_p}\) if and only if \(H_p(t_0) = 0\), see e.g., [21, Chap. V, Theorem 4.1].
For our purpose, we need to study the \(x^{p2}\)coefficient \(\delta _{p2}(t)\) of \(g(x)^e\). Specifically, we shall show Proposition 4.3 below, which asserts that \(\delta _{p1}(t)\) and \(\delta _{p2}(t)\) are coprime. We start with giving an explicit formula of \(\delta _{p2}(t)\).
Lemma 4.1
We have
Proof
By the binomial theorem, \(g(x)^e = x^e(x1)^e(xt)^e\) is equal to
The \(x^{p2}\)coefficient of this is the sum over (i, j) with \(j=i+1\), since \(j=i+1\) if \(2e+ij=p2\). Thus, we have the desired equation. \(\square \)
The next lemma is the essential part of the proof of the coprimeness of \(\delta _{p1}(t)\) and \(\delta _{p2}(t)\).
Lemma 4.2
We have the equality
Proof
The lemma follows from the equality
in characteristic p for \(i=0,1,\ldots ,e\) with \(\left( {\begin{array}{c}e\\ e+1\end{array}}\right) =0\). Indeed, multiplying both sides of (22) by \((1)^{e1}t^{i+1}\) and taking the sum over \(i=0,1,\ldots ,e\), we have
Since \(2(ei) \equiv 12i\ \mathrm{mod}\ p\), we have
The equation (22) is obtained by multiplying both sides by \(\displaystyle \left( {\begin{array}{c}e\\ i\end{array}}\right) \left( {\begin{array}{c}e\\ i+1\end{array}}\right) \). \(\square \)
Now we prove
Proposition 4.3
The polynomials \(\delta _{p1}(t)\) and \(\delta _{p2}(t)\) are coprime, i.e., have no common root in \(\overline{\mathbb {F}_p}\).
Proof
Note that \(\delta _{p1}(0)\) and \(\delta _{p1}(1)\) are not zero, since \(\delta _{p1}(t) = (1)^e H_p (t)\) by (19) and \(H_p(0)=1\) and \(H_p(1)=(1)^e\), see the proof of [21, Chap. V, Theorem 4.1 (c)]). If \(\delta _{p1}(\alpha ) =\delta _{p2}(\alpha ) = 0\) for some \(\alpha \in \overline{\mathbb {F}_p} \smallsetminus \{ 0, 1 \}\), then one has \(\left( \frac{d}{dt} \delta _{p1}\right) (\alpha ) = 0\) by Lemma 4.2. We have \(H_{p}(\alpha )=\left( \frac{d}{dt} H_{p}\right) (\alpha ) = 0\), which means that \(\alpha \) is a double root of \(H_p (t)\). This contradicts the fact shown by Igusa [12] that all roots of \(H_p(t)\) are simple (cf. [21, Chap. V, Theorem 4.1 (c)]). \(\square \)
The following corollary derived from Proposition 4.3 will be used in the next section.
Corollary 4.4
Let K be an algebraically closed field of characteristic p. Let \(E_0\) be an elliptic curve \(y^2 = g_0(x)\) over K, where \(g_0(x)\) is a cubic separable polynomial in K[x]. Assume that \(E_0\) is supersingular, i.e., the \(x^{p1}\)coefficient of \(g_0(x)^e\) is zero. Then \(x^{p2}\)coefficient of \(g_0(x)^e\) is not zero.
Proof
There exist \(u \ne 0\) and v in K such that \(g_0 (x) = g (X)\) with \(X := ux + v\), where \(g(X)=X(X1)(Xt)\) for some \(t \in K \smallsetminus \{ 0, 1 \}\). Let \(\epsilon _i\) (resp. \(\epsilon _i'\)) be the \(X^i\)coefficients (resp. \(x^i\)coefficients) of \(g(X)^{(p1)/2}\) and \(g_0(x)^{(p1)/2}\) respectively for \(0 \le i \le 3 (p1)/2\). Similarly to the proof of Lemma 2.4, we have
for \(0 \le i \le 3 (p1)/2\), and in particular \(\epsilon _{p1}^{\prime }\) and \(\epsilon _{p2}^{\prime }\) are
It follows from \(\epsilon '_{p1}=0\) and (23) that \(\epsilon _{p1}=0\). By Proposition 4.3, \(\epsilon _{p1}=0\) implies \(\epsilon _{p2}\ne 0\). By (24) we obtain \(\epsilon '_{p2}\ne 0\). \(\square \)
The second aim of this section is to show that the eth derivative \((g_0(x)^e)^{(e)}\) of \(g_0(x)^e\) is separable for any cubic separable polynomial \(g_0(x)\). We start with reviewing an elementary congruence relation for the reader’s convenience.
Lemma 4.5
We have
for \(n=0,1,\ldots ,p1\).
Proof
We have
where we used Wilson’s theorem \((p1)!\equiv 1 \pmod {p}\). \(\square \)
The next proposition gives a factorization of \((g(x)^e)^{(e)}\) for \(g(x)=x(x1)(xt)\). Let \(H_p(t)\) be the polynomial defined by (18). Let \(a_i \in \overline{\mathbb {F}_p}\) \((i=1,\ldots ,e)\) be the roots of \(H_p(t)=0\), i.e.,
Proposition 4.6
We have
where \((g(x)^e)^{(e)}\) denotes the eth derivative of \(g(x)^{e}\) with respect to x.
Proof
Replacing j by \(ej\) in (21) and taking its eth derivative, we have
Hence \(x^{i+j}t^{ej}\)coefficient of the left side of (26) is \((1)^{i+j}\left( {\begin{array}{c}e\\ i\end{array}}\right) \left( {\begin{array}{c}e\\ j\end{array}}\right) \) times
Put \(P:=\prod _{i=1}^e\left( (ta_i)x(1a_i)t\right) \). We have
by (25) and (18). Using the binomial theorem, one obtains
where we used \(\left( {\begin{array}{c}k\\ i\end{array}}\right) =0\) for \(i>k\) and \(\left( {\begin{array}{c}ek\\ j\end{array}}\right) =0\) for \(j>ek\) in the second equality. Since
the \(x^{i+j}t^{ej}\)coefficient of P is
By applying Vandermonde’s identity to the sum, (30) is equal to \((1)^{i+j}\left( {\begin{array}{c}e\\ i\end{array}}\right) \left( {\begin{array}{c}e\\ j\end{array}}\right) \) times
If \(i+j > e\), then both of (28) and (31) are zero. Otherwise, applying Lemma 4.5 to \(n=i+j\) and \(n=eij\), we have that (28) is equal to (31). Thus the equation (26) holds. \(\square \)
As a corollary of Proposition 4.6, we have
Corollary 4.7
Let \(g_0(x)\) be a separable cubic polynomial. Then \((g_0(x)^e)^{(e)}\) is a separable polynomial.
Proof
A linear coordinate change makes \(g_0\) a Legendre form \(g(x)=x(x1)(xt)\) with \(t\ne 0,1\). It is clear that \((g_0(x)^e)^{(e)}\) is separable if and only if \((g(x)^e)^{(e)}\) is separable. Let \(a_1,\ldots ,a_e\) be the roots of \(H_p(t)\) as above. Note that \(a_i\ne 0,1\) for \(i=1,2,\ldots ,e\) as \(H_p(0)=1\) and \(H_p(1)=(1)^e\) (cf. the proof of [21, Chap. V, Theorem 4.1 (c)]). If \(t \not \in \{a_1,\ldots ,a_e\}\), the roots of \((g(x)^e)^{(e)}\) are \((1a_i)t/(ta_i)\) for \(i=1, \ldots , e\). Otherwise, we may assume \(t=a_1\) without loss of generality, and then the roots of \((g(x)^e)^{(e)}\) are \((1a_i)a_1/(a_1a_i)\) for \(i=2,\ldots ,e\). In both cases, it is straightforward to see that the roots are mutually distinct, using the fact that \(H_p(t)\) is separable, i.e., \(a_1,\ldots ,a_e\) are mutually distinct (cf. [12] and [21, Chap. V, Theorem 4.1 (c)]). \(\square \)
Proof of the main theorem
Assume \(p>3\). Let K be an algebraically closed field of characteristic p. We use the same notation as in Sect. 2, i.e., \(A_1,B_1,A_2,B_2\), \(\lambda , \mu , \nu \), \(f_1, f_2, f\), a, b, c, d are as in Sect. 2. We choose \(A_1,B_1,A_2,B_2\) in K so that (1) and (2) are supersingular. By Proposition 2.3, it suffices to show that there exists \((\lambda ,\mu ,\nu )\in K^3\) of Howe type such that \(adbc=0\), \(ab^{p1}+d^p=0\) and \(a^p+c^{p1}d=0\).
From now on, we regard \(\lambda \), \(\mu \) and \( \nu \) as indeterminates, and consider a, b, c, d as polynomials in \(\lambda \), \(\mu \), \(\nu \). Note that a, b, c and d are homogeneous polynomials in \(\lambda ,\mu ,\nu \) of degrees \(2p2\), \(p2\), \(2p1\) and \(p1\) respectively.
We start with studying the factors of \(\mu \) and \(\nu \) in \(adbc\). More precisely, we prove that \(adbc\) is a multiple of \((\mu \nu )^{(p+1)/2}\) in Lemma 5.3 below. Set \(e := (p1)/2\) as in Section 3. Let \(\alpha _k\) and \(\beta _k\) be the \(x^k\)coefficients of \(f_1(x)\) and \(f_2(x)\) respectively, i.e.,
Let \(\gamma _i\) be the \(x^i\)coefficient of \(f(x)^{(p1)/2}\) with \(f(x)=f_1(x)f_2(x)\) as in (8). We have
and recall that \(a=\gamma _{p1}\), \(b=\gamma _{2p1}\), \(c=\gamma _{p2}\) and \(d=\gamma _{2p2}\). Put
Note \(f_2(x) = F_2(x  \lambda )\). We denote by \(\beta _n^{\prime }\) the \(x^n\)coefficient of \(F_2(x)^{e}\), i.e.,
with \(\beta _{3e}^{\prime }=1\). Looking at the \(x^k\)coefficients of both sides of \(f_2(x)^{e} = F_2(x  \lambda )^{e}\), we have
From this, we find the highest terms in \(\lambda \) of \(\beta _k\) for some k, making use of \(\left( {\begin{array}{c}n\\ k\end{array}}\right) = 0\) in \(\mathbb {F}_p\) if \(0\le k \le p1\) and \(p\le n \le p+k1\):
Lemma 5.1
The coefficient \(\beta _k\) is a homogeneous polynomial in \(K[\lambda ,\nu ]\) of degree \(3ek\). Moreover

(1)
the highest term in \(\lambda \) of \(\beta _0\) is \((\lambda )^{3 e}\),

(2)
the highest term in \(\lambda \) of \(\beta _1\) is \(3e(\lambda )^{3 e  1}\),

(3)
the highest term in \(\lambda \) of \(\beta _e\) is \(\left( {\begin{array}{c}p2\\ e\end{array}}\right) \beta _{p2}^{\prime }\cdot (\lambda )^{\frac{p3}{2}}\) and

(4)
the highest term in \(\lambda \) of \(\beta _{e+1}\) is \(\left( {\begin{array}{c}p2\\ e+1\end{array}}\right) \beta _{p2}^{\prime }\cdot (\lambda )^{\frac{p5}{2}}\).
Proof
Note that \(\beta '_n\) is a monomial in \(K[\nu ]\) of degree \(3 e  n\), since \(F_2(x)^{e}\) is homogeneous of degree 3e as a polynomial in \(K[x, \nu ] \). Hence, by (34) the coefficient \(\beta _k\) is a homogeneous polynomial in \(K[\lambda ,\nu ]\) of degree \(3ek\).
(1) and (2): For \(k=0,1\), the highest term in \(\lambda \) of \(\beta _k\) is the term of \(n=3e\) of (34), which is
(3) and (4): Let k be e or \(e+1\). Since \(\left( {\begin{array}{c}n\\ k\end{array}}\right) = 0\) for \(p\le n \le p+k1\) with \(p+k1 \ge 3e\), it follows from (34) that
Since \(E_2\) is supersingular, we have \(\beta '_{p1}=0\) and \(\beta _{p2}^{\prime } \ne 0\) by Corollary 4.4. Hence, the highest term in \(\lambda \) of (35) is
Note that \(p2\ge k\) for \(k=e\) and \(e+1\) if \(p\ge 5\). \(\square \)
Remark 5.2
The assumption \(p>3\) is used in Lemma 5.1 (2) and (4). This is the reason why the proof of the main theorem does not work for \(p=3\). In fact, there is no superspecial Howe curve in characteristic 3, see Remark 5.8.
In the next lemma, we study the order \(\mathrm{ord}_\mu (a)\) of a with respect to \(\mu \) and so on, where \(\mathrm{ord}_\mu (a)\) is the largest number k such that a is a mulpliple of \(\mu ^k\) in \(K[\lambda ,\mu ,\nu ]\).
Lemma 5.3
We have the following:

(1)
\(\mathrm{ord}_\mu (a) = \frac{p+1}{2}\).

(2)
\(\mathrm{ord}_\mu (c) = \frac{p+1}{2}\).

(3)
\(\mathrm{ord}_\mu (adbc)=\mathrm{ord}_\nu (adbc)=\frac{p+1}{2}\).

(4)
\(\frac{a d  b c}{(\mu \nu )^{\frac{p+1}{2}} } \equiv B \lambda ^{2 p  4} \bmod {(\mu , \nu )}\) for some constant \(B \in K^{\times }\).
Proof
(1) We claim that there exists \(\tilde{\alpha }_{k} \in K\) such that \(\alpha _k = \mu ^{3e  k} \tilde{\alpha }_{k}\) for each \(0 \le k \le 3e\) with \(e=(p1)/2\). Indeed, we have
and thus \(\alpha _{k}\) is a multiple of \(\mu ^{3 e  k}\). Putting \(\tilde{\alpha }_{k} := \alpha _{k} / (\mu ^{3 e  k})\), we also have \(\tilde{\alpha }_k \in K\) for \(0 \le k \le 3e\). Since both the elliptic curves \(E_1 : z^2 = f_1(x)\) and \(E_2 : w^2 = f_2(x)\) are supersingular, we have \(\alpha _{p1} = 0\) and \(\beta _{p1} = 0\), whereas \(\alpha _{p2} \ne 0\) (and thus \(\tilde{\alpha }_{p2} \ne 0\)) and \(\beta _{p2} \ne 0\) by Corollary 4.4. It follows that
where \(\beta _k\) is a polynomial in \(K [\lambda , \nu ]\) for each \(0 \le k \le 3 e\). Since \(\beta _1 \ne 0\) by Lemma 5.1 (2), we have \(\tilde{\alpha }_{3 e  j} \beta _{je} = \tilde{\alpha }_{p2} \beta _{1} \ne 0 \) for \(j = (p+1)/2\), and thus \(\mathrm{ord}_{\mu } (a) = (p+1)/2\).
(2) Similarly to the proof of (1), one has
Since we have \(\tilde{\alpha }_{p2}\ne 0\), and \(\beta _0 \ne 0\) by Lemma 5.1 (1), we also have \(\tilde{\alpha }_{3 e  j} \beta _{je1} = \tilde{\alpha }_{p2} \beta _{0} \ne 0 \) for \(j = (p+1)/2\), and thus \(\mathrm{ord}_{\mu } (c) = (p+1)/2\).
(3) Similarly to the proof of (1), remaining two entries of the CartierManin matrix M are written as
both of which are not a multiple of \(\mu \) since \(\alpha _{3 e} = 1\) and since \(\beta _{e}, \beta _{e+1} \ne 0\) by Lemma 5.1 (3) and (4). Thus, if the coefficient of \(\mu ^{(p+1)/2}\) in \(adbc\) is not zero, then we have \(\mathrm{ord}_{\mu }(a d  b c) = (p+1)/2\). By straightforward computation, the coefficients of \(\mu ^{(p+1)/2}\) in ad and bc are \(\tilde{\alpha }_{p2} \beta _1 \tilde{\alpha }_{3 e} \beta _e\) and \(\tilde{\alpha }_{p2} \beta _0 \tilde{\alpha }_{3 e} \beta _{e+1}\), respectively. Here we have
where \(\tilde{\alpha }_{p2} \ne 0\) and \(\tilde{\alpha }_{3 e} = \alpha _{3 e} = 1\). If \(\beta _1 \beta _e  \beta _0 \beta _{e+1} \ne 0\), we have \(\mathrm{ord}_{\mu }(ad  b c)= (p+1)/2\). By Lemma 5.1, the highest term of \(\beta _1 \beta _{e}\) in \(\lambda \) is
and that of \(\beta _{0} \beta _{e+1}\) is
Since \(\beta _{p2}' \ne 0\) by Corollary 4.4, it suffices to show
in \(\mathbb {F}_p\). If the equality held, we would have \(3e/(p2e) = 1/(e+1)\) in \(\mathbb {F}_p\). Multiplying the denominators and the numerators of both sides by 2, we get \(3(p1)/(p3)= 2/(p+1)\) in \(\mathbb {F}_p\). But the left hand side is equal to 1 and the right hand side is equal to 2. This is a contradiction.
Next, we show \(\mathrm{ord}_{\nu } (a d  b c) = (p+1)/2\). To show this, we consider the transformation \(X = x  \lambda \). The polynomials \(f_1 (x)\) and \(f_2 (x)\) are rewritten as
respectively. Let \(\gamma _{l}^{\prime }\) denote the coefficient of \(X^{l}\) in \((F_1(X) F_2(X))^{(p1)/2}\). Putting \(a^{\prime } = \gamma ^{\prime }_{p1}\), \(c^{\prime } = \gamma ^{\prime }_{p2}\), \(b^{\prime } = \gamma ^{\prime }_{2p1}\) and \(d^{\prime } = \gamma ^{\prime }_{2p2}\), we have \(ad  bc =a^{\prime } d^{\prime } b^{\prime } c^{\prime }\) by the second assertion of Lemma 2.4. By the same argument as in the proof of \(\mathrm{ord}_{\mu }(ad  bc) = (p+1)/2\) for \(f_1(x)\) and \(f_2(x)\), we have \(\mathrm{ord}_{\nu }(a' d'  b' c') = (p+1)/2\) for \(F_2 (X)\) and \(F_1(X)\), and thus \(\mathrm{ord}_{\nu }(adbc) = (p+1)/2\).
(4) From the first part of the proof of (3), the coefficient of \((\mu \nu )^{\frac{p+1}{2}}\) in \(a d  b c\) is \(B \lambda ^{2 p  4} \) with
which is not zero. Recall from the proof of (1) that \(\tilde{\alpha }_{p2}\) and \(\tilde{\alpha }_{3 e}\) are nonzero constants in k. Recall also from the beginning of this section that \(\beta _{p2}^{\prime }\) is a monomial in \(K[\nu ]\) of degree \(3 e  (p2) = (p+1)/2\), and thus \(( \beta _{p2}^{\prime } / \nu ^{\frac{p+1}{2}} ) \in K^\times \). Thus, the assertion of (4) holds. \(\square \)
Let \(R:=K[\lambda ,\mu ,\nu ]\) and put
which belong to R. Since a, b, c and d are homogeneous polynomials in \(\lambda ,\mu ,\nu \) of degrees \(2p2\), \(p2\), \(2p1\) and \(p1\) respectively, we have that \(h_0\), \(h_1\) and \(h_2\) are homogeneous of degrees \(2p4\), \(p(p1)\) and \(2p(p1)\) respectively. We have the stratification
as mentioned in Sect.3. Our aim is to prove that any point \((\lambda : \mu : \nu )\) of \(V(h_0,h_1,h_2)\) is of Howe type. In particular, we need to prove that \(\mu \) and \(\nu \) are not zero. First we show:
Lemma 5.4
Assume \(p>3\). For any point \((\lambda _0:\mu _0:\nu _0)\) on \(V(h_0)\) in \(\mathbf{P }^2\), we have \(\mu _0\ne 0\) or \(\nu _0\ne 0\).
Proof
It suffices to show that \(\mu _0=\nu _0=0\) implies \(\lambda _0=0\) for \((\lambda _0: \mu _0: \nu _0)\in V(h_0)\). This immediately follows from Lemma 5.3 (4). \(\square \)
Thanks to Lemma 5.4, it suffices to consider the set of points with \(\mu \ne 0\) or \(\nu \ne 0\). We study only the case of \(\nu \ne 0\), since a similar argument works also for \(\mu \ne 0\) (exchange the roles of \(\mu \) and \(\nu \) by the coordinate change \(X=x\lambda \) as in (40) and (41)). From now on, we substitute 1 for \(\nu \) and consider a, b, c, d as polynomials in \(\lambda , \mu \), i.e., we work over the ring
Set \(a'=a/\mu ^{(p+1)/2}\) and \(c'=c/\mu ^{(p+1)/2}\), which belong to S by Lemma 5.3 (1) and (2). Let \(a'_0,c'_0, b_0,d_0\) be the constant terms of \(a',c', b,d\) as polynomials in \(\mu \), which are polynomials in \(\lambda \).
Lemma 5.5
As polynomials in \(\lambda \), we have

(1)
\(c'_0\) and \(d_0\) are coprime.

(2)
\(b_0\) and \(d_0\) are coprime.
Proof
Since \(\beta _k\) is the \(x^k\)coefficient of “the Maclaurin series expansion” of \(f_2(x)^{e} = F_2(x  \lambda )^{e}\), we have
if \(k < p\). We use the fact that \({\tilde{\alpha }}_k := \alpha _k/\mu ^{3ek}\) belongs to K for \(0\le k \le 3e\) with \(\tilde{\alpha }_{p2}\ne 0\) and \({\tilde{\alpha }}_{3e}\ne 0\), as seen in the proof of Lemma 5.3 (1).
As \({\tilde{\alpha }}_{p2}\) and \({\tilde{\alpha }}_{3e}\) are nonzero constants, it suffices to see that \(\beta _0\) and \(\beta _e\) are coprime as polynomials in \(\lambda \). Since
and \(F_2\) is a separable polynomial, \(\beta _0\) and \(\beta _e\) are coprime by the next claim. For any separable \(Q(\lambda )\in K[\lambda ]\) and for any integer n with \(0\le n <p\), we claim that \(Q(\lambda )^n\) and \((Q(\lambda )^n)^{(n)}\) are coprime. Indeed, let r be any root of \(Q(\lambda )\) and write \(Q(\lambda )=(\lambda r)Q_0(\lambda )\) with \(Q_0(r)\ne 0\). By the generalized Leibniz rule, we have
Hence \((Q(\lambda )^n)^{(n)}_{\lambda =r}=n!Q_0(r)^n \ne 0\).
As \({\tilde{\alpha }}_{3e}\) is a nonzero constant, it suffices to see that \(\beta _e\) and \(\beta _{e+1}\) are coprime. This follows from
since \((F_2^{e})^{(e)}(\lambda )\) is separable by Corollary 4.7. \(\square \)
Now let us show that if \((\lambda _0,\mu _0,\nu _0)\) satisfies \(h_i=0\) for \(i=0,1,2\), then \((\lambda _0,\mu _0,\nu _0)\) is of Howe type. This is a key ingredient of this paper, as we explained in Sect. 3.
Proposition 5.6
Suppose that \(p>3\). Then any point \((\lambda _0:\mu _0:\nu _0)\) on \(V(h_0,h_1,h_2)\) is of Howe type.
Proof
Let \((\lambda _0:\mu _0:\nu _0)\) be a point of \(V(h_0,h_1,h_2)\). By Lemma 5.4, we have \(\mu _0\ne 0\) or \(\nu _0 \ne 0\). Consider the case of \(\nu _0\ne 0\). We study the open affine subscheme of \(V(h_0,h_1,h_2)\) defined by \(\nu \ne 0\). Substituting 1 for \(\nu \), it is isomorphic to \({\text {Spec}}S/J\), where \(S=K[\lambda ,\mu ]\) and J is the ideal \(\langle h_0(\lambda ,\mu ,1), h_1(\lambda ,\mu ,1), h_2(\lambda ,\mu ,1) \rangle _S\) of S. On S/J, we have
By Lemma 5.5, \(a'_0 d_0  b_0 c'_0\) and \(d_0^p\) are coprime as polynomials in \(\lambda \), and therefore there exist polynomials u, v in \(\lambda \) such that \(u(a'_0 d_0  b_0 c'_0)+v d_0^p = 1\). Hence we have \(1\equiv 0 \quad (\mathrm{mod}\ \mu )\). This means that \(\mu \) is unit in S/J. Thus we have shown that \(\nu _0\ne 0\) implies \(\mu _0\ne 0\) for any point \((\lambda _0:\mu _0:\nu _0)\) on \(V(h_0,h_1,h_2)\). A similar argument shows that \(\mu _0\ne 0\) implies that \(\nu _0\ne 0\) (exchange the roles of \(\mu \) and \(\nu \) by the coordinate change \(X=x\lambda \) as in (40) and (41)). Hence we conclude that both of \(\mu _0\) and \(\nu _0\) are not zero.
It remains to show that \(f_1(x)\) and \(f_2(x)\) defined in (3) and (4) for \((\lambda _0,\mu _0, \nu _0)\) are coprime. As \(\mu _0\) and \(\nu _0\) are not zero, \(f_1(x)\) and \(f_2(x)\) are separable polynomials. Suppose that \(f_1(x)\) and \(f_2(x)\) were not coprime. After taking a linear coordinate change, they are written as \(f_1(x) = x(x1)(xt_1)\) and \(f_2(x) = x(xt_2)(xt_3)\). Then
Then the CartierManin matrix \(M=\begin{pmatrix}a &{}\quad b\\ c &{}\quad d\end{pmatrix}\) of the curve \(y^2=f_1(x)f_2(x)\) becomes a upper triangular matrix (i.e., \(c=0\)) with \(a = (t_1t_2t_3)^e\). Since \(f_1(x)\) and \(f_2(x)\) are separable, we have \(t_i\ne 0\) for \(i=1,2,3\) and therefore \(h_2=a^p+c^{p1}d \ne 0\). This is a contradiction. \(\square \)
Finally we show that \(V(h_0,h_1,h_2)\) is not empty. The fact is reminiscent of the quasiaffineness (cf. [18, (6.5). Theorem]) of EkedahlOort strata in the case of the moduli space of principally polarized abelian varieties.
Proposition 5.7
Assume \(p>3\). Then \(V(h_0,h_1,h_2)\) is nonempty and finite.
Proof
We suppose that \(V(h_0,h_1,h_2) = \emptyset \) were true. Set \(X:=V(h_0)=V(h_0)\smallsetminus V(h_0,h_1,h_2)\). Note that (a, d) can not be (0, 0) on X, since \((a,d)=(0,0)\) implies \(h_1=h_2=0\). Consider the morphism
sending \((\lambda :\mu :\nu )\) to \((a^p+c^{p1}d:c^{p1}d)\) for the part of \(a\ne 0\) and \((ab^{p1}+d^p:d^p)\) for the part of \(d\ne 0\), which is welldefined, i.e., \((a^p+c^{p1}d,c^{p1}d)\ne (0,0)\) for \(a\ne 0\) and \((ab^{p1}+d^p,d^p)\ne (0,0)\) for \(d\ne 0\) with the equality \((a^p+c^{p1}d:c^{p1}d)=(ab^{p1}+d^p:d^p)\) for \(a\ne 0\) and \(d\ne 0\) deduced from (16). Let Y be the schemetheoretic image of \(\varphi \) ( [8, Chap. II, Exercise 3.11, (d)]). Note that Y is affine, as it is a closed subscheme of \(\mathbf{A }^1\). Since the schemetheoretic image of a proper scheme is proper (cf. [8, Chap. II, Exercise 4.4]), Y has to be finite. But, this is absurd, since the image of \(\varphi \) is not finite by the following two facts. Firstly X is connected, since it is a hypersurface in \(\mathbf{P }^2\) (cf. [8, Chap. III, Cor. 7.9]). Secondly the image of \(\varphi \) contains two distinct points. Indeed, (1 : 1) (resp. (1 : 0)) is in the image of \(\varphi \), since there exists a point of \(\mathbf{P }^2\) where \(h_0=0\) and \(a = 0\) (resp. \(h_0=0\) and \(d = 0\)), by [8, Chap. I, Theorem 7.2].
In the proof of Proposition 5.6, we have seen that \(V(h_0,h_1,h_2)\) is contained in \(\mathbf{P }^2\smallsetminus V(\mu ) \simeq \mathbf{A }^2\), whence \(V(h_0,h_1,h_2)\) is affine. Thus, \(V(h_0,h_1,h_2)\) is finite, as it is projective. \(\square \)
Finally, we look at the case of \(p=3\).
Remark 5.8
Assume \(p=3\). There exists a unique supersingular elliptic curve \(y^2=x^3x\). We use the notation in Sect. 2. Remark that we assumed \(p>3\) in Sect. 2 only to assure that any elliptic curve can be written as \(y^2=x^3+Ax+B\) and that all arguments in Sect. 2 work for \(A_1=A_2=1\) and \(B_1=B_2=0\) with \(p=3\). It is straightforward to see that the (1, 1)entry of \(MM^\sigma \) is equal to \((\mu \nu )^8\). Hence the supersingularity of C associated to \((\lambda ,\mu ,\nu )\) implies that \(\mu \) or \(\nu \) is zero, whence \((\lambda ,\mu ,\nu )\) is not of Howe type. Thus, we conclude that there is no supersingular Howe curve for \(p=3\).
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Acknowledgements
The authors thank Everett Howe, Tomoyoshi Ibukiyama and Rachel Pries for their valuable helpful comments to earlier versions of this paper. The authors are also grateful to the anonymous referees for their careful reading of this article and their comments. This work was supported by JSPS GrantinAid for Research Activity Startup Grant Nos. 18H05836, 19K21026, 20K14301 and JSPS Grantin Aid for Scientific Research (C) 17K05196.
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Kudo, M., Harashita, S. & Senda, H. The existence of supersingular curves of genus 4 in arbitrary characteristic. Res. number theory 6, 44 (2020). https://doi.org/10.1007/s4099302000217x
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