An Explicit Sextic Twist of a Supersingular Elliptic Curve in Characteristic Two

Abstract

Let $E:y^2+y=x^3+x$ over ${\mathbb{F}}_2$, a supersingular curve whose automorphism group is isomorphic to $SL(2,3)$, of order twenty-four. Using an automorphism of order six, we construct the sextic twist of E along the fiber product of an Artin-Schreier and a Kummer covering, obtaining the family $Z^2+Z=B\left(t\right)X^3+A\left(t\right)+c$, in which the constant c is not removable by a constant translation of Z over ${\mathbb{F}}_{16}$ when A is fixed. We analyze two members. For $A\left(t\right)=t^3+t$ and $B\left(t\right)=t^2(t+1)$, the associated elliptic surface is K3, the L-polynomial has degree eight, and the Mordell-Weil rank of the generic fiber is zero over ${\mathbb{F}}_{16}\left(t\right)$ and four over ${\mathbb{F}}_{256}\left(t\right)$. For $A\left(t\right)=t^7+t$ and $B\left(t\right)=t^{17}$, the surface is again K3, the L-polynomial has degree fourteen, and the ranks are zero and two. In both cases, the rank follows from an exact factorization of the zeta numerator of the auxiliary curve, isolating in the non-invariant part of its cohomology the factors geometrically isogenous to E, the isogeny being defined over ${\mathbb{F}}_{256}$ but not over ${\mathbb{F}}_{16}$.

1. Introduction

Ulmer [1] constructed non-isotrivial elliptic curves of arbitrarily large rank over ${\mathbb{F}}_q\left(t\right)$ by relating them to quotients of Fermat surfaces, and the pattern of much subsequent work on ranks over function fields has been to take a curve with a large automorphism group, twist it along a covering adapted to that group, and read the rank from the resulting decomposition of cohomology. Three distinct things are at play, and it is worth keeping them apart. The automorphism group of the constant curve together with the covering produces the twist. The factors of the Jacobian of the covering that are geometrically isotypic to that curve produce the rank. Finally, the domination of the resulting surface by a product of curves, quite independently of either, gives the Tate conjecture and hence, via the Artin-Tate formalism [2] and the theorem of Kato and Trihan [3], the full Birch-Swinnerton-Dyer conjecture. In particular, no supersingularity hypothesis is needed for the last of these.

Griffon and Ulmer [4] carried this out in detail for $y^2=x^3+t^q-t$ over ${\mathbb{F}}_r\left(t\right)$, computing rank, regulator, and Tate-Shafarevich group, all in characteristic $p>3$. The constructions of [5,6] in characteristic two and that of [7] in characteristic three are the immediate antecedents; Section 2 sets out the comparison in detail.

What distinguishes characteristic two is the size of $\mathrm{Aut}\left(E\right)$. For $p>3$, a $j=0$ curve has $\left|\mathrm{Aut}\right(E\left)\right|=6$; for $p=2$, the curve $E:y^2+y=x^3+x$ has $\mathrm{Aut}\left(E\right)\cong SL(2,3)$ of order 24, with elements of orders one, two, three, four, and six. The elements of order four generate the quaternion subgroup used in [6].

It is important to delimit what is and is not new here. That E admits automorphisms of order six, and that the corresponding twists are trivialized over degree-six extensions, is part of the general theory and is covered by [5]; the present study claims no novelty for it. This paper makes two contributions: first, the explicit realization of such a twist along a composite covering, namely, the fiber product of an Artin-Schreier covering with a Kummer covering, carried out in coordinates over a fixed model of ${\mathbb{F}}_{16}$; and second, for two resulting surfaces, both K3, the determination of the zeta data of the auxiliary curves, the exact factorizations isolating the geometrically E-isotypic factors, and the resulting Mordell-Weil ranks over ${\mathbb{F}}_{16}\left(t\right)$ and ${\mathbb{F}}_{256}\left(t\right)$. This study does not compute regulators, Tate-Shafarevich groups, or local terms, and so it makes no claim to a complete arithmetic determination. The general theory guarantees that such twists exist; it does not supply these surfaces or their invariants. It is also important to note the limitations on the scale of the results: the ranks four and two over ${\mathbb{F}}_{256}\left(t\right)$ are small, and no claim of large or record rank is made. What makes the examples interesting is that, in characteristic two, the surfaces, the zeta data, and the ranks are obtained in closed form, and every step can be verified by exact computation using the supplied scripts.

This paper’s contributions are organized as follows. Section 3 gives $\mathrm{Aut}\left(E\right)$ in explicit form and presents automorphisms of orders three and six defined over ${\mathbb{F}}_{16}$. In Section 4, the twist equation is derived using invariant theory, fixing the orientation of the Kummer action so that the descent yields $B\left(t\right)X^3$ rather than $X^3/B\left(t\right)$, and the constant c and its trace are determined. Section 5 gives integral local models for two members of the family, both K3. Section 6 computes the zeta numerators of the corresponding auxiliary curves, decomposes them under the group action, identifies the relevant invariant subspace, and determines the ranks exactly, obtaining ranks of four and two over ${\mathbb{F}}_{256}\left(t\right)$. Section 7 records two structural constraints and describes what actually governs the rank.

2. Related Work

The literature on this construction is divided along three axes: the characteristic, the degree of the twist, and the type of trivializing covering. Herein, the discussion of the literature is organized according to these principles rather than chronologically, since what distinguishes the present work is a particular combination of the three.

Large ranks over function fields. A central modern antecedent is [1], which constructs non-isotrivial elliptic curves of arbitrarily large rank over ${\mathbb{F}}_q\left(t\right)$ through quotients of Fermat surfaces. Constructions of unbounded rank over function fields are older, going back to Tate and Shafarevich and, in characteristic two, to Elkies [8]; see the Introduction in [6] and the references therein. The technique of reading ranks off an equivariant decomposition of the cohomology of a covering, which is used herein, descends from that work and from the Mordell-Weil lattice formalism of [9]. The descent that is used herein, relating Mordell-Weil groups over $k\left(t\right)$ to homomorphisms between Jacobians over k, is the subject of [10], and a survey of elliptic curves over function fields and of the Birch-Swinnerton-Dyer conjecture in this setting can be found in [11]. For Artin-Schreier coverings, the closest framework is that of Pries and Ulmer [12], who give an Artin-Schreier variant of Berger’s construction of Jacobians dominated by products of curves [13], prove the Birch-Swinnerton-Dyer conjecture for the resulting Jacobians, and express the rank through homomorphisms between Jacobians of auxiliary Artin-Schreier curves.

Twisted constant curves. A closely related construction is [4], which analyzes $y^2=x^3+t^q-t$ over ${\mathbb{F}}_r\left(t\right)$ and computes rank, regulator and Tate-Shafarevich group. That analysis is confined to characteristic $p>3$, and the restriction is structural rather than technical: the Weierstrass form $y^2=x^3+s$ does not describe a curve of j-invariant zero when $p\in \{2,3\}$, and so neither the equations nor the automorphism group carries over.

The classification of twists. The existence of sextic twists in characteristic two and their capitulation over extensions of degree six belong to the general theory of [5], which classifies twists of elliptic curves including the characteristic-two case.

Explicit twists in small characteristics. Three works are directly comparable. In characteristic two, Ref. [6] treats quartic twists of the same curve $y^2+y=x^3+x$, using the quaternion subgroup of order eight; the trivializing covering is a cyclic wild covering of degree four in Artin-Schreier standard form, presented by the tower $r^2+r=A$, $s^2+s=rA+B$, and it is their treatment of Mordell-Weil descent that we follow. In odd characteristics with $p\equiv 5\left(\mathrm{mod}6\right)$, Ref. [14] treats quartic and sextic twists; thus, they are in a characteristic where $\left|\mathrm{Aut}\right(E,O\left)\right|=6$ and both factors of a sextic covering are tame. In characteristic three, ref. [7] treats twists of degrees two, three, four, and six and derives the ranks from zeta functions of Kummer and Artin-Schreier extensions; the sextic case there has its wild part in the degree-three Artin-Schreier extension and its tame part in the degree-two Kummer extension. A related explicit computation in characteristic three, for $y^2=x^3+bx+t^{3^n+1}$, obtains the L-function from the zeta function of a superelliptic curve [15]. The present paper is the degree-six characteristic-two counterpart, in which those roles are reversed. Table 1 summarizes the comparison.

Table 1. Explicit twist constructions in small characteristics.

Work	Char.	Twist Degree	Trivializing Covering	Rank Method
[4]	$p>3$	6	tame Kummer cover $u^6=t^q-t$	explicit L-function and cohomological decomposition; BSD
[14]	$p\equiv 5\left(\mathrm{mod}6\right)$	4, 6	both factors tame	maximal curves
[6]	2	4	cyclic wild cover of degree 4, Artin-Schreier-Witt	$\mathrm{Hom}(\mathrm{Jac},E)$ descent
[7]	3	2, 3, 4, 6	wild part of degree 3, tame part of degree 2	zeta of Kummer and AS extensions
This work	2	6	wild part of degree 2, tame part of degree 3	$\mathrm{Hom}(\mathrm{Jac},E)$ descent; exact zeta factorization

Newton polygons of Artin-Schreier curves. The auxiliary curves here are Artin-Schreier. Their Newton polygons identify the part with slope $1/2$, which is necessary but not sufficient for positive rank: the rank is determined by the $C_3$-compatible geometrically E-isotypic component, its Frobenius phase, and the constant field. The p-rank stratification of the moduli of such curves is due to [16]; as shown by the examples herein, p-rank zero does not determine the polygon, and so the explicit computation cannot be replaced by an appeal to that stratification. Explicit L-polynomials are available for the van der Geer-van der Vlugt curves, a class of Artin-Schreier curves that are all supersingular [17], including a recent formula in characteristic two [18], and the zeta functions of quadratic Artin-Schreier curves in characteristic two are studied in [19]. The curves D and H attached to $S_2$ are not supersingular (Proposition 6), and so they lie outside that class, which is why their numerators are obtained here from point counts. The current state of knowledge on Newton polygons of curves is surveyed in [20].

From Tate to Birch-Swinnerton-Dyer. For surfaces dominated by a product of curves, the Tate conjecture holds [1] (Proposition 6.4 and its proof); the comparison of the Brauer group with the Tate-Shafarevich group of the generic fiber is [2], and the passage from finiteness of one ℓ-primary component to the full conjecture in a positive characteristic is [3]. Section 6.5 sets out the chain with its hypotheses.

3. The Automorphism Group

Let $E:y^2+y=x^3+x$ over ${\mathbb{F}}_2$, so that $a_3=a_4=1$ and $a_6=0$. Substituting $x=u^{2}X+r$ and $y=u^{3}Y+u^{2}sX+\tau$ and requiring that the result again be E forces $r=s^2$ and gives the following.

Proposition 1.

The automorphisms of E over ${\overline{\mathbb{F}}}_2$ fixing the origin, that is, the elements of the geometric automorphism group $\mathrm{Aut}(E_{{\overline{\mathbb{F}}}_2},O)$, are the maps

$${\phi }_{u,s,\tau }(x,y)=\left(u^{2}x+s^2,u^{3}y+u^{2}sx+\tau \right),u^3=1,s^4+s=u+1,{\tau }^2+\tau =s^6+s^2.$$

There are $3·4·2=24$ such triples. The group is $Q_8⋊C_3$, the order-four elements with $u=1$ forming the quaternion subgroup and $u=\omega$ acting on it by cyclic permutation of $i,j,k$; this is $SL(2,3)$. Throughout, “automorphism” means an automorphism fixing the origin.

The distribution of orders, verified by direct composition, is as follows. For $u=1$, one obtains the identity, one element of order two, and six elements of order four, forming the quaternion group $Q_8$. For each of $u=\omega$ and $u={\omega }^2$, one obtains four elements of order three and four of order six. The totals $8+8+8=24$ match the order distribution (one, one, six, eight, and eight) of $SL(2,3)$ for orders one, two, four, three, and six.

We fix the model ${\mathbb{F}}_{16}={\mathbb{F}}_2\left[\theta \right]/({\theta }^4+\theta +1)$ and set $\omega ={\theta }^5=\theta +{\theta }^2$, so that ${\omega }^3=1$ and ${\omega }^2+\omega =1$. With $u=\omega$, the equation $s^4+s=\omega +1={\omega }^2$ has the four solutions

$$s\in \{{\theta }^3,1+{\theta }^3,\theta +{\theta }^2+{\theta }^3,1+\theta +{\theta }^2+{\theta }^3\}\subset {\mathbb{F}}_{16},$$

and for each of these, ${\tau }^2+\tau =s^6+s^2$ has two solutions in ${\mathbb{F}}_{16}$.

The maps ${\phi }_{u,s,\tau }$ of Proposition 1 are therefore not ${\mathbb{F}}_2$-rational in general, and this study records which of them are defined over which field. Enumerating the triples in the model of ${\mathbb{F}}_{16}$ just fixed, all 24 of them have $u,s,\tau \in {\mathbb{F}}_{16}$, and so $\mathrm{Aut}(E_{{\overline{\mathbb{F}}}_2},O)$ is defined over ${\mathbb{F}}_{16}$. Exactly four triples are defined over ${\mathbb{F}}_2$, and the same four are defined over ${\mathbb{F}}_4$: those with $u=1$ and $s,\tau \in \{0,1\}$. They form the cyclic group of order four generated by $(x,y)\mapsto (x+1,y+x)$. The elements of orders three and six have $u\in \{\omega ,{\omega }^2\}\not\subset {\mathbb{F}}_2$ and $s\notin {\mathbb{F}}_4$ (since $s^4+s=0$ on ${\mathbb{F}}_4$), and so ${\mathbb{F}}_{16}$ is their field of definition. Only the automorphisms $\psi$ and $\chi$ of Lemma 1, taken over ${\mathbb{F}}_{16}$, are used from here on.

Lemma 1.

For $u=\omega$ and s any root of $s^4+s={\omega }^2$, the two roots τ of ${\tau }^2+\tau =s^6+s^2$ give automorphisms of order six and order three, respectively. Writing ψ for the former and χ for the latter, one has ${\psi }^3=[-1]$, ${\chi }^3=\mathrm{id}$ and $\psi =\chi \circ [-1]$, in accordance with the fact that in $SL(2,3)$ every element of order six is the product of an element of order three with the central involution. Of u, s, and τ, all three lie in ${\mathbb{F}}_{16}$.

Proof of Lemma 1.

Write $\phi ={\phi }_{\omega ,s,\tau }$. Composing, ${\phi }^3$ has u-component ${\omega }^3=1$ and s-component $s(1+{\omega }^2+{\omega }^4)=0$, since $1+\omega +{\omega }^2=0$ and ${\omega }^4=\omega$; an explicit composition gives ${\phi }^3(x,y)=(x,y+{\tau }^{\prime }),{\tau }^{\prime }=\tau +\omega s^3$, and ${\tau }^{\prime 2}+{\tau }^{\prime }=0$, and so ${\tau }^{\prime }\in \{0,1\}$ and ${\phi }^3\in \{\mathrm{id},[-1]\}$. Thus, $\phi$ has an order divisible by six and divisible by three. Evaluating ${\tau }^{\prime }=\tau +\omega s^3$ on the two roots of ${\tau }^2+\tau =s^6+s^2$ gives ${\tau }^{\prime }=1$ for one and ${\tau }^{\prime }=0$ for the other; thus, one has order six and the other order three. Neither can have order 1 or 2, since the u-component is $omegane1$.    □

For definiteness, fix

$$u=\omega =\theta +{\theta }^2,s={\theta }^3,{\tau }_{\psi }={\theta }^3,{\tau }_{\chi }=1+{\theta }^3.$$

(1)

4. The Sextic Twist

Let $A\in {\mathbb{F}}_{16}\left(t\right)$ and $B\in {\mathbb{F}}_{16}{\left(t\right)}^{\times }$, and let D be the normalization of the fiber product over ${\mathbb{P}}_t^1$ of

$$C_2:r^2+r=A\left(t\right)\mathrm{and}{C}_3:w^3=B\left(t\right).$$

Herein, D is equipped with the automorphism $\sigma =({\sigma }_2,{\sigma }_3)$ given by

$${\sigma }_2:r\mapsto r+1,{\sigma }_3:w\mapsto {\omega }^{2}w.$$

(2)

The choice ${\omega }^2$ rather than $\omega$ in (2) is not cosmetic; it is what makes the descent below produce $B\left(t\right)X^3$ directly, and it fixes the pairing of characters used in Section 6. Since $\gcd (2,3)=1$, one has $⟨\psi \times \sigma ⟩\cong \mathbb{Z}/2\times \mathbb{Z}/3$, and so the quotient by the diagonal group may be formed in either order.

Proposition 2.

With the orientation (2), the quotient $(E\times D)/⟨\psi \times \sigma ⟩\to {\mathbb{P}}_t^1$ has generic fiber

$$Z^2+Z=B\left(t\right)X^3+A\left(t\right)+c,c=x_0^3+x_0,x_0=s^2{\omega }^2,$$

(3)

over ${\mathbb{F}}_{16}\left(t\right)$. For the branch (1), one has $x_0=\theta$ and $c=\theta +{\theta }^3$.

Proof. 

The order-three automorphism $\chi$ fixes the line $x=x_0$ with $x_0=s^2{\omega }^2$ and, after the translation $x^{\prime }=x+x_0$, acts by $\chi (x^{\prime },y)=({\omega }^2{x}^{\prime },y+{\omega }^{2}s{x}^{\prime })$. The function $Y=y+\omega s{x}^{\prime }$ is $\chi$-invariant. With the orientation (2), the function

$$X=x^{\prime }/w$$

is invariant under $\chi \times {\sigma }_3$, since $x^{\prime }/w\mapsto {\omega }^2{x}^{\prime }/\left({\omega }^{2}w\right)=x^{\prime }/w$, and $X^3=x^{\prime 3}/B\left(t\right)$, so that

$$x^{\prime 3}=B\left(t\right)X^3.$$

Substituting into the translated Weierstrass equation, the coefficients of $x^{\prime 2}$ and of $x^{\prime }$ vanish identically as a consequence of $x_0=s^2{\omega }^2$ and $s^4+s={\omega }^2$, leaving

$$Y^2+Y=B\left(t\right)X^3+c,c=x_0^3+x_0.$$

For the Artin-Schreier step, $Z=Y+r$ is invariant under the diagonal action of $[-1]\times {\sigma }_2$ and satisfies $Z^2+Z=(Y^2+Y)+(r^2+r)=B\left(t\right)X^3+c+A\left(t\right)$.    □

Remark 1.

Had ${\sigma }_3\left(w\right)=\omega w$ been taken instead, the invariant function would be $X=x^{\prime }w$ and the descent would give $Z^2+Z=X^3/B\left(t\right)+A\left(t\right)+c$, a twist by $B^{-1}$. The two are, of course, related by replacing the Kummer datum B with $B^{-1}$, but that is a change of hypothesis, not a step in a proof, and the two choices pair with conjugate characters of $\mathbb{Z}/3$.

Remark 2.

The constant c satisfies ${\mathrm{Tr}}_{{\mathbb{F}}_{16}/{\mathbb{F}}_2}\left(c\right)=1$. Indeed, $\#\{y^2+y=x^3+x\}\left({\mathbb{F}}_{16}\right)=25$ while $\#\{y^2+y=x^3\}\left({\mathbb{F}}_{16}\right)=9$, and so the two curves are not isomorphic over ${\mathbb{F}}_{16}$, and the coordinate change above is defined over ${\mathbb{F}}_{16}$. Since the kernel of the trace on ${\mathbb{F}}_{16}$ coincides with the image of $\wp \left(z\right)=z^2+z$, both of cardinality eight, no substitution $Z\mapsto Z+h$ with $h\in {\mathbb{F}}_{16}$ removes c while leaving A fixed. One may of course absorb c into the Artin-Schreier datum by writing the auxiliary covering as $r^2+r=A\left(t\right)+c$; what is not permitted is to mix the two conventions. The four admissible branches of s give

$$c\in \{\theta +{\theta }^3,{\theta }^3,{\theta }^2+{\theta }^3,1+\theta +{\theta }^2+{\theta }^3\},$$

each of trace one, any two differing by an element of $\wp \left({\mathbb{F}}_{16}\right)$, hence giving isomorphic curves.

5. Two Elliptic Surfaces

Two members of the family (3) will be analyzed, with $S_1$ being written for the one with $A\left(t\right)=t^3+t$, $B\left(t\right)=t^2(t+1)$ and $S_2$ for the one with $A\left(t\right)=t^7+t$, $B\left(t\right)=t^{17}$, both over ${\mathbb{F}}_{16}$. In each case, setting $\overline{X}=BX$ and $\overline{Z}=BZ$ in (3) gives the long Weierstrass form

$${\overline{Z}}^2+B\overline{Z}={\overline{X}}^3+(A+c)B^2,$$

so that $a_1=a_2=a_4=0$, $a_3=B$, $a_6=(A+c)B^2$, and $\Delta =a_3^4=B^4$ with $c_4=0$ identically. Since $\Delta$ is a unit away from the zeros of B and from $t=\infty$, the reduction is good everywhere else; the bad fibers, and only those, are additive, as $c_4$ vanishes identically. The equality $j=0$ does not by itself force bad reduction.

Two criteria are used for minimality at a place v with uniformizer $\pi$. First, a non-minimal model becomes minimal after a substitution lowering $v(\Delta )$ by a positive multiple of twelve, and so $v(\Delta )<12$ already forces minimality. Second, when $v(\Delta )=12$, an argument is made directly, using the following observation.

Lemma 2.

Let R be a complete discrete valuation ring of equal characteristic two, with uniformizer π, and consider $y^2+a_{3}y=x^3+a_6$ over R. Suppose $h:=v\left(a_3\right)\ge 3$ and $m:=v\left(a_6\right)$ is odd with $m<6$. Then the model is minimal.

Proof. 

Every admissible change of variables is, up to scaling, of the form $x=x^{\prime }+r$, $y=y^{\prime }+s{x}^{\prime }+u$ with $r,s,u\in R$. Since $2=0$ in R, this yields

$$a_2^{\prime }=r+s^2,a_3^{\prime }=a_3,a_4^{\prime }=r^2+a_{3}s,a_6^{\prime }=r^3+a_6+u^2+a_{3}u.$$

A subsequent division by $({\pi }^2,{\pi }^3)$ requires $v\left(a_2^{\prime }\right)\ge 2$, $v\left(a_4^{\prime }\right)\ge 4$ and $v\left(a_6^{\prime }\right)\ge 6$. If $v\left(s\right)=0$, then $v\left(s^2\right)=0$, and so $v\left(a_2^{\prime }\right)\ge 2$ forces $v\left(r\right)=0$, whence $v\left(r^2\right)=0$ while $v\left(a_{3}s\right)=h\ge 3$, giving $v\left(a_4^{\prime }\right)=0$, a contradiction. Hence $v\left(s\right)\ge 1$, and so $v\left(s^2\right)\ge 2$ and $v\left(r\right)\ge 2$, and therefore $v\left(r^3\right)\ge 6$. It remains to show $v(a_6+u^2+a_{3}u)<6$. The three terms have valuations m, $2v\left(u\right)$, and $h+v\left(u\right)$, and the minimum of these can fail to be attained uniquely only if two of them coincide. Now m is odd and $2v\left(u\right)$ is even, and so $m\ne 2v\left(u\right)$. If $m=h+v\left(u\right)$, then $v\left(u\right)=m-h$ and $2v\left(u\right)=2m-2h<m$, since $m<6\le 2h$; the minimum is then $2v\left(u\right)$, attained uniquely, and $v\left(a_6^{\prime }\right)=2v\left(u\right)<m<6$. If $2v\left(u\right)=h+v\left(u\right)$, then $v\left(u\right)=h\ge 3$, and so both of these valuations are at least six while $m<6$, and the minimum m is attained uniquely. In every case, $v\left(a_6^{\prime }\right)<6$; therefore, no admissible change permits the division, and the model is minimal.    □

Remark 3.

The naive criterion that a model with $a_1=a_2=a_4=0$ is non-minimal exactly when $v\left(a_3\right)\ge 3$ and $v\left(a_6\right)\ge 6$ is false in characteristic two, because it ignores translations. For instance, $y^2+{\pi }^{3}y=x^3+{\pi }^2+{\pi }^4$ has $v\left(a_6\right)=2<6$, yet $y\mapsto y+\pi$ gives $y^2+{\pi }^{3}y=x^3$, which reduces to $Y^2+Y=X^3$. In Lemma 2, it is the parity of $v\left(a_6\right)$, not its size alone, that blocks the translation.

5.1. The Surface $S_1$

Here, $a_3=t^2(t+1)$ and $a_6=t^4{(t+1)}^2(t^3+t+c)$, both already integral; thus, $\Delta =t^8{(t+1)}^4$, and the bad places are $t=0$, $t=1$ and $t=\infty$.

Proposition 3.

For $S_1$, the minimal models and discriminant valuations are as follows. At $t=0$, the given model is minimal, since ${\mathrm{ord}}_t(\Delta )=8<12$, and

$$y^2+t^2(t+1)y=x^3+t^4{(t+1)}^2(t^3+t+c),{\mathrm{ord}}_t\left({\Delta }_{\min }\right)=8.$$

At $t=1$, the given model is minimal, since ${\mathrm{ord}}_{t+1}(\Delta )=4<12$, and ${\mathrm{ord}}_{t+1}\left({\Delta }_{\min }\right)=4$. At $t=\infty$, with $\varsigma =1/t$ one, has ${\mathrm{ord}}_{\varsigma }\left(a_3\right)=-3$ and ${\mathrm{ord}}_{\varsigma }\left(a_6\right)=-9$, and so integrality forces the substitution $\overline{X}={\varsigma }^{-4}x$, $\overline{Z}={\varsigma }^{-6}y$, giving

$$y^2+{\varsigma }^3(1+\varsigma )y=x^3+{\varsigma }^3{(1+\varsigma )}^2(1+{\varsigma }^2+c{\varsigma }^3),{\mathrm{ord}}_{\varsigma }\left({\Delta }_{\min }\right)=12,$$

which is minimal by Lemma 2 with $h=3$ and $m=3$. Consequently, $\mathrm{deg}{\Delta }_{\min }=8+4+12=24$, and so $\mathrm{deg}\mathcal{L}=\chi \left({\mathcal{O}}_{S_1}\right)=2$. The fibration has a section–hence, no multiple fibers–and the canonical bundle formula gives $K_{S_1}={\pi }^{*}(\mathcal{L}\otimes {\omega }_{{\mathbb{P}}^1})={\mathcal{O}}_{S_1}$. Then $K_{S_1}\simeq {\mathcal{O}}_{S_1}$ forces $p_g=1$, and $\chi \left({\mathcal{O}}_{S_1}\right)=1-q+p_g=2$ gives $q=0$; therefore, $S_1$ is a K3 surface.

Note that ${\mathrm{ord}}_t\left(a_6\right)=4$ at $t=0$ because ${\mathrm{ord}}_t(A+c)=0$, which holds precisely because $c\ne 0$; minimality there, however, already follows from ${\mathrm{ord}}_t(\Delta )=8<12$.

5.2. The Surface $S_2$

Fix $B\left(t\right)=t^{17}$ and $A\left(t\right)=t^7+t$. The long Weierstrass form is

$${\overline{Z}}^2+t^{17}\overline{Z}={\overline{X}}^3+t^{41}+t^{35}+c{t}^{34},$$

so that $a_3=t^{17}$, $a_6=t^{41}+t^{35}+c{t}^{34}$ and $\Delta =t^{68}$, the bad places being $t=0$ and $t=\infty$.

Proposition 4.

For $S_2$, the integral local models are as follows. At $t=0$, the substitution $\overline{X}=t^{10}x$, $\overline{Z}=t^{15}y$ gives

$$y^2+t^{2}y=x^3+t^{11}+t^5+c{t}^4,{\mathrm{ord}}_t\left({\Delta }_{\min }\right)=8.$$

At $t=\infty$, with $\varsigma =1/t$ and $\overline{X}=t^{14}x$, $\overline{Z}=t^{21}y$,

$$y^2+{\varsigma }^{4}y=x^3+\varsigma +{\varsigma }^7+c{\varsigma }^8,{\mathrm{ord}}_{\varsigma }\left({\Delta }_{\min }\right)=16.$$

Consequently, $\mathrm{deg}{\Delta }_{\min }=24$ and $\chi \left({\mathcal{O}}_{S_2}\right)=2$, and the argument given for $S_1$ applies verbatim; therefore, $S_2$ is a K3 surface.

Minimality holds at $t=0$ because ${\mathrm{ord}}_t(\Delta )=8<12$. At $\varsigma =0$, one has ${\mathrm{ord}}_{\varsigma }\left(a_3\right)=4$ and ${\mathrm{ord}}_{\varsigma }\left(a_6\right)=1$, and so Lemma 2 applies with $h=4$ and $m=1$. Since $K_{S_i}\simeq {\mathcal{O}}_{S_i}$, the K3 surfaces $S_1$ and $S_2$ contain no $(-1)$-curves and are therefore minimal, and the elliptic fibrations are relatively minimal because the Weierstrass models above are minimal at every place.

Herein, no Kodaira symbols are assigned to the bad fibers of either surface. Nothing below depends on them: the degree of the L-polynomial and the rank are both obtained in Section 6 from the cohomology of the auxiliary curve, with no reference to fiber types or component counts. Readers wishing to classify them may apply Tate’s algorithm [21] in the form valid in residue characteristic two, where the $j=0$ case requires the wild subcases.

6. Cohomology and Rank

6.1. The Auxiliary Curve of $S_1$

For $S_1$, the Kummer covering is $C_3:w^3=t^2(t+1)$. It is ramified exactly at $t=0$ and $t=1$, since $\mathrm{ord}\left(B\right)$ is two and one there, while ${\mathrm{ord}}_{\infty }\left(B\right)=-3$ is divisible by three, such that $C_3$ is unramified above $t=\infty$; Riemann-Hurwitz gives $g\left(C_3\right)=0$ with three geometric points above infinity. On $C_3$, the function t has a simple pole at each of those three points; therefore, $A=t^3+t$ has a pole of order three there, and the Artin-Schreier covering $D\to C_3$ is wild there with a different exponent $(2-1)(3+1)=4$ at each point. Therefore,

$$2g\left(D\right)-2=2(2·0-2)+3·4=8,g\left(D\right)=5.$$

Over ${\mathbb{F}}_2$, the three points above infinity split as one rational place together with one closed place of degree two. Indeed, with $\varsigma =1/t$ and $u=\varsigma w$, one has $u^3=1+\varsigma$, and so the points of $C_3$ above $t=\infty$ correspond to $u\in \{1,\omega ,{\omega }^2\}$, the three cube roots of unity. They are all rational over ${\mathbb{F}}_{2^m}$ exactly when $\omega \in {\mathbb{F}}_{2^m}$, that is, when m is even, and for odd m only $u=1$ is rational. Since the Artin-Schreier covering $D\to C_3$ is totally ramified above each of them, as the pole of A has odd order three, the same count holds for D: one rational place for odd m and three for even m.

Proposition 5.

For $S_1$, the numerator of the zeta function of D over ${\mathbb{F}}_2$ is

$$P_D\left(T\right)=1+2T+3T^2+2T^3-2T^4-8T^5-4T^6+8T^7+24T^8+32T^9+32T^{10},$$

with Newton slopes 0, $1/2$, and one of multiplicities two, six, and two. The quotient $D/⟨{\sigma }_3⟩$ is the supersingular elliptic curve $r^2+r=t^3+t$, with numerator $1+2T+2T^2$, and

$$Q_1\left(T\right)={\displaystyle \frac{P_D\left(T\right)}{1+2T+2T^2}}={(1+2T^2)}^2(4T^4-3T^2+1),$$

(4)

of degree eight, whose Newton slopes are 0, $1/2$, and one of multiplicities two, four, and two.

The factor $1+2T^2$ has Frobenius eigenvalues $\alpha$ with ${\alpha }^2=-2$ (hence ${\alpha }^4=4$) whereas E over ${\mathbb{F}}_{16}$ has Frobenius eigenvalues $-4$. The corresponding elliptic factor is therefore not isogenous to E over ${\mathbb{F}}_{16}$; it becomes so over ${\mathbb{F}}_{256}$. This is precisely why the rank is zero over ${\mathbb{F}}_{16}\left(t\right)$ and positive over ${\mathbb{F}}_{256}\left(t\right)$, and we accordingly speak of geometrically E-isotypic factors.

The reconstruction of $P_D$ from point counts satisfies the functional equation, the Riemann hypothesis, and the predictive test in Section 9, which, for $S_1$, is available because counts beyond $m=2g$ were computed. The resulting genus agrees with the Riemann-Hurwitz value above. The essential point in (4) is that the factor $1+2T^2$ occurs squared in the non-invariant part, so that the non-invariant slope-$1/2$ multiplicity is four rather than two.

6.2. The Auxiliary Curve of $S_2$

Since $\gcd (17,3)=1$, the covering $C_3:w^3=t^{17}$ has rational normalization: with $v=t^6/w$, one obtains $v^3=t^{18}/t^{17}=t$, such that $t=v^3$ and $w=v^{17}$, and $C_3\cong {\mathbb{P}}_v^1$. Under the orientation (2), $w\mapsto {\omega }^{2}w$ induces

$$v=t^6/w⟼{\omega }^{-2}v=\omega v,$$

and so ${\sigma }_3$ acts on ${\mathbb{P}}_v^1$ by $v\mapsto \omega v$. The fiber product therefore normalizes to the Artin-Schreier curve

$$D:r^2+r=v^{21}+v^3,$$

(5)

which is preserved by $v\mapsto \omega v$ because $3\mid 21$ and $3\mid 3$, and by ${\sigma }_2:r\mapsto r+1$. As 21 is odd, D has a single place above $v=\infty$, and $g\left(D\right)=10$.

6.3. Zeta Data

Proposition 6.

Over ${\mathbb{F}}_2$, the numerator of the zeta function of D is

$$\begin{array}{cc}\hfill P_D\left(T\right)=& 1+2T+4T^2+6T^3+12T^4+16T^5+28T^6\hfill \\ & +32T^7+56T^8+72T^9+128T^{10}+\dots +1024T^{20},\hfill \end{array}$$

the remaining coefficients being determined by $a_{20-i}=2^{10-i}{a}_i$. Its Newton polygon with respect to $q=2$ has slopes $1/3$, $1/2$, and $2/3$ with multiplicities nine, two, and nine.

The quotient $H=D/⟨{\sigma }_3⟩$ is $r^2+r={\varsigma }^7+\varsigma$ with $\varsigma =v^3$, of genus three, with

$$P_H\left(T\right)=1+2T+4T^2+6T^3+8T^4+8T^5+8T^6$$

and Newton slopes $1/3$ and $2/3$, each of multiplicity three. Neither D nor H is supersingular.

Proposition 7.

$P_H$ divides $P_D$ exactly, and the quotient

$$Q\left(T\right)=P_D\left(T\right)/P_H\left(T\right)=1+4T^4+4T^6+8T^8+32T^{10}+128T^{14}$$

of degree 14 admits the exact factorization

$$Q\left(T\right)=(1+2T^2)R\left(T\right),R\left(T\right)=1-2T^2+8T^4-12T^6+32T^8-32T^{10}+64T^{12},$$

(6)

in which R has Newton slopes $1/3$ and $2/3$ only, each of multiplicity six.

Thus, $H^1\left(D\right)$ decomposes under ${\sigma }_3$ into the invariant part $H^1\left(H\right)$, of dimension six, and the sum of the two nontrivial eigenspaces, of dimension fourteen; within the latter, (6) isolates a single supersingular elliptic factor. It is important to note that the multiplicity of slope $1/2$ does not by itself identify an elliptic factor; the explicit factorization (6) is what does so here.

6.4. The Invariant Subspace

The two involved statements are separated, since only the first is needed for the ranks.

Proposition 8

(Mordell-Weil descent). Let $E_{\mathrm{tw}}$ be the generic fiber of (3). Then,

$$E_{\mathrm{tw}}\left({\overline{\mathbb{F}}}_2\left(t\right)\right)\otimes \mathbb{Q}\cong {\left(\mathrm{Hom}(\mathrm{Jac}D,E)\otimes \mathbb{Q}\right)}^{C_6}.$$

Proof. 

Over ${\overline{\mathbb{F}}}_2\left(t\right)$, the twist becomes trivial after the base change to the function field of D, by construction of D as the covering trivializing the cocycle defining $\psi \times \sigma$. Galois descent for the torsor therefore identifies $E_{\mathrm{tw}}\left({\overline{\mathbb{F}}}_2\left(t\right)\right)$ with the group of $C_6$-equivariant morphisms $D\to E$, where $C_6$ acts on D through $\sigma$ and on E through $\psi$. The Albanese sequence

$$0⟶E\left({\overline{\mathbb{F}}}_2\right)⟶\mathrm{Mor}(D,E)⟶\mathrm{Hom}(\mathrm{Jac}D,E)⟶0$$

is $C_6$-equivariant, and $E\left({\overline{\mathbb{F}}}_2\right)$ is torsion; therefore, it vanishes upon tensoring with $\mathbb{Q}$. Only invertibility of $|C_6|$ in $\mathbb{Q}$ is needed here; taking $C_6$-invariants is exact on $\mathbb{Q}$-vector spaces; and the claim follows. This is the sextic analog of the descent used in §2 in [6], whose rank decomposition is carried out in §4 in [6]; the decomposition of a Mordell-Weil group into isotypic pieces under a group action goes back to [9]; for the form with homomorphisms between Jacobians, see [10,12].    □

Combined with Tate’s isogeny theorem [22], Proposition 8 is what we use to prove Theorem 1; the argument refers neither to the surface nor to Birch-Swinnerton-Dyer. Indeed, Tate’s theorem [22] identifies

$${\mathrm{Hom}}_{{\mathbb{F}}_{{16}^n}}(\mathrm{Jac}D,E)\otimes {\mathbb{Q}}_{\ell }\cong {\mathrm{Hom}}_{{\mathbb{Q}}_{\ell }\left[F_{16}^n\right]}\left(V_{\ell }\left(\mathrm{Jac}D\right),V_{\ell }\left(E\right)\right),$$

the homomorphisms of ℓ-adic Tate modules commuting with Frobenius, so that an isogeny factor of $\mathrm{Jac}D$ contributes exactly when its Frobenius eigenvalues agree with those of E over the field in question. Taking $C_6$-invariants imposes the corresponding character compatibility. The two descriptions used in this paper agree. After extending scalars to ${\overline{\mathbb{Q}}}_{\ell }$, so that the nontrivial characters of $C_3$ are defined, one has

$$H^1\left(D,{\overline{\mathbb{Q}}}_{\ell }\right)\simeq V_{\ell }{\left(\mathrm{Jac}D\right)}^{\vee },V_{\ell }\left(E\right)\simeq H^1\left(E,{\overline{\mathbb{Q}}}_{\ell }\right)\left(1\right),$$

the second by the polarization of E, whence

$$\mathrm{Hom}\left(V_{\ell }\left(\mathrm{Jac}D\right),V_{\ell }\left(E\right)\right)\simeq H^1\left(D\right)\otimes H^1\left(E\right)\left(1\right).$$

Since the Tate twist is trivial as a $C_6$-representation, the two descriptions have the same $C_6$-isotypic multiplicities; the twist does affect the Frobenius action, and Frobenius eigenvalues are therefore compared separately, through Tate’s theorem and (7). Passing to a fixed constant field,

$$E_{\mathrm{tw}}\left({\mathbb{F}}_{{16}^n}\left(t\right)\right)\otimes \mathbb{Q}\cong {\left[{\left({\mathrm{Hom}}_{{\overline{\mathbb{F}}}_2}(\mathrm{Jac}D,E)\otimes \mathbb{Q}\right)}^{C_6}\right]}^{F_{16}^n},$$

(7)

where $F_{16}$ denotes the geometric Frobenius of ${\mathbb{F}}_{16}$. Therefore, the rank over ${\mathbb{F}}_{16}\left(t\right)$ and over ${\mathbb{F}}_{256}\left(t\right)$ is the dimension of the subspace fixed by $F_{16}$ and by $F_{16}^2$, respectively.

The identification of the L-polynomial is a separate matter, and it is stated as such herein.

Proposition 9.

Let $U\subset {\mathbb{P}}_t^1$ be the complement of the branch locus of $\overline{f}:D\to {\mathbb{P}}_t^1$; in both examples, this branch locus is exactly the bad-reduction locus. Let π be the elliptic fibration over U, let $f:D_U\to U$ be the restriction of $\overline{f}$, and let $j:U\hookrightarrow {\mathbb{P}}_t^1$ be the inclusion. Then,

$$R^1{\pi }_{*}{\mathbb{Q}}_{\ell }\cong {\left(H^1(E_{{\overline{\mathbb{F}}}_2},{\mathbb{Q}}_{\ell })\otimes f_{*}{\mathbb{Q}}_{\ell }\right)}^{C_6},$$

and

$$H^1\left({\mathbb{P}}_{{\overline{\mathbb{F}}}_2}^1,j_{*}{R}^1{\pi }_{*}{\mathbb{Q}}_{\ell }\right)\cong {\left(H^1\left(E\right)\otimes H^1\left(D\right)\right)}^{C_6},$$

(8)

so that the L-polynomial of $E_{\mathrm{tw}}$ is the characteristic polynomial of Frobenius on the right-hand side of (8).

Proof. 

Over U, the fibration $\pi$ is the twist of the constant curve $E\times U$ by the $C_6$-torsor f, and so $R^1{\pi }_{*}{\mathbb{Q}}_{\ell }$ is obtained from the constant sheaf $H^1\left(E\right)\otimes {\mathbb{Q}}_{\ell }$ by descent along f, the invariants being taken for the diagonal action; this is the first isomorphism. Fix $\ell \ne 2$. Since ${\mathbb{Q}}_{\ell }$ has characteristic zero, taking $C_6$-invariants is exact, and it commutes with pushforward.

Since $\overline{f}$ is finite, ${\overline{f}}_{*}$ is exact and $j_{*}{f}_{*}{\mathbb{Q}}_{\ell }\simeq {\overline{f}}_{*}{\mathbb{Q}}_{\ell }$, whence

$$R\mathsf{\Gamma }\left({\mathbb{P}}^1,j_{*}{R}^1{\pi }_{*}{\mathbb{Q}}_{\ell }\right)\simeq {\left(H^1\left(E\right)\otimes R\mathsf{\Gamma }(D,{\mathbb{Q}}_{\ell })\right)}^{C_6}.$$

Taking $H^1$ gives (8).    □

The degree-zero and degree-two parts of that complex vanish, since they are $H^i\left({\mathbb{P}}^1\right)\otimes H^1{\left(E\right)}^{C_6}$ with $H^1{\left(E\right)}^{C_6}=0$: indeed, ${\psi }^3=[-1]$, such that for $\ell \ne 2$ the representation of $\psi$ on $H^1(E,{\mathbb{Q}}_{\ell })$ has no eigenvalue of one, and the degree-two part vanishes by Poincaré duality. Consequently, the Grothendieck product reduces to the determinant in degree one, which identifies the characteristic polynomial of Frobenius in (8) with the L-polynomial. The geometric Frobenius is used throughout.

The character decomposition is as follows. The $C_2$-factor acts on $H^1\left(E\right)$ through $[-1]$ and on $H^1\left(D\right)$ through ${\sigma }_2$; since $g\left(C_3\right)=0$ for $S_1$ and $S_2$, so that $H^1{\left(D\right)}^{{\sigma }_2}=H^1\left(C_3\right)=0$, the whole of $H^1\left(D\right)$ is anti-invariant, and the $C_2$-condition in (8) is satisfied identically (for a Kummer curve of positive genus, this fails; see Section 7). The $C_3$-factor acts on $H^1\left(E\right)$ through $\chi$ by the two nontrivial characters of $\mathbb{Z}/3$, one on each of the two Frobenius eigenlines, and on $H^1\left(D\right)$ through ${\sigma }_3$. Invariance therefore pairs each eigenline of $H^1\left(E\right)$ with the conjugate eigenspace of $H^1\left(D\right)$, so that (8) has dimension equal to that of the sum of the two nontrivial ${\sigma }_3$-eigenspaces, namely, $7+7=14$ for $S_2$ and $4+4=8$ for $S_1$.

We do not use the quotient surface in the proof of Theorem 1. This matters here: the quotient $(E\times D)/C_6$ has fixed points for the order-two subgroup, whose order is divisible by the characteristic; therefore, the quotient singularities are potentially wild, and their resolution cannot be assumed to behave as in the tame case. The restriction $\ell \ne 2$ comes from the characteristic of the base field, while the exactness of $C_6$-invariants follows from $\mathrm{char}{\mathbb{Q}}_{\ell }=0$. Neither fact controls the geometry of the wild quotient singularities.

Before this theorem is proven, the equivariant step is isolated, so that the rank appears as an equality of dimensions of spaces that are simultaneously $C_6$-equivariant and Frobenius-fixed.

Lemma 3.

Work over ${\mathbb{Q}}_{\ell }\left(\omega \right)$ and assume $H^1{\left(D\right)}^{{\sigma }_2}=0$, which holds for $S_1$ and $S_2$ because $C_3$ has genus zero. Write $H^1{\left(D\right)}_{\mathrm{noninv}}=V_{\omega }\oplus V_{{\omega }^2}$ for the decomposition of the non-invariant part into the two nontrivial ${\sigma }_3$-eigenspaces, and write $H^1\left(E\right)=H^1{\left(E\right)}_{\omega }\oplus H^1{\left(E\right)}_{{\omega }^2}$ for the corresponding decomposition under χ. Then,

$${\left(H^1\left(E\right)\otimes H^1\left(D\right)\right)}^{C_3}=\left(H^1{\left(E\right)}_{\omega }\otimes V_{{\omega }^2}\right)\oplus \left(H^1{\left(E\right)}_{{\omega }^2}\otimes V_{\omega }\right),$$

the $C_2$-condition being automatic because $H^1{\left(D\right)}^{{\sigma }_2}=0$. Frobenius over ${\mathbb{F}}_{16}$ preserves each summand. Each copy of the factor $1+2T^2$ in the non-invariant zeta factor Q contributes one dimension to $V_{\omega }$ and one to $V_{{\omega }^2}$-hence, one dimension to each summand. A reciprocal root α occurring there is Frobenius-fixed over ${\mathbb{F}}_{{16}^n}$ precisely when

$${\alpha }^{4n}={(-4)}^n.$$

(9)

Proof. 

Invariance under ${\sigma }_3$ acting diagonally forces the $\chi$-character on $H^1\left(E\right)$ and the ${\sigma }_3$-character on $H^1\left(D\right)$ to be inverses of one another, which gives the displayed decomposition; the trivial character contributes nothing, since $\chi$ has order three and $H^1{\left(E\right)}^{\chi }=0$. For the $C_2$-condition, $H^1{\left(D\right)}^{{\sigma }_2}=H^1\left(C_3\right)=0$, and so every class is anti-invariant and pairs with the action of $[-1]$ on $H^1\left(E\right)$. Frobenius over ${\mathbb{F}}_{16}$ commutes with the group action, since $\omega \in {\mathbb{F}}_{16}$ and the action is defined over ${\mathbb{F}}_{16}$, and therefore preserves each isotypic summand; Frobenius over ${\mathbb{F}}_2$ instead interchanges $V_{\omega }$ and $V_{{\omega }^2}$, which is why a primary subspace corresponding to ${(1+2T^2)}^r$ has dimension r in each of the two characters. Two representations must be kept apart here. The L-polynomial is read on $H^1\left(E\right)\otimes H^1\left(D\right)$, whereas the fixed points of Hom are read on the twisted representation $H^1\left(D\right)\otimes H^1\left(E\right)\left(1\right)$. Now, $\#E\left({\mathbb{F}}_{16}\right)=25$; therefore, Frobenius acts on $H^1\left(E\right)$ by the scalar $-4$ over ${\mathbb{F}}_{16}$ and on an $\alpha$-eigenline of $H^1\left(D\right)$ by ${\alpha }^4$. On the twisted representation, it therefore acts by $(-4){\alpha }^4/16$ over ${\mathbb{F}}_{16}$ (hence, by ${\left((-4){\alpha }^4/16\right)}^n$ over ${\mathbb{F}}_{{16}^n}$), and

$${\displaystyle \frac{{(-4)}^n{\alpha }^{4n}}{{16}^n}}=1⟺{\alpha }^{4n}={\displaystyle \frac{{16}^n}{{(-4)}^n}}={(-4)}^n,$$

which is (9). Omitting the twist would give the incorrect condition ${\alpha }^{4n}={(-4)}^{-n}$.    □

Theorem 1.

With the same notation as above,

$$\mathrm{rank}{E}_{\mathrm{tw},1}\left({\mathbb{F}}_{16}\left(t\right)\right)=0,\mathrm{rank}{E}_{\mathrm{tw},1}\left({\mathbb{F}}_{256}\left(t\right)\right)=4,$$

$$\mathrm{rank}{E}_{\mathrm{tw},2}\left({\mathbb{F}}_{16}\left(t\right)\right)=0,\mathrm{rank}{E}_{\mathrm{tw},2}\left({\mathbb{F}}_{256}\left(t\right)\right)=2,$$

Proof. 

The argument proceeds from (7). By Proposition 8 and Tate’s theorem, the rank over ${\mathbb{F}}_{{16}^n}\left(t\right)$ is the dimension of the $F_{16}^n$-fixed subspace of ${\left(\mathrm{Hom}(\mathrm{Jac}D,E)\otimes \mathbb{Q}\right)}^{C_6}$, which by Lemma 3 is the number of reciprocal roots $\alpha$ of Q, counted in the two isotypic summands, satisfying (9).

This argument is the proof of the theorem. The order of vanishing of the L-polynomial in Section 6.5, via the Artin-Tate formalism and the theorem of Kato and Trihan, is a second, independent route that serves only as confirmation. In passing from the number of reciprocal roots to a dimension, we use the fact that Frobenius acts semisimply on $V_{\ell }\left(\mathrm{Jac}D\right)$ for an abelian variety over a finite field [22]. Hence every power $F_{16}^n$ is again semisimple, and on each $C_6$-isotypic summand, which Frobenius preserves by Lemma 3, the dimension of the $F_{16}^n$-fixed subspace equals the algebraic multiplicity of the eigenvalue 1 on the twisted representation, that is, the number of reciprocal roots $\alpha$ satisfying (9), counted with multiplicity.

Direct counting gives $\#E\left({\mathbb{F}}_{16}\right)=25$–hence, $a_E=-8$ and ${\beta }_1={\beta }_2=-4$: Frobenius acts on both lines of $H^1\left(E\right)$ by $-4$. A factor of $\mathrm{Jac}D$ therefore contributes over ${\mathbb{F}}_{{16}^n}$ precisely when the corresponding $\alpha$ satisfies (9).

The reciprocal roots of $1+2T^2$ satisfy ${\alpha }^2=-2$; hence, ${\alpha }^4=4$. For $n=1$, this gives ${\alpha }^4=4\ne -4$; thus, nothing is contributed, and the rank over ${\mathbb{F}}_{16}\left(t\right)$ is zero. For $n=2$, it gives ${\alpha }^8=16={(-4)}^2$, and so each such $\alpha$ contributes. By (4), the factor $1+2T^2$ occurs squared in Q for $S_1$, accounting for four reciprocal roots, and by (6), it occurs once for $S_2$, accounting for two.

The remaining factors contribute nothing for either n. For $S_2$, the complementary factor R has Newton slopes $1/3$ and $2/3$, such that $v_2\left(\alpha \right)\in \{1/3,2/3\}$ and $v_2\left({\alpha }^{4n}\right)\in \{4n/3,8n/3\}$, whereas $v_2\left({(-4)}^n\right)=2n$; these never coincide, and so ${\alpha }^{4n}={(-4)}^n$ is impossible. For $S_1$, the complementary factor $4T^4-3T^2+1$ has Newton slopes zero and one, and so its reciprocal roots are a unit and two times a unit; in either case, ${\alpha }^{4n}$ has 2-adic valuation 0 or $4n$, while ${(-4)}^n$ has valuation $2n$, and so the equality fails again. The ranks are therefore zero and four for $S_1$ and zero and two for $S_2$.    □

Explicitly, writing $Q^{\left[4\right]}\left(U\right)={\prod }_{\alpha }(1-{\alpha }^{4}U)$ for the fourth-power transform of the non-invariant factor, one has $L\left(T\right)=Q^{\left[4\right]}(-4T)$, and for $S_1$,

$$L_{S_1}\left(T\right)={(1+16T)}^4\left(1+8T+528T^2+2048T^3+65536T^4\right),$$

which does not vanish at $T={16}^{-1}$. Over the quadratic constant extension, the eigenvalues are squared, such that $L_{S_1/{\mathbb{F}}_{256}}\left(T\right)={\prod }_i(1-{\lambda }_i^{2}T)$, and one computes

$$L_{S_1/{\mathbb{F}}_{256}}\left(T\right)={(1-256T)}^4{\left(1+496T+65536T^2\right)}^2,$$

which vanishes at $T={256}^{-1}$ to an order of exactly four, in agreement with Theorem 1.

The reciprocal roots of Q have absolute value $\sqrt{2}$, as they must; the roots of Q itself have absolute value $1/\sqrt{2}$.

For $S_2$, as an auxiliary check and not as part of the proof, a bounded-degree search is performed for sections over ${\mathbb{F}}_{16}\left(t\right)$; none are found. Since $Z\mapsto Z^2+Z$ is ${\mathbb{F}}_2$-linear, for each candidate $X\left(t\right)$, the existence of a corresponding $Z\left(t\right)$ is an ${\mathbb{F}}_2$-linear system that can be decided exactly; this was carried out for X polynomial of degrees one and three and for $X=Y\left(t\right)/t^k$ with $k\le 5$ and $\mathrm{deg}Y\le 3$. The degree bound is set as follows: writing $\mathrm{deg}X=a$, the right-hand side of (3) has degree $17+3a$, which must be even for Z to be a polynomial; therefore, only odd a occur, and $\mathrm{deg}Z=(17+3a)/2$. Thus, $a=1$ and $a=3$ give $\mathrm{deg}Z=10$ and 13, while the next case $a=5$ already needs $\mathrm{deg}Z=16$, the linear system growing as $4(\mathrm{deg}Z+1)$ over ${\mathbb{F}}_2$; that is where this study stopped, and nothing is claimed outside this range. The absence of such sections is consistent with Theorem 1 but is not part of its proof.

For $S_2$, the same transforms give

$$\begin{array}{cc}\hfill L_{S_2/{\mathbb{F}}_{16}}\left(T\right)& ={(1+16T)}^{2}g{\left(T\right)}^2,\hfill \\ \hfill g\left(T\right)& =1-48T+1280T^2-23552T^3+327680T^4-3145728T^5+16777216T^6,\hfill \end{array}$$

which does not vanish at $T={16}^{-1}$, and

$$\begin{array}{cc}\hfill L_{S_2/{\mathbb{F}}_{256}}\left(T\right)& ={(1-256T)}^{2}h{\left(T\right)}^2,\hfill \\ \hfill h\left(T\right)& =1+256T+32768T^2+15728640T^3\hfill \\ & +2147483648T^4+1099511627776T^5+281474976710656T^6,\hfill \end{array}$$

which vanishes at $T={256}^{-1}$ to order exactly two. The sextic factors g and h are the fourth- and eighth-power transforms of R, whose Newton slopes $1/3$ and $2/3$ keep their reciprocal roots away from the central value; the orders of vanishing zero and two therefore match Theorem 1 for $S_2$ exactly as they did for $S_1$.

6.5. Birch-Swinnerton-Dyer

The proof of Theorem 1 above does not use this subsection, and readers interested only in the ranks may skip it. The chain is set out in the order in which it is used, with the hypotheses made explicit.

First, the minimal regular model $S_i$ of $E_{\mathrm{tw},i}$ over ${\mathbb{P}}_t^1$ is birationally dominated by $E\times D$: the quotient map $E\times D\to (E\times D)/C_6$ is generically finite, and $(E\times D)/C_6$ is birational to $S_i$, since the two have the same function field. Birational domination is all that is required, and so the wild quotient singularities, which are not analyzed here, do not enter.

Second, the Tate conjecture for divisors is a birational invariant of smooth projective surfaces over a finite field and passes from a product of curves to any surface it dominates; for elliptic surfaces dominated by a product of curves, this is Proposition 6.4, and its proof in [1]. It holds over every finite extension of the constant field, since the domination is defined over ${\mathbb{F}}_{16}$. Explicitly, E (with its model over ${\mathbb{F}}_{16}$ and the automorphism $\psi$), the curve D (with ${\sigma }_2$ and ${\sigma }_3$) and, hence, the group $C_6=⟨\psi \times \sigma ⟩$ are all defined over ${\mathbb{F}}_{16}$, because $\omega ,\theta \in {\mathbb{F}}_{16}$, $A,B\in {\mathbb{F}}_2\left[t\right]$ in both examples, and the formulas of Lemma 1 and (2) have coefficients in ${\mathbb{F}}_{16}$. The quotient map $E\times D\to (E\times D)/C_6$ and the birational identification of $(E\times D)/C_6$ with $S_i$, whose generic fiber (3) lies over ${\mathbb{F}}_{16}\left(t\right)$, are therefore defined over ${\mathbb{F}}_{16}$, and base change exhibits $S_i\otimes {\mathbb{F}}_{{16}^n}$ as dominated by the product of curves $(E\times D)\otimes {\mathbb{F}}_{{16}^n}$ for every $n\ge 1$.

Third, by Theorem 4.1 in [2], the Tate conjecture for $S_i$ is equivalent to the finiteness of $\mathrm{Br}\left(S_i\right)$, and under the Artin-Tate comparison, the ℓ-primary component of $\mathrm{Br}\left(S_i\right)$ is identified, for $\ell \ne p$, with that of $\mathrm{III}(E_{\mathrm{tw},i}/{\mathbb{F}}_{{16}^n}\left(t\right))$. This discussion uses $\ell \ne 2$ throughout.

Fourth, by the theorem of Kato and Trihan [3], if the ℓ-primary part of $\mathrm{III}$ is finite for a single prime ℓ, then the full Birch-Swinnerton-Dyer conjecture holds for $E_{\mathrm{tw},i}$ over the function field in question, including the leading-term formula. In particular, the algebraic rank equals the order of vanishing of $L(E_{\mathrm{tw},i},T)$ at the central point, which, in the variable T used here, is $T=q^{-1}$, with $q=16$ or $q=256$ according to the constant field.

Consequently, the L-polynomials computed above provide independent confirmation of the ranks stated in Theorem 1.

7. Two Constraints and a Comparison

7.1. Monomial Kummer Coverings Have Rational Normalization

For $w^3=t^d$ with $\gcd (d,3)=1$, choose $\lambda ,\mu$ with $\lambda d+3\mu =1$ and set $v=t^{\mu }{w}^{\lambda }$; then $v^3=t$ and w is a power of v, so the normalization is ${\mathbb{P}}_v^1$ and $H^1\left(C_3\right)=0$. No divisibility relation between d and $q+1$ affects this. It is important to note what this does and does not say. The covering remains of degree three, and the associated cubic base change still contributes to the construction: in the case treated above, $C_3$ has rational normalization, yet the nontrivial part of the ${\sigma }_3$-action on $H^1\left(D\right)$ has dimension fourteen. It is the interaction of the cubic action with the Artin-Schreier covering that produces this cohomology, not $H^1\left(C_3\right)$ itself.

7.2. Artin-Schreier Curves Need Not Be Supersingular

Propositions 6 and 7 give explicit counterexamples: both D and H have Newton slopes $1/3$ and $2/3$. Both have p-rank zero, which already shows that the p-rank does not determine the Newton polygon; the stratification by p-rank of the moduli of Artin-Schreier curves [16] is therefore not by itself decisive here, and the explicit computation settles the question.

7.3. What Actually Governs the Rank

Newton slope $1/2$ identifies candidate factors only. It does not govern the rank, as $S_1$ itself shows: its non-invariant part has slope-$1/2$ multiplicity four, yet the rank over ${\mathbb{F}}_{16}\left(t\right)$ is zero. What governs the rank is the multiplicity of the geometrically E-isotypic component of the part of $H^1\left(D\right)$ on which ${\sigma }_2$ acts as $-1$ and ${\sigma }_3$ acts through a nontrivial character, these being the two conditions imposed by the actions of $[-1]$ and of $\chi$ on $H^1\left(E\right)$, together with the phase of its Frobenius eigenvalues and the constant field: the rank counts those eigenvalues that reach the central value q over the field considered. Slope $1/2$ is a necessary condition on the factor, but not a sufficient one, and a search measuring only the slope gives misleading signals. A third family member makes the point. For $A\left(t\right)=t^3+t$ and $B\left(t\right)=t(t+1)$ one finds $g\left(D\right)=6$ and

$$P_D\left(T\right)=(1+2T+2T^2)(1+2T^2)(16T^8-8T^6+2T^4-2T^2+1),$$

of total slope-$1/2$ multiplicity four, against six for $S_1$ (Proposition 5). The two constructions are nevertheless not comparable, because here the factor $1+2T+2T^2$ is the numerator of the invariant quotient $D/⟨{\sigma }_3⟩$, again the supersingular curve $r^2+r=t^3+t$, and contributes nothing to (8), while the remaining degree-eight factor has slopes $1/4$ and $3/4$ only. The comparison must be read with both group factors in view. Here, unlike for $S_1$ and $S_2$, the Kummer curve $C_3:w^3=t(t+1)$ is ramified at $t=0$, $t=1$ and $t=\infty$, so that $2g\left(C_3\right)-2=3(-2)+3·2=0$ and $C_3$ is an elliptic curve, isomorphic to $y^2+y=x^3$, with zeta numerator $1+2T^2$. Hence, $H^1{\left(D\right)}^{{\sigma }_2}=H^1\left(C_3\right)\ne 0$, and the assertion of Section 6 that all of $H^1\left(D\right)$ is ${\sigma }_2$-anti-invariant does not hold here. Decomposing $H^1\left(D\right)$ under the characters of $⟨{\sigma }_2⟩\times ⟨{\sigma }_3⟩$, the ${\sigma }_3$-invariant part is $H^1(D/⟨{\sigma }_3⟩)$, with numerator $1+2T+2T^2$ and with ${\sigma }_2$ acting as $[-1]$; the ${\sigma }_2$-invariant part is $H^1\left(C_3\right)$, with numerator $1+2T^2$ and with ${\sigma }_3$ acting nontrivially; and the degree-eight factor is the part non-invariant under both, with slopes $1/4$ and $3/4$ only. Now (8) requires ${\sigma }_2=-1$, because $[-1]$ acts as $-1$ on $H^1\left(E\right)$, and ${\sigma }_3\ne 1$, because $\chi$ has no nonzero fixed vector on $H^1\left(E\right)$. The factor $1+2T+2T^2$ is ${\sigma }_3$-invariant and fails the second condition. The factor $1+2T^2$, although supersingular and ${\sigma }_3$-non-invariant, is ${\sigma }_2$-invariant and fails the first, and so it does not enter $\mathrm{Hom}{(\mathrm{Jac}D,E)}^{C_6}$. The degree-eight factor has no constituent geometrically isogenous to E. Therefore $\mathrm{Hom}{(\mathrm{Jac}D,E)}^{C_6}=0$ and the rank of this third member is zero over ${\mathbb{F}}_{{16}^n}\left(t\right)$ for every n, in particular over ${\mathbb{F}}_{256}\left(t\right)$. What distinguishes $S_1$ is not a larger total multiplicity but the fact that in (4) the geometrically E-isotypic factor occurs squared within the part of $H^1\left(D\right)$ on which ${\sigma }_2$ acts as $-1$ and ${\sigma }_3$ acts nontrivially.

Remark 4.

No systematic survey is reported here. Counting points on the affine model $\{r^2+r=A\left(t\right),w^3=B\left(t\right)\}$ agrees with counting on the smooth projective normalization of D only with care: the places above $t=\infty$ need not all be rational, and their splitting type varies with the parameters. For $S_1$, they split into one rational place and one closed place of degree two, and using the wrong splitting yields a numerator that fails the functional equation, which is how we first mistook this case for an anomaly. Each reported numerator was required to satisfy the functional equation and the Riemann hypothesis, and in each case the genus agrees with the Riemann-Hurwitz value. Out-of-sample validation was available only for $S_1$, using the reserved counts $11\le m\le 14$. A systematic search over larger parameter ranges should use a Kedlaya-style algorithm; this is left open.

8. Conclusions

What is proved here is the following. The curve $E:y^2+y=x^3+x$ over ${\mathbb{F}}_2$ admits automorphisms of order six defined over ${\mathbb{F}}_{16}$, given explicitly in Proposition 1 and Lemma 1. Twisting along the fiber product of an Artin-Schreier and a Kummer covering yields the family (3), in which the constant c has absolute trace one and is not removable by a constant translation of Z when A is fixed. Two members give K3 elliptic surfaces with $\mathrm{deg}{\Delta }_{\min }=24$, and their auxiliary curves have zeta numerators that factor exactly, isolating the geometrically E-isogenous parts; the resulting Mordell-Weil ranks of the generic fibers are 0 over ${\mathbb{F}}_{16}\left(t\right)$ and 4 and 2, respectively, over ${\mathbb{F}}_{256}\left(t\right)$.

The limitations are those stated in the Introduction and in Section 5: regulators, Tate-Shafarevich groups, local terms, Kodaira types and the wild quotient singularities are not treated, and nothing in the proof of Theorem 1 depends on them.

Two questions remain open. The first is whether the multiplicity of the $C_3$-compatible, geometrically E-isogenous component whose Frobenius phase satisfies (9) can exceed the two copies occurring for $S_1$. A larger non-invariant slope-$1/2$ multiplicity alone would not imply a larger rank: slope $1/2$ is necessary but not sufficient, as the comparison in Section 7 shows, where a supersingular elliptic factor that is non-invariant under ${\sigma }_3$ nevertheless contributes nothing because it is invariant under ${\sigma }_2$. The second is a systematic survey over larger parameter ranges, which requires point counting on the normalization of the fiber product at higher genus, and for which a Kedlaya-style algorithm is the appropriate tool.

9. Computational Appendix

All computations use exact arithmetic: ${\mathbb{F}}_{2^m}$ is represented by bitmasks modulo the irreducible polynomials listed below, and zeta numerators are reconstructed over $\mathbb{Q}$ by Newton’s identities. Scripts accompany the submission as Supplementary File S1 (see the Supplementary Materials Section): S1_auxiliary.py reproduces the auxiliary curve of $S_1$ and Theorem 1 for it, assemble2.py the auxiliary curve of $S_2$, referee_check.py the numerator of H, pd_fast.py and pd_np.py the point counts, explicit_F16.py the automorphism data, sections.py the auxiliary search for sections, and sweep_standalone.py the comparison of Section 7; a README maps each file to the statement it reproduces and lists the expected output. The file README.md of the supplementary archive contains this mapping in tabular form, and RUN_LOG.txt records the interpreter and library versions together with the full output of one complete run of each script. They require Python 3.12 with numpy and sympy, and are run as python3 <script>.py. For each curve, the numerator was reconstructed from the counts with $m\le 2g$, after which the functional equation and the Riemann hypothesis were checked. For $S_1$, with $g=5$, the additional counts $11\le m\le 14$ were reserved for out-of-sample validation. For $S_2$ with $g=10$ and for H, where the published counts stop at $m=2g=6$, no such reserved counts are available; verification rests on the functional equation, the Riemann hypothesis, and the exactness of the factorizations.

9.1. Field Models

This study uses ${\mathbb{F}}_{2^m}={\mathbb{F}}_2\left[X\right]/\left(f_m\right)$ with $f_m$ as in Table 2. Each is irreducible over ${\mathbb{F}}_2$; the zeta numerator does not depend on the choice. Indeed, for both $S_1$ and $S_2$ the curve D is defined over ${\mathbb{F}}_2$, since $A,B\in {\mathbb{F}}_2\left[t\right]$, and so is H. The count $N_m=\#D\left({\mathbb{F}}_{2^m}\right)$ is then the number of ${\mathbb{F}}_{2^m}$-points of a fixed curve, and it does not depend on the polynomial $f_m$ used to represent ${\mathbb{F}}_{2^m}$: any two representations are isomorphic as fields, and the isomorphism fixes the equations. The model of ${\mathbb{F}}_{16}$ fixed in Section 3 enters only through $\omega$, s, $\tau$ and c, and the only invariant of c that is used is ${\mathrm{Tr}}_{{\mathbb{F}}_{16}/{\mathbb{F}}_2}\left(c\right)=1$ (Remark 2), which does not depend on the model.

Table 2. Irreducible polynomials $f_m$ used to represent ${\mathbb{F}}_{2^m}$.

m	${\mathit{f}}_{\mathit{m}}$	m	${\mathit{f}}_{\mathit{m}}$
1	$X+1$	11	$X^{11}+X^2+1$
2	$X^2+X+1$	12	$X^{12}+X^6+X^4+X+1$
3	$X^3+X+1$	13	$X^{13}+X^4+X^3+X+1$
4	$X^4+X+1$	14	$X^{14}+X^{10}+X^6+X+1$
5	$X^5+X^2+1$	15	$X^{15}+X+1$
6	$X^6+X+1$	16	$X^{16}+X^{12}+X^3+X+1$
7	$X^7+X+1$	17	$X^{17}+X^3+1$
8	$X^8+X^4+X^3+X+1$	18	$X^{18}+X^7+1$
9	$X^9+X^4+1$	19	$X^{19}+X^5+X^2+X+1$
10	$X^{10}+X^3+1$	20	$X^{20}+X^3+1$

9.2. Point Counts

For $S_2$, with $D:r^2+r=v^{21}+v^3$ of genus ten and a single rational place above $v=\infty$, the projective counts $N_m=\#D\left({\mathbb{F}}_{2^m}\right)$ for $m=1,\dots ,20$ are

$$\begin{array}{cc}& 5,9,11,33,25,93,89,321,569,1089,\hfill \\ & 1985,3825,8481,16,641,33,121,65,025,129,409,258,753,525,697,1,044,993.\hfill \end{array}$$

For the quotient $H:r^2+r={\varsigma }^7+\varsigma$ of genus three, again with one rational place at infinity, the counts for $m=1,\dots ,6$ are $5,9,11,17,25,69$.

For $S_1$, with D of genus five, the places above $t=\infty$ consist of one rational place and one closed place of degree two, and so the number of rational places at infinity is one when m is odd and three when m is even. This splitting is not optional: using three rational places for all m produces a numerator failing the functional equation. The affine counts for $m=1,\dots ,14$ are

$$4,4,4,4,24,112,144,220,544,1264,1984,4060,8064,16720.$$

9.3. Reconstruction and Certificates

From $S_m=q^m+1-N_m$ one obtains the coefficients $c_k$ of the numerator by $k{c}_k={\sum }_{j=1}^{k}j{e}_j{c}_{k-j}$ with $e_j=-S_j/j$. The full coefficient list for $S_2$, in ascending order, is

$$1,2,4,6,12,16,28,32,56,72,128,144,224,256,448,512,768,768,1024,1024,1024,$$

with 2-adic valuations $0,1,2,1,2,4,2,5,3,3,7,4,5,8,6,9,8,8,10,10,10$ and lower convex hull with vertices $(0,0)$, $(9,3)$, $(11,4)$, $(20,10)$. For $S_1$, the list is

$$1,2,3,2,-2,-8,-4,8,24,32,32,$$

with hull vertices $(0,0)$, $(2,0)$, $(8,3)$, $(10,5)$. Note the negative coefficients: positivity is not a property of zeta numerators and must never be used as a filter.

The divisions are exact and may be checked by multiplying out:

$$P_H·(1+2T^2)R=P_{D,S_2},(1+2T+2T^2)·{(1+2T^2)}^2(4T^4-3T^2+1)=P_{D,S_1}.$$

9.4. Consistency Checks

Three checks were applied to every reconstructed numerator: the functional equation $a_{2g-i}=q^{g-i}{a}_i$; the Riemann hypothesis $|{\alpha }_i|=\sqrt{q}$ for the reciprocal roots; and the requirement that the numerator reproduce the point counts not used in its construction. The third is essential, since a numerator of too small a degree can satisfy the first two and still be wrong. These are consistency checks on a computation, not a substitute for it, which is why the counts and field models are recorded above.