CM theory, maximal hyperelliptic curves, and Chebyshev polynomials

TL;DR

This paper investigates hyperelliptic curves defined by Chebyshev polynomials over finite fields, identifying new maximal cases and conditions that prevent maximality, using advanced algebraic and computational techniques.

Contribution

It introduces new maximality results and criteria for hyperelliptic curves associated with Chebyshev polynomials, expanding understanding of their properties over finite fields.

Findings

New maximal cases for hyperelliptic curves over finite fields.

Conditions ruling out maximality for certain (d,q) pairs.

A novel method for proving maximality in specific characteristics.

Abstract

This paper studies hyperelliptic curves $\cH_d$ corresponding to $y^2=\varphi_d(x)$ over finite fields, with $\varphi_d(x)$ a Chebyshev polynomial. Starting from the case where $d=\ell$ is an odd prime number, new cases $(d,q)$ are presented where $\cH_d$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. In addition, new conditions ruling out the possibility that $\cH_d/\FF_{q^2}$ is maximal for given $(d,q)$, are presented. The arguments involve a mix of results on slopes of Frobenius, explicit descriptions of abelian subvarieties of the jacobian of $\cH_d$ with complex multiplication, and a technique from the theory of $2$-descent on jacobians of hyperelliptic curves. In particular, the method used here to prove maximality in characteristics $p\equiv 1\bmod 4$ for $d\equiv 1\bmod 4$ a prime number, deserves attention, as it differs from earlier maximality arguments for other curves. Using the new results as well as extensive calculations with Magma, we pose some questions. A positive answer would completely classify the pairs $(q,d)$ resulting in maximality.

Figures

Equations

$${\Phi }_d\left(x+\frac{1}{x}\right)=x^d+\frac{1}{x^d}$$

$$\begin{array}[]{ccc}\mathbb{P}^{1}&\stackrel{{\scriptstyle t\mapsto t^{d}}}{{\longrightarrow}}&\mathbb{P}^{1}\\ {\Big{\downarrow}}{\scriptstyle\beta}&&{\Big{\downarrow}}{\scriptstyle\beta}\\ \mathbb{P}^{1}&\stackrel{{\scriptstyle t\mapsto\varphi_{d}(t)}}{{\longrightarrow}}&\mathbb{P}^{1}:\end{array}$$

$$y^2={\phi }_d(x).$$

$$\#{\mathcal{H}}_d({\mathbb{F}}_{q^2})=q^2+1+2g({\mathcal{H}}_d)q.$$

$${\Phi }_{\ell }=\left\{(1,\pm 1),(-1,\pm 2),(1,\pm 3),\dots ,\left((-1{)}^{(\ell -3)/2},(\ell \pm 1)/2\right)\right\}.$$

$$D_p=⟨(p\mathrm{mod}4,\pm p)⟩\subset {\left(\mathbb{Z}/4\mathbb{Z}\right)}^{\times }\times {\mathbb{F}}_{\ell }^{\times }/(\pm 1).$$

$$\frac{\#\left({\Phi }_{\ell }\cap D_p\cdot \tau \right)}{\#D_p}\text{taken with multiplicity}[\text{Gal}(K_{\ell }/\mathbb{Q}):D_p],$$

$${\zeta }_{\ell }\in {\mathbb{F}}_{p^2}({\zeta }_{4\ell }+{\zeta }_{4\ell }^{-1})\subseteq {\mathbb{F}}_{q^2}$$

$$\mathcal{J}(\mathcal{X})\sim \mathcal{J}({\mathcal{H}}_{\ell })\times \mathcal{J}({\mathcal{H}}_{\ell })\times E$$

$$X^{2g}+n{X}^g+p^g$$

$$X^{2g}+m{p}^{g/2}{X}^g+p^g$$

$$\#\mathcal{J}({\mathcal{H}}_{\ell })({\mathbb{F}}_p)=1+m{p}^{g/2}+p^g\equiv (2+m)\mathrm{mod}4.$$

$$\delta :\mathcal{J}({\mathcal{H}}_{\ell })({\mathbb{F}}_p)/2\mathcal{J}({\mathcal{H}}_{\ell })({\mathbb{F}}_p)⟶{\mathbb{F}}_p^{\times }/{{\mathbb{F}}_p^{\times }}^2$$

$$\delta\colon[(\alpha,\beta)-\infty]\mapsto\left\{\begin{array}[]{rl}\alpha{{\mathbb{F}}_{p}^{\times}}^{2}&\textrm{if}\;\;\alpha\neq 0,\\ \ell\,{{\mathbb{F}}_{p}^{\times}}^{2}&\textrm{if}\;\;\alpha=0;\end{array}\right.$$

$$X^{2g}+p^g.$$

$$\begin{array}[]{rcl}(1,\pm a)\in\Phi_{\ell}&\Longleftrightarrow&\langle\!\langle{a}\rangle\!\rangle\;\textrm{is odd},\\ (-1,\pm a)\in\Phi_{\ell}&\Longleftrightarrow&\langle\!\langle{a}\rangle\!\rangle\;\textrm{is even}.\end{array}$$

$$\frac{\#\left({\Phi }_{\ell }\cap D_p\tau \right)}{\#D_p}\in \{0,1/2,1\}$$

$$D_p\cdot (1,\pm 2)=\{(1,\pm 2),(1,\pm 2a)\}$$

$$D_p\cdot (1,\pm 4)=\{(1,\pm 4),(1,\pm 4a)\}.$$

$$D_p\cdot (1,\pm 5)=\{(1,\pm 5),(1,\pm 5a)\}.$$

$$D_p\cdot (1,\pm 3)=\{(1,\pm 3),(1,\pm 3a)\}.$$

$$D_p=\{(1,\pm 1),(1,\pm 2),(1,\pm 4),(1,\pm 8)\}.$$

$$D_p=\{(1,\pm 1),(1,\pm 6),(1,\pm 8),(1,\pm 10),(1,\pm 11),(1,\pm 14)\}.$$

$$D_p=\{(1,\pm 1),(1,\pm 3),(1,\pm 9),(1,\pm 14)\}$$

$$D_p=\left\{(1,\pm a):a\in \{1,2,4,5,8,9,10,16,18,20\}\right\},$$

$${\mathcal{H}}_d/{\mathbb{F}}_{q^2}\text{is maximal}⟺n=km\text{with}m\in {\mathbb{Z}}_{\ge 1}\text{odd}⟺q\equiv -1,1+2d\mathrm{mod}4d.$$

$$p\equiv -1,31\mathrm{mod}60.$$

$$\mathcal{J}({\mathcal{H}}_{{\ell }_1{\ell }_2})\sim \mathcal{J}({\mathcal{H}}_{{\ell }_1})\times \mathcal{J}({\mathcal{H}}_{{\ell }_2})\times A$$

$$\Phi =\left\{\left((-1{)}^{m+1},\pm m\right):1\le m\le (d-1)/2,\gcd (m,d)=1\right\}.$$

$$\left\{{\omega }_n:=(x^{n-1}-x^{-n-1})\frac{dx}{y}:1\le n\le ({\ell }_1{\ell }_2-1)/2\right\}$$

Full text

Keywords: finite field, maximal curves, hyperelliptic curves, complex multiplication, Chebyshev polynomials, Newton polygon, 2-descent.

2000 Mathematics Subject Classification: 11G20, 11M38, 14G15, 14H25, 14K22.

1. Introduction

Given $d\in\mathbb{Z}_{>0}$, the $d$-th Chebyshev polynomial $\Phi_{d}(x)\in\mathbb{Z}[x]$ is the unique polynomial such that in $\mathbb{Z}[x,x^{-1}]$ the equality

[TABLE]

holds. For a finite field ${\mathbb{F}}_{q}$ of cardinality $q$ and characteristic $p$, denote the reduction of $\Phi_{d}(x)$ modulo $p$ by $\varphi_{d}(x)\in{\mathbb{F}}_{p}[x]\subseteq{\mathbb{F}}_{q}[x]$. Immediate consequences of the definition include

• $*$

$\varphi_{d}(x)$ is monic and of degree $d$;

• $*$

\varphi_{d}(0)=\left\{\begin{array}[]{rl}0&\textrm{for}\;d\;\textrm{odd},\\ -2&\textrm{for}\;d\equiv 2\bmod 4,\\ 2&\textrm{for}\;d\equiv 0\bmod 4;\end{array}\right.

• $*$

$\varphi_{d}(-x)=(-1)^{d}\varphi_{d}(x)$;

• $*$

$\varphi_{de}(x)=\varphi_{d}(\varphi_{e}(x))$;

• $*$

$\varphi_{d+2}(x)=x\cdot\varphi_{d+1}(x)-\varphi_{d}(x)$;

• $*$

the derivative \varphi^{\prime}_{d}(0)=\left\{\begin{array}[]{rl}0&\textrm{for}\;d\;\textrm{even},\\ d&\textrm{for}\;d\equiv 1\bmod 4,\\ -d&\textrm{for}\;d\equiv 3\bmod 4.\end{array}\right.

An additional property follows (with $\beta\colon t\mapsto t+\frac{1}{t}$) from commutativity of the diagram

[TABLE]

• $*$

$\varphi_{d}(x)\in{\mathbb{F}}_{q}[x]$ is separable $\Longleftrightarrow\;\;d=1\;\;\text{or}\;\;\gcd(q,2d)=1$.

In the remainder of this paper we consider the complete, regular and absolutely irreducible curve ${\mathcal{H}}_{d}$ over ${\mathbb{F}}_{q}$ corresponding to the affine equation

[TABLE]

Here we assume $q$ is odd and $\varphi_{d}(x)$ is separable; in other words $\gcd(q,2d)=1$. This implies that ${\mathcal{H}}_{d}$ has genus $g=g({\mathcal{H}}_{d})=\lfloor(d-1)/2\rfloor$. Our aim is to describe the pairs $(d,q)$ such that ${\mathcal{H}}_{d}$ is maximal over ${\mathbb{F}}_{q^{2}}$; by definition this means that $\#{\mathcal{H}}_{d}({\mathbb{F}}_{q^{2}})$ attains the Hasse-Weil upper bound for the number of points on a curve over a finite field, i.e.,

[TABLE]

One of the main results presented here is Thm. 3.4.1: it states that if $\ell\equiv p\equiv 1\bmod 4$, then ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{p^{\ell-1}}$. The proof is a mixture of rather different tools, namely permutation polynomials, some theory of complex multiplication, and $2$-descent in the arithmetic of jacobians of hyperelliptic curves.

Maximality results for more general ${\mathcal{H}}_{d}$ are discussed as well. In characteristics $\equiv 3\bmod 4$ a complete characterization for odd $d$ is provided by Prop. 4.1.1 (the case $2|d$ is discussed in [13, Thm. 1.5 and §5]). In characteristics $p\equiv 1\bmod 4$, some necessary conditions for maximality of ${\mathcal{H}}_{d}$ are presented in §4.2. Computer experiments resulted in a simple observation, namely that the only examples where we have seen maximality of ${\mathcal{H}}_{d}$ in characteristic $p\equiv 1\bmod 4$, is for $d=\ell^{n}$ a power of a prime $\ell\equiv 1\bmod 4$ and moreover $p\bmod d$ a generator of the cyclic group $(\mathbb{Z}/d\mathbb{Z})^{\times}$.

The text is organized as follows. Section 2 discusses necessary background material. In Section 3 the curves ${\mathcal{H}}_{d}$ are discussed for $d=\ell\geq 3$ a prime number. Next, §4.1.1 contains complete results for odd $d$ and characteristic $\equiv 3\bmod 4$, and §4.2 presents an approach towards the characteristic $\equiv 1\bmod 4$ case for general $d$, culminating besides various examples in Question 4.2.9. Finally, the Appendix contains and briefly describes Magma code used in this paper.

2. Preliminaries

Maximality of any curve ${\mathcal{C}}/{\mathbb{F}}_{q^{2}}$ of genus $g$ is equivalent to the $q^{2}$-Frobenius endomorphism $F$ of the jacobian ${\mathcal{J}}({\mathcal{C}})$ being equal to $[-q]$, the multiplication by $-q$. In this case ${\mathcal{J}}({\mathcal{C}})({\mathbb{F}}_{q^{2}})=\textrm{Ker}(F-\textit{id})=\textrm{Ker}([q+1])\cong\left(\mathbb{Z}/(q+1)\mathbb{Z}\right)^{2g}$. So in particular when $q$ is odd and the maximal curve ${\mathcal{C}}$ is a hyperelliptic curve obtained from an equation $y^{2}=f(x)$ with $f(x)$ separable and of odd degree, then $f(x)$ splits completely in ${\mathbb{F}}_{q^{2}}[x]$: indeed, if $f(\alpha)=0$, then the divisor $(\alpha,0)-\infty$ yields a point of order $2$ in ${\mathcal{J}}({\mathcal{C}})$, hence $\alpha\in{\mathbb{F}}_{q^{2}}$. Maximality of ${\mathcal{C}}/{\mathbb{F}}_{q^{2}}$ is also equivalent to the statement that the characteristic polynomial of $F\in\textrm{End}({\mathcal{J}}({\mathcal{C}}))$ equals $(X+q)^{2g}$. In this case the slopes of the Newton polygon of this polynomial consist of the singleton set $\{\frac{1}{2}\}$. Recall that for a polynomial $a_{0}X^{n}+a_{1}X^{n-1}+\ldots+a_{n-1}X+a_{n}\in\mathbb{Z}[X]$ and $q$ a power of the prime $p$, these slopes are the slopes of the lower boundary of the convex hull in $\mathbb{R}^{2}$ of the points $\left(j,\textrm{ord}_{p}(a_{j})/\textrm{ord}_{p}(q)\right)$. The vertical scaling factor $\textrm{ord}_{p}(q)$ here makes the slopes of Frobenius independent of extensions of the finite field and corresponding Frobenius. In particular, as will be used below, if ${\mathcal{C}}/{\mathbb{F}}_{p}$ is maximal over some extension ${\mathbb{F}}_{p^{2e}}$ then $\frac{1}{2}$ is the only slope of the $p$-Frobenius on ${\mathcal{J}}({\mathcal{C}})$.

Regarding the polynomials $\varphi_{d}(x)$, a well-known property is that under the condition $\gcd(p^{2n}-1,d)=1$, the map $t\mapsto\varphi_{d}(t)$ regarded as a map $\varphi_{d}\colon{\mathbb{F}}_{p^{n}}\to{\mathbb{F}}_{p^{n}}$ is a permutation. At least in the case $d$ is odd, this was already shown in 1896 by L.E. Dickson, see [5, §54] (note the misprint in the statement of the result, writing $p^{2n-1}$ instead of $p^{2n}-1$ as is clearly used in the proof on his next page). Dickson’s result in particular implies that with $m$ the order of $p\bmod d\in\left(\mathbb{Z}/d\mathbb{Z}\right)^{\times}$ one has $\#{\mathcal{H}}_{d}({\mathbb{F}}_{p^{n}})=p^{n}+1$ for every $n<m\cdot(3-\gcd(m,2))$. As a consequence, for these $n$ the coefficient of $X^{2g-n}$ in the characteristic polynomial of the $p$-Frobenius is [math].

A final observation that will be important for us, concerns the case that $d=\ell\geq 3$ is a prime number. As explained in [16, Proposition 4], the jacobian ${\mathcal{J}}({\mathcal{H}}_{\ell})$ contains the CM field $K_{\ell}:=\mathbb{Q}(i,\zeta_{\ell}+\zeta_{\ell}^{-1})$ in its endomorphism algebra; here $\zeta_{\ell}$ denotes a primitive $\ell$-th root of unity. The action of $K_{\ell}$ on regular translation-invariant $1$-forms on ${\mathcal{J}}({\mathcal{H}}_{\ell})$ (the jacobian now considered as an abelian variety in characteristic [math]) defines a CM type $\Phi_{\ell}$. Since $K_{\ell}$ is a Galois extension of $\mathbb{Q}$, this CM type can be described as a subset of $\textrm{Gal}(K_{\ell}/\mathbb{Q})$. Explicitly, $\mathbb{Q}(i,\zeta_{\ell})=\mathbb{Q}(\zeta_{4\ell})$ has Galois group $\textrm{Gal}(\mathbb{Q}(\zeta_{4\ell})/\mathbb{Q})\cong\left(\mathbb{Z}/4\ell\mathbb{Z}\right)^{\times}\cong\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times\left(\mathbb{Z}/\ell\mathbb{Z}\right)^{\times}$. The subfield $K_{\ell}$ corresponds to the order two subgroup $\langle(1,-1)\rangle\subset\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times\left(\mathbb{Z}/\ell\mathbb{Z}\right)^{\times}$. Hence $\textrm{Gal}(K_{\ell}/\mathbb{Q})\cong\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$. As a subset of this, the CM type $\Phi_{\ell}$ is given as

[TABLE]

Based on work of Shimura and Taniyama one can describe, as e.g., shown in [1, Theorem 3.1], at any prime $p\not\in\{2,\ell\}$ the slopes (of the Newton polygon of the characteristic polynomial) of Frobenius on the reduction modulo $p$ of ${\mathcal{J}}({\mathcal{H}}_{\ell})$. Explicitly, the decomposition group $D_{p}\subset\textrm{Gal}(K_{\ell}/\mathbb{Q})$ at the prime $p$ equals the cyclic group

[TABLE]

In terms of this, the slopes of the $p$-Frobenius on ${\mathcal{J}}({\mathcal{H}}_{\ell})$ are the fractions

[TABLE]

with $\tau$ ranging over a set of representatives of the cosets $D_{p}\backslash\textrm{Gal}(K_{\ell}/\mathbb{Q})$. A straightforward example illustrating how this is used, is provided below in Prop. 3.1.1. We note in passing that CM results for ${\mathcal{J}}({\mathcal{H}}_{\ell})$ were also used by Sugiyama, see in particular [11, Example 5.3, Corollary 1.2]. The next session includes, notably in the results of §3.4, additional applications of this kind of argument. It is also used extensively in the calculations of §4.2.

3. Prime degree

Let $\ell$ be an odd prime number. We discuss the curve ${\mathcal{H}}_{\ell}$ over finite fields of characteristic $p\not\in\{2,\ell\}$. Its genus is $g:=g({\mathcal{H}}_{\ell})=(\ell-1)/2$. The tools for this that were introduced in Section 2 make it natural to distinguish cases according to the possibilities of the pair $\ell\bmod 4,\,p\bmod 4$. In the discussion an important role is played by the CM type $\Phi_{\ell}$.

3.1. Case $\ell\equiv 3\bmod 4,\;p\equiv 1\bmod 4$

Proposition 3.1.1**.**

If $\ell\equiv 3\bmod 4$ and $p\equiv 1\bmod 4$, then ${\mathcal{H}}_{\ell}$ is not maximal over any finite field of characteristic $p$.

Proof.

With notations as introduced in Section 2, the condition $p\equiv 1\bmod 4$ implies that in $\textrm{Gal}(K_{\ell}/\mathbb{Q})\cong\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$, the decomposition group $D_{p}=\langle(1,\pm p)\rangle$. The condition $\ell\equiv 3\bmod 4$ implies that $\#D_{p}$ is odd. Hence for any $\tau\in\textrm{Gal}(K_{\ell}/\mathbb{Q})$, the corresponding slope $\frac{\#(\Phi_{\ell}\,\cap\,D_{p}\cdot\tau)}{\#D_{p}}$ differs from $1/2$. This violates maximality of ${\mathcal{H}}_{\ell}$ in characteristic $p$. ∎

3.2. Case $\ell\equiv 3\bmod 4,\;p\equiv 3\bmod 4$

In this situation we have the following result.

Proposition 3.2.1**.**

Let $\ell\equiv 3\bmod 4$ be a prime number. For every prime $p\neq\ell$ with $p\equiv 3\bmod 4$, the powers $q=p^{n}$ such that ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{q^{2}}$ are given by $n=km$ with $m$ the order of $\pm p\bmod\ell$ in the group ${\mathbb{F}}_{\ell}^{\times}/(\pm 1)$ and $k$ odd.

Proof.

Take $\ell,p,m$ as in the formulation of the theorem. Since the group ${\mathbb{F}}_{\ell}^{\times}/(\pm 1)$ has odd order, also $m$ is odd. By definition $q:=p^{m}\equiv\pm 1\bmod\ell$, and since $m$ is odd, $q\equiv-1\bmod 4$. As shown in [13, Theorem 1.6], this implies that ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{q^{2}}$.

We claim that the finite fields ${\mathbb{F}}_{p^{2n}}$ over which ${\mathcal{H}}_{\ell}$ is maximal, are precisely the ones with $n$ an odd multiple of $m$. Indeed, maximality over ${\mathbb{F}}_{p^{2m}}$ means that the $p^{2m}$-Frobenius $F$ on ${\mathcal{J}}({\mathcal{H}}_{\ell})$ equals $[-p^{m}]$. So if $n=km$ with odd $k$, then the $p^{2n}$-Frobenius $F^{k}$ equals $[-p^{m}]^{k}=[-p^{n}]$, implying that ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{p^{2n}}$. Vice versa, maximality over ${\mathbb{F}}_{p^{2n}}$ for some $n$ shows, using the same argument and the fact that $m$ is odd, that ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{p^{2nm}}$. The Frobenius in this case is $F^{n}=[-p^{m}]^{n}$, and since it has to be equal to $[-p^{mn}]$ we conclude that $n$ is odd. Now [4, Theorem 3.6, (ii)$\Rightarrow$(iii)] applied to $q=p^{n}$ shows $p^{n}\equiv\pm 1\bmod\ell$ and therefore $n$ is a multiple of the order $m$ of $\pm p\bmod\ell$ in the group ${\mathbb{F}}_{\ell}^{\times}/(\pm 1)$. This completes the proof. ∎

Reformulating Proposition 3.2.1, in the given situation the powers $q$ of $p$ such that ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{q^{2}}$, are precisely those satisfying the two conditions $q\equiv 3\bmod 4$ and $q\equiv\pm 1\bmod\ell$.

3.3. Case $\ell\equiv 1\bmod 4,\;p\equiv 3\bmod 4$

Proposition 3.3.1**.**

Suppose $\ell\equiv 1\bmod 4$ is a prime number. A necessary condition for maximality of ${\mathcal{H}}_{\ell}$ over some finite field of characteristic $p\equiv 3\bmod 4$, is that the order $m$ of $\pm p\bmod\ell\in{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$ is odd.

In case $m$ is odd and $q=p^{n}$, the curve ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{q^{2}}$ precisely when $n$ is an odd multiple of $m$.

Proof.

Consider primes $\ell,p$ satisfying $\ell\equiv 1\bmod 4$ and $p\equiv 3\bmod 4$. The arguments presented below for this case are partially similar to those already used in §3.2.

Suppose $q=p^{n}$ is such that ${\mathcal{H}}_{\ell}/{\mathbb{F}}_{q^{2}}$ is maximal. As explained in Section 2, this implies in particular that $\varphi_{\ell}(x)\in{\mathbb{F}}_{q^{2}}[x]$ splits completely. With $\zeta_{4\ell}\in\overline{{\mathbb{F}}}_{p}$ a primitive $4\ell$-th root of unity, this means that ${\mathbb{F}}_{p}(\zeta_{4\ell}+\zeta_{4\ell}^{-1})\subseteq{\mathbb{F}}_{q^{2}}$. Since clearly ${\mathbb{F}}_{p^{2}}\subseteq{\mathbb{F}}_{q^{2}}$, then [4, Lemma 3.5] yields

[TABLE]

and therefore $\ell|(q^{2}-1)$. Using [4, Theorem 3.2], the latter divisibility gives rise to an over ${\mathbb{F}}_{q^{2}}$ defined isogeny

[TABLE]

where ${\mathcal{X}}$ denotes the smooth projective curve corresponding to $y^{2}=x^{2\ell+1}+x$ and $E$ is the elliptic curve given by $y^{2}=x^{3}+x$. Maximality of ${\mathcal{H}}_{\ell}/{\mathbb{F}}_{q^{2}}$ together with the observation that since $p\equiv 3\bmod 4$, we have that $E$ is supersingular, now shows that ${\mathcal{X}}$ is supersingular, i.e., any zero of the characteristic polynomial of the $p$-Frobenius on ${\mathcal{J}}({\mathcal{X}})$ is of the form $\zeta\sqrt{p}$ with $\zeta$ a root of unity. By [7, Theorem 2], one concludes that the order $m$ of $p\in{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$ is odd. This means $\#D_{p}\equiv 2\bmod 4$, and $p^{m}\equiv-1$ or $1+2\ell\bmod 4\ell$. Now [4, Theorem 3.6, (iii)$\Rightarrow$(ii)] shows that ${\mathcal{H}}_{\ell}/{\mathbb{F}}_{p^{2m}}$ is maximal. Maximality over odd degree extensions of this field then holds as well.

As $m$ is odd, maximality of ${\mathcal{H}}_{\ell}/{\mathbb{F}}_{q^{2}}$ implies that ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{p^{2nm}}$. Viewing the latter field as an extension of ${\mathbb{F}}_{p^{2m}}$ then reveals as in §3.2 that $n$ is odd. We have $p^{2n}=q^{2}\equiv 1\bmod\ell$, hence $p^{n}\equiv\pm 1\bmod\ell$ which shows that $n$ is an odd multiple of the order $m$ of $p\in{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$. ∎

Remark 3.3.2**.**

A generalization of Prop. 3.3.1 to ${\mathcal{H}}_{d}$ for general odd $d$ is presented below in Prop. 4.1.1.

Remark 3.3.3**.**

The maximal curves ${\mathcal{H}}_{\ell}$ described in Proposition 3.3.1 are covered by the Hermitian curve. This follows as a special case of composing maps presented in the proofs of [12, Thm. 1] and [4, Thm. 3.2].**

3.4. Case $\ell\equiv 1\bmod 4,\;p\equiv 1\bmod 4$

Given primes $\ell\neq p$ with $\ell\equiv p\equiv 1\bmod 4$, consider the decomposition group $D_{p}=\langle(1,\pm p)\rangle\subset\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$. As remarked in Section 2, a necessary condition for existence of a finite extension of ${\mathbb{F}}_{p}$ over which ${\mathcal{H}}_{\ell}$ is maximal, is that $\#D_{p}$ is even. In the case under consideration this is equivalent to the order of $p\bmod\ell\in{\mathbb{F}}_{\ell}^{\times}$ being a multiple of $4$. We now consider the two extreme cases of this.

Maximal order. The result here is as follows; note the novel technique to show maximality in the given situation.

Theorem 3.4.1**.**

Let $\ell\equiv p\equiv 1\bmod 4$ be distinct prime numbers and assume $p$ is a primitive root modulo $\ell$.

Then for $q=p^{a}$ the curve ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{q^{2}}$ if and only if $a=b\cdot(\ell-1)/2$ with $b$ a positive odd integer.

Proof.

The conditions on $\ell,p$ mean that $D_{p}=\{1\}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$. Then $\#(\Phi_{\ell}\cap D_{p})=(\ell-1)/4$ (count the odd integers in $1,2,\ldots,(\ell-1)/2$). Similarly $\#\left(\Phi_{\ell}\cap D_{p}\cdot(-1,\pm 1)\right)=(\ell-1)/4$. Hence the Newton polygon of the $p$-Frobenius on ${\mathcal{J}}({\mathcal{H}}_{\ell})$ has $\frac{1}{2}$ as its only slope. We will determine the corresponding characteristic polynomial.

The assumption on the order of $p\bmod\ell$ implies that $\varphi_{\ell}(x)$ is a permutation polynomial on ${\mathbb{F}}_{p^{a}}$ and hence $\#{\mathcal{H}}_{\ell}({\mathbb{F}}_{p^{a}})=p^{a}+1$ for every $a$ satisfying $1\leq a<(\ell-1)/2=g$. So the characteristic polynomial of the $p$-Frobenius has the form

[TABLE]

with $g=g({\mathcal{H}}_{\ell})=(\ell-1)/2$ and $n\in\mathbb{Z}$. Moreover, since its Newton polygon has $\frac{1}{2}$ as its only slope, $p^{(\ell-1)/4}|n$. The characteristic polynomial therefore is

[TABLE]

for some integer $m$. All zeros in $\mathbb{C}$ of this polynomial have absolute value $\sqrt{p}$, and hence $m^{2}-4\leq 0$, so $m\in\{-2,-1,0,1,2\}$. Evaluating at $X=1$ and using $p\equiv 1\bmod 4$ yields

[TABLE]

We claim that $\#{\mathcal{J}}({\mathcal{H}}_{\ell})({\mathbb{F}}_{p})\equiv 2\bmod 4$ (which shows $4|m$ and hence $m=0$). Indeed, the divisor $(0,0)-\infty$ on ${\mathcal{H}}_{\ell}$ yields a rational point of order $2$ in ${\mathcal{J}}({\mathcal{H}}_{\ell})$. Any other rational point of order $2$ would correspond to a factorization of $\varphi_{\ell}(x)\in{\mathbb{F}}_{p}[x]$ into two factors, with neither factor a constant multiple of $x$. Write $\phi_{\ell}(x)=x\cdot\psi_{\ell}(x)$. Any zero of $\psi_{\ell}(x)$ is of the form $\zeta_{4\ell}+\zeta_{4\ell}^{-1}$ with $\zeta_{4\ell}$ a primitive $4\ell$-th root of unity. We will show that $[{\mathbb{F}}_{p}(\zeta_{4\ell}+\zeta_{4\ell}^{-1})\,:\,{\mathbb{F}}_{p}]=\ell-1$. Indeed, using the same notation for primitive roots of unity in characteristic [math], the subfield $\mathbb{Q}(\zeta_{4\ell}+\zeta_{4\ell}^{-1})\subset\mathbb{Q}(\zeta_{4\ell})$ has Galois group $(\mathbb{Z}/4\ell\mathbb{Z})^{\times}/(\pm 1)\cong\left((\mathbb{Z}/4\mathbb{Z})^{\times}\times{\mathbb{F}}_{\ell}^{\times}\right)/\langle(-1,-1)\rangle$. Our assumption that $p\equiv 1\bmod 4$ is a primitive root modulo $\ell$ means that the decomposition group $\tilde{D}_{p}=\langle(1,p)\rangle$ at $p$ in this Galois group has size $\ell-1$. As $\tilde{D}_{p}\cong\textrm{Gal}({\mathbb{F}}_{p}(\zeta_{4\ell}+\zeta_{4\ell}^{-1})\,/\,{\mathbb{F}}_{p})$, this proves the assertion regarding the degree of ${\mathbb{F}}_{p}(\zeta_{4\ell}+\zeta_{4\ell}^{-1})$. So $\psi_{\ell}(x)\in{\mathbb{F}}_{p}[x]$ is irreducible, which means that $(0,0)-\infty$ gives rise to the only point of order $2$ in ${\mathcal{J}}({\mathcal{H}}_{\ell})({\mathbb{F}}_{p})$. Hence to show $\#{\mathcal{J}}({\mathcal{H}}_{\ell})({\mathbb{F}}_{p})\equiv 2\bmod 4$, it remains to prove that ${\mathcal{J}}({\mathcal{H}}_{\ell})({\mathbb{F}}_{p})$ contains no point of order $4$, i.e., no point $P$ such that $2P$ corresponds to the divisor class of $(0,0)-\infty$. To verify that indeed no such $P$ exists, a well-known technique from $2$-descent computations can be used: there is a homomorphism

[TABLE]

given, in terms of divisor classes, by

[TABLE]

see [9, Lemma 2.2] or [10, Lemma 4.3.(2)(ii)]. Here we use that $p\equiv 1\bmod 4$ and therefore the properties of Chebyshev polynomials listed in Section 1 yield $\psi_{\ell}(0)=\varphi^{\prime}_{\ell}(0)=\ell$. Since $p\equiv 1\bmod 4$ is a primitive root modulo $\ell$, the Legendre symbol satisfies $\genfrac{(}{)}{}{}{\ell}{p}=\genfrac{(}{)}{}{}{p}{\ell}\equiv p^{(\ell-1)/2}\bmod\ell=-1\bmod\ell$. This shows that $\delta([(0,0)-\infty])$ is nontrivial, and one concludes that $[(0,0)-\infty]\not\in 2{\mathcal{J}}({\mathcal{H}}_{\ell})({\mathbb{F}}_{p})$.

As a consequence, the $p$-Frobenius on ${\mathcal{J}}({\mathcal{H}}_{\ell})$ has characteristic polynomial

[TABLE]

Hence for $q=p^{a}$ (and primes $p\neq\ell$ such that $p\equiv\ell\equiv 1\bmod 4$ with $p$ a primitive root modulo $\ell$) the curve ${\mathcal{H}}_{\ell}$ is maximal over ${\mathbb{F}}_{q^{2}}$ precisely when $a=bg=b(\ell-1)/2$ with $b$ a positive odd integer, which is what we wanted to show. ∎

Remark 3.4.2**.**

As shown in [3, Prop. 2.3], the shape of the characteristic polynomial in the above proof (namely $X^{2g}+nX^{g}+p^{g}$) implies that ${\mathcal{J}}({\mathcal{H}}_{\ell})$ is ${\mathbb{F}}_{p^{g}}$-isogenous to the $g$-fold product of some elliptic curve. Indeed, this is a simple consequence of classical work of Tate [14, Thm. 1(c)].

One of the ingredients in our proof is to use the fact that $\varphi_{\ell}(x)$ is a permutation polynomial over various ${\mathbb{F}}_{p^{a}}$, in order to restrict the possibilities for the characteristic polynomial of the $p$-Frobenius. This idea also appears in work of Özbudak [8, Section 2].

Minimal order. Here we discuss the case that the prime number $p\equiv 1\bmod 4$ has order $4$ in ${\mathbb{F}}_{\ell}^{\times}$. For convenience when studying the CM-type $\Phi_{\ell}$, the following notation is introduced.

Notation 3.4.3**.**

Given $a\in{\mathbb{F}}_{\ell}^{\times}$, denote by $\langle\!\langle{a}\rangle\!\rangle\in\mathbb{Z}_{>0}$ the smallest positive integer $n$ with the property that $n\bmod\ell\in\{a,\,-a\}$.

Note that for $a\in{\mathbb{F}}_{\ell}^{\times}$ one has $1\leq\langle\!\langle{a}\rangle\!\rangle=\langle\!\langle{-a}\rangle\!\rangle\leq(\ell-1)/2$. The CM type $\Phi_{\ell}$ can be described, as a subset of $\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$, by

[TABLE]

Proposition 3.4.4**.**

Given distinct prime numbers $\ell\equiv p\equiv 1\bmod 4$ such that $\ell\neq 5$ and $p\bmod\ell\in{\mathbb{F}}_{\ell}^{\times}$ has order $4$, the curve ${\mathcal{H}}_{\ell}$ is not maximal over any finite extension of ${\mathbb{F}}_{p}$.

Proof.

Write $i:=p\bmod\ell$. The decomposition group $D_{p}$ equals $\{(1,\pm 1),\,(1,\pm i)\}$ and we are interested in the possible slopes

[TABLE]

for $\tau\in\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$. We claim that for any $p,\ell$ as described here, with $\ell>5$, a slope $\neq 1/2$ occurs.

Indeed, write $a:=\langle\!\langle{i}\rangle\!\rangle<\ell/2$, the smallest integer $>0$ such that $a^{2}\equiv-1\bmod\ell$. If $a$ is odd, then $D_{p}\subset\Phi_{\ell}$ provides a slope $1$. Hence we may and will assume $a$ is even. Then

[TABLE]

where $2<\ell/2$ and $2<2a<\ell$. Hence if $2a<\ell/2$ we obtain a slope [math]. This reduces our claim to the case of an even $a$ with $\ell/4<a<\ell/2$. Here, take

[TABLE]

Since $\ell>5$ and $\ell\equiv 1\bmod 4$, we have $\langle\!\langle{4}\rangle\!\rangle=4$. The smallest representants of the classes $\pm 4a\bmod\ell$ are $4a-\ell$ and $2\ell-4a$. This again yields a slope [math], unless $4a-\ell<2\ell-4a$. So what remains is the case $a$ even with $\frac{\ell}{4}<a<\frac{3}{8}\ell$. Under these conditions we consider

[TABLE]

As before, $\langle\!\langle{5}\rangle\!\rangle=5$. The smallest positive representants of $\pm 5a\bmod\ell$ are $5a-\ell$ and $2\ell-5a$. This results in a slope $1$ unless $\frac{3}{10}\ell<a<\frac{3}{8}\ell$. Assuming these inequalities, take

[TABLE]

In case $3a<\ell$, the smallest positive representant of $\pm 3a\bmod\ell$ is $\ell-3a$ and this leads to a slope $1$. In the remaining case $\ell<3a<\frac{9}{8}\ell$, clearly $3a-\ell<2\ell-3a$ so also here a slope $1$ is obtained. This proves the claim. The proposition is an immediate consequence. ∎

Corollary 3.4.5**.**

Let $\ell$ be a prime number of the form $\ell=4k+1$ with $k=1$ or $k$ prime. The only characteristics $p\neq\ell$ with $p\equiv 1\bmod 4$ such that ${\mathcal{H}}_{\ell}$ is maximal over some finite extension of ${\mathbb{F}}_{p}$ are the ones with the property that $p$ is a primitive root modulo $\ell$.

Example 3.4.6**.**

The smallest primes $\ell\equiv 1\bmod 4$ are $5$ and $13$. These cases are covered as instances of Propositions 3.4.1, 3.4.4 and Corollary 3.4.5.

For $\ell=17$ and $17\neq p\equiv 1\bmod 4$, the only case of a decomposition group $D_{p}$ of even order not discussed in the Propositions 3.4.1, 3.4.4 is

[TABLE]

This occurs whenever $p\equiv 25,\,45,\,49,\,53\bmod 68$. Note that $D_{p}\cap\Phi_{\ell}=\{(1,\pm 1)\}$, so one of the resulting slopes is $1/4$. So ${\mathcal{H}}_{17}$ is not maximal over any finite extension of ${\mathbb{F}}_{p}$ for such $p$. In fact, $1/4$ and $3/4$ are the only slopes in this case, each with length $8$. So the $p$-rank of ${\mathcal{J}}({\mathcal{H}}_{17})$ is [math] in the described situation, whereas this Jacobian is not supersingular.

A next case not covered completely by the results obtained so far, is $\ell=37$. Here apart from a maximal and minimal even case, also $\#D_{p}=6$ occurs, explicitly

[TABLE]

This is the subgroup of $\{1\}\times{\mathbb{F}}_{\ell}^{\times}/(\pm 1)$ generated by $(1,\pm 8)$. Since $D_{p}\cap\Phi_{\ell}$ leads to a slope $1/3$, also here a necessary (and sufficient) condition for maximality of ${\mathcal{H}}_{\ell}$ in characteristic $p\equiv 1\bmod 4$ is that $p$ is a primitive root modulo $\ell$.

As a last example, consider $\ell=41$. Here the additional cases

[TABLE]

and

[TABLE]

coming from an element of order $4$ resp. $10$ in ${\mathbb{F}}_{41}^{\times}/(\pm 1)$, turn up. The first one leads to a Newton polygon with slopes $1/4,\,3/4$ and the second one results in slopes $3/20,\,17/20$. Hence again $p$ a primitive root modulo $\ell$ is a necessary and sufficient condition for maximality of ${\mathcal{H}}_{\ell}$ over some field of characteristic $p\equiv 1\bmod 4$.

We do not know whether this is true in general. Using the slopes Magma code presented on p. Appendix: some Magma code we verified that it holds for every prime $\ell\equiv 1\bmod 4$, $\ell\leq 101$.

Remark 3.4.7**.**

Similar to the proof of Prop. 3.4.4, we verified for primes $\ell$, $p\equiv 1\bmod 4$ such that $p\bmod\ell\in{\mathbb{F}}_{\ell}^{\times}$ has order $8$ (i.e., the decomposition group $D_{p}\subset\text{Gal}(\mathbb{Q}(i,\zeta_{\ell}+\zeta_{\ell}^{-1})/\mathbb{Q})$ has order $4$), a slope $\neq 1/2$ at $p$ on ${\mathcal{J}}({\mathcal{H}}_{\ell})$ occurs. Hence also in these cases maximality of ${\mathcal{H}}_{\ell}$ in characteristic $p$ does not happen. The results motivate the following question.

Question 3.4.8**.**

Given distinct primes $\ell\equiv p\equiv 1\bmod 4$, is it true that maximality of ${\mathcal{H}}_{\ell}$ in characteristic $p$ is only possible in the situation of Thm. 3.4.1 (i.e., when $p$ is a primitive root modulo $\ell$)?

Remark 3.4.9**.**

When $p\equiv 1\pmod{4}$ is a primitive root modulo the prime $\ell\equiv 1\bmod 4$ and $q=p^{(\ell-1)/2}$, we do not know whether the maximal curve ${\mathcal{H}}_{\ell}/{\mathbb{F}}_{q^{2}}$ is covered by the Hermitian curve. As a small, explicit example: consider $\ell=5$ and $q=13^{2}=169$. The genus $2$ curve ${\mathcal{H}}_{5}\colon y^{2}=x^{5}-5x^{3}+x$ is maximal over ${\mathbb{F}}_{13^{4}}$. Is it over this field covered by the Hermitian curve?

Note that Giulietti and Korchmáros [6] constructed the first example of a maximal curve over a finite field which is not covered by the Hermitian curve.**

4. General odd degree (some results and calculations)

Throughout this section, $d\in\mathbb{Z}_{\geq 3}$ will be odd. We will first discuss necessary and sufficient conditions for maximality of ${\mathcal{H}}_{d}$ over finite fields of characteristic $p\equiv 3\bmod 4$. The case $p\equiv 1\bmod 4$ is then discusses in a separate subsection.

4.1. Characteristic $3$ modulo $4$

Combining results from our papers [13] and [4] with work by Kodama and Washio [7] leads to the following.

Proposition 4.1.1**.**

Let $d\in\mathbb{Z}_{\geq 1}$ be odd and let $p\equiv 3\bmod 4$ be a prime number.

Then ${\mathcal{H}}_{d}$ is maximal over some finite field of characteristic $p$ if and only if the subgroup $\langle\bar{p}\rangle\subseteq(\mathbb{Z}/4d\mathbb{Z})^{\times}$ contains either $\overline{-1}$ or $\overline{1+2d}$.

If these equivalent conditions are satisfied, then writing $q=p^{n}$ and $2k=\text{ord}(\bar{p})$, it holds that $k$ is an odd integer and one has

[TABLE]

Proof.

We start by showing the two implications that appear in the first assertion.

$\Leftarrow$: Take $q:=p^{n}$ such that $\bar{q}\in\{\overline{-1},\overline{1+2d}\}$. Then by [13, Thm. 1.6], ${\mathcal{H}}_{d}/{\mathbb{F}}_{q^{2}}$ is maximal.

$\Rightarrow$: Let ${\mathcal{X}}$ be the curve defined by $y^{2}=x^{2d+1}+x$ and let $E$ be the elliptic curve with equation $y^{2}=x^{3}+x$. As is shown in [13, Rem. 5.5], over $\overline{{\mathbb{F}}}_{p}$ an isogeny ${\mathcal{J}}({\mathcal{X}})\sim{\mathcal{J}}({\mathcal{H}}_{d})^{2}\times E$ exists. The assumptions that ${\mathcal{H}}_{d}$ is maximal and that $p\equiv 3\bmod 4$ therefore imply that ${\mathcal{J}}({\mathcal{X}})$ is supersingular. Then [7, p. 199, Cor. to Thm. 2] implies that maximality occurs for ${\mathcal{X}}$ over some finite field of characteristic $p$, and moreover either $\overline{-1}$ or $\overline{1+2d}$ is in $\langle\bar{p}\rangle$.

To show the remaining parts, first note that $(\mathbb{Z}/4d\mathbb{Z})^{\times}\cong(\mathbb{Z}/4\mathbb{Z})^{\times}\times(\mathbb{Z}/d\mathbb{Z})^{\times}$; under the standard isomorphism $\bar{p}$ corresponds to the pair $(p\bmod 4,\,p\bmod d)$ in the latter group. Since $p\equiv 3\bmod 4$, this means that $\text{ord}(\bar{p})$ is even. Writing this order as $2k$, it follows that $\bar{p}^{k}$ is the unique element of order $2$ in $\langle\bar{p}\rangle$, which by assumption equals one of $\overline{-1},\,\overline{1+2d}$. In particular $p^{k}\equiv 3\bmod 4$, so $k$ is odd. Furthermore, $q=p^{n}\equiv p^{k}\bmod 4d$ precisely when $n$ is an odd multiple of $k$. The proof of “$\Leftarrow$” shows that ${\mathcal{H}}_{d}/{\mathbb{F}}_{q^{2}}$ is maximal. If ${\mathcal{H}}_{d}$ is maximal over some ${\mathbb{F}}_{p^{2f}}$, then $\varphi_{d}(x)$ splits completely in ${\mathbb{F}}_{p^{2f}}[x]$, hence ${\mathbb{F}}_{p^{2}}(\zeta_{4d}+\zeta_{4d}^{-1})\subseteq{\mathbb{F}}_{p^{2f}}$. Using [4, Lemma 3.5] it follows that $p^{2f}\equiv 1\bmod d$. Since also $p^{2f}\equiv 1\bmod 4$, one concludes that $2k|2f$, hence $k|f$. Maximality of ${\mathcal{H}}_{d}$ over both ${\mathbb{F}}_{p^{2k}}$ and its extension ${\mathbb{F}}_{p^{2f}}$ implies that $f$ is an odd multiple of $k$. This completes the proof. ∎

Remark 4.1.2**.**

A crucial ingredient in the proof presented here is the result by Kodama and Washio [7] concerning ${\mathcal{X}}\colon y^{2}=x^{2d+1}+x$. They showed that the three assertions

• (a)

${\mathcal{J}}({\mathcal{X}})$ is supersingular in characteristic $p$;

• (b)

${\mathcal{X}}$ is minimal over some finite field of characteristic $p$;

• (c)

${\mathcal{X}}$ is maximal over some finite field of characteristic $p$,

are equivalent. Their proof involves calculations with Jacobi sums. Similar ideas may be found in the short classical note [15] by Tate and Shafarevich, who attribute it to A. Weil.

If $\ell|d$ for some prime $\ell\equiv 3\bmod 4$, then the equality $\varphi_{d}(x)=\varphi_{\ell}(\varphi_{d/\ell}(x))$ leads to a nonconstant morphism ${\mathcal{H}}_{d}\to{\mathcal{H}}_{\ell}$. As a consequence, using Proposition 3.1.1, maximality of ${\mathcal{H}}_{d}$ cannot occur over finite fields of characteristic $p\equiv 1\bmod 4$. Hence for such odd $d$, Proposition 4.1.1 describes all cases of maximality in every characteristic.

Example 4.1.3**.**

Consider $d=15$. The discussion above combined with Proposition 4.1.1 shows that maximality of ${\mathcal{H}}_{15}$ occurs in characteristic $p$ precisely when $p\equiv 3\bmod 4$ and $\langle\bar{p}\rangle\subset(\mathbb{Z}/60\mathbb{Z})^{\times}$ contains one of $\overline{-1},\overline{31}$. This translates into

[TABLE]

In these cases ${\mathcal{H}}_{15}/{\mathbb{F}}_{p^{n}}$ is maximal if and only if $n\equiv 2\bmod 4$. Note in particular that, for example, when $p\equiv 11\bmod 60$ then both ${\mathcal{H}}_{3}$ and ${\mathcal{H}}_{5}$ are maximal over ${\mathbb{F}}_{p^{2}}$, although maximality of ${\mathcal{H}}_{15}$ cannot occur in such characteristic. Some CM theory discussed in the next section turns out to explain this.

4.2. Characteristic $1$ modulo $4$

Combining [13, Thm. 1.5] and Prop. 3.1.1 results in the observation that a necessary condition for maximality of ${\mathcal{H}}_{d}$ in characteristic $p\equiv 1\bmod 4$, provided $d>2$, is that all prime factors of $d$ are $1\bmod 4$. As the case where $d$ itself is prime, is already discussed in Section 3, this means that new such cases have either $\ell_{1}\ell_{2}|d$ or $\ell^{n}|d$ with $\ell_{1}\neq\ell_{2},\ell$ primes congruent to $1\bmod 4$ and $n\geq 2$. The elementary properties of the polynomials $\varphi_{d}(x)$ then yield either ${\mathcal{H}}_{d}\to{\mathcal{H}}_{\ell_{1}\ell_{2}}$ or ${\mathcal{H}}_{d}\to{\mathcal{H}}_{\ell^{n}}$ a non-constant morphism. Therefore, a necessary condition for maximality of ${\mathcal{H}}_{d}$ in characteristic $p$ is (depending on the given $d$) maximality of ${\mathcal{H}}_{\ell_{1}\ell_{2}}$ or of ${\mathcal{H}}_{\ell^{n}}$. We focus on these two types of curves.

Lemma 4.2.1**.**

Let $\ell_{1}\neq\ell_{2}$ be odd prime numbers. The maps ${\mathcal{H}}_{\ell_{1}\ell_{2}}\to{\mathcal{H}}_{\ell_{1}}$ and ${\mathcal{H}}_{\ell_{1}\ell_{2}}\to{\mathcal{H}}_{\ell_{2}}$ coming from $\varphi_{\ell_{1}\ell_{2}}(x)=\varphi_{\ell_{1}}(\varphi_{\ell_{2}}(x))=\varphi_{\ell_{2}}(\varphi_{\ell_{1}}(x))$ lead to a $\mathbb{Q}$-isogeny

[TABLE]

for some abelian variety $A/\mathbb{Q}$ of dimension $(\ell_{1}-1)(\ell_{2}-1)/2$.

Write $d=\ell_{1}\ell_{2}$. In characteristic [math], this $A$ has complex multiplication with endomorphism algebra $\mathbb{Q}(i,\zeta_{d}+\zeta_{d}^{-1})$. Identifying $\mbox{\rm{Gal}}(\mathbb{Q}(i,\zeta_{d}+\zeta_{d}^{-1})/\mathbb{Q})$ with $(\mathbb{Z}/4\mathbb{Z})^{\times}\times(\mathbb{Z}/d\mathbb{Z})^{\times}/(\pm 1)$, the corresponding CM type $\Phi$ is described by the following subset of this Galois group:

[TABLE]

Proof.

Given $d>0$, denote by ${\mathcal{X}}_{d}$ the (smooth, complete) hyperelliptic curve corresponding to $y^{2}=x^{d}+x^{-d}$. We have a commutative diagram

where the maps ${\mathcal{X}}_{nm}\to{\mathcal{X}}_{m}$ come from $(x,y)\mapsto(x^{n},y)$ and ${\mathcal{X}}_{n}\to{\mathcal{H}}_{n}$ from $(x,y)\mapsto(x+1/x,y)$ and ${\mathcal{H}}_{nm}\to{\mathcal{H}}_{m}$ from $(x,y)\mapsto(\varphi_{n}(x),y)$.

We now reason in the spirit of [16, §3.1]. Define $\sigma,\zeta,\iota\in\text{Aut}({\mathcal{X}}_{\ell_{1}\ell_{2}})$ via $\sigma(x,y)=(1/x,y)$ and $\zeta(x,y)=(\zeta_{\ell_{1}\ell_{2}}x,y)$ and $\iota(x,y)=(-x,iy)$. Here $\zeta_{n}$ denotes a primitive $n$-th root of unity and $i^{2}=-1$. Via pull-back, the spaces of regular $1$-forms on ${\mathcal{H}}_{\ell_{j}}$ and on ${\mathcal{H}}_{\ell_{1}\ell_{2}}$ will be considered as $\sigma$-invariant differentials on ${\mathcal{X}}_{\ell_{1}\ell_{2}}$. The latter space has basis

[TABLE]

and it equals the space of pull-backs of regular $1$-forms on ${\mathcal{H}}_{\ell_{1}\ell_{2}}$. For $j=1,2$, the differentials related to ${\mathcal{H}}_{\ell_{j}}$ are precisely those spanned by $\{\omega_{n}\colon\ell_{2-j}|n\}$. Hence $\{\omega_{n}\colon\gcd(n,\ell_{1}\ell_{2})=1\}$ spans a subspace of (the pull-back of) $H^{0}({\mathcal{H}}_{\ell_{1}\ell_{2}},\Omega^{1})$ complementary to the differentials from both ${\mathcal{H}}_{\ell_{j}}$. Note that

[TABLE]

and

[TABLE]

Since $\text{span}\{\omega_{n}\colon\ell_{1}|n\}\cap\text{span}\{\omega_{n}\colon\ell_{2}|n\}=\{0\}$, upto isogeny ${\mathcal{J}}({\mathcal{H}}_{\ell_{1}\ell_{2}})$ contains the product ${\mathcal{J}}({\mathcal{H}}_{\ell_{1}})\times{\mathcal{J}}({\mathcal{H}}_{\ell_{2}})$. This results in the asserted decomposition (defined over $\mathbb{Q}$) of ${\mathcal{J}}({\mathcal{H}}_{\ell_{1}\ell_{2}})$, with $\dim(A)=g({\mathcal{H}}_{\ell_{1}\ell_{2}})-g({\mathcal{H}}_{\ell_{1}})-g({\mathcal{H}}_{\ell_{2}})=(\ell_{1}-1)(\ell_{2}-1)/2$.

The action of $\mathbb{Z}[\iota,\zeta+\zeta^{-1}]\subset\text{End}({\mathcal{J}}({\mathcal{X}}_{\ell_{1}\ell_{2}}))$ fixes the parts coming from ${\mathcal{J}}({\mathcal{H}}_{\ell_{1}}),{\mathcal{J}}({\mathcal{H}}_{\ell_{2}})$ and ${\mathcal{J}}({\mathcal{H}}_{\ell_{1}\ell_{2}})$. As a consequence, comparing fixed subspaces, the abelian variety $A$ corresponds to $\text{span}\{\omega_{n}\colon\gcd(n,\ell_{1},\ell_{2})=1\}.$ In terms of the standard identification

[TABLE]

(with $a\equiv 1\bmod 4$ and $a\equiv-1\bmod\ell_{1}\ell_{2}$), the expressions for $\iota^{*}\omega_{n}$ and $\left(\zeta^{*}+(\zeta^{-1})^{*}\right)\omega_{n}$ imply that $A$ has CM with CM-field $\mathbb{Q}(i,\zeta_{\ell_{1}\ell_{2}}+\zeta^{-1}_{\ell_{1}\ell_{2}})$, and the corresponding CM type is the asserted one. ∎

Lemma 4.2.1 results in an additional condition for maximality of ${\mathcal{H}}_{\ell_{1}\ell_{2}}$ over some finite field of characteristic $p\equiv 1\bmod 4$. The conditions we have so far, are:

• (a)

${\mathcal{H}}_{\ell_{1}}$ and ${\mathcal{H}}_{\ell_{2}}$ are maximal over some common ${\mathbb{F}}_{p^{n}}$, so in particular $\ell_{1},\ell_{2}\equiv 1\bmod 4$;

• (b)

The CM abelian variety $A$ appearing in Lemma 4.2.1 has supersingular reduction in characteristic $p$, in other words, the only slope of Frobenius at $p$ is $1/2$.

An immediate consequence of condition (a) is the following.

Lemma 4.2.2**.**

If ${\mathcal{H}}_{\ell_{1}}$ is maximal over ${\mathbb{F}}_{p^{2m}}$ and ${\mathcal{H}}_{\ell_{2}}$ is maximal over ${\mathbb{F}}_{p^{2n}}$, then maximality of ${\mathcal{H}}_{\ell_{1}\ell_{2}}$ in characteristic $p$ implies that $n$ and $m$ have equal $2$-adic valuation.

Proof.

Denote the $2$-adic valuation of a nonzero integer $a$ by $v_{2}(a)$. If ${\mathcal{H}}_{\ell_{1}\ell_{2}}/{\mathbb{F}}_{p^{2k}}$ is maximal, then so is ${\mathcal{H}}_{\ell_{1}}$ and therefore $v_{2}(k)=v_{2}(m)$. By the same argument $v_{2}(k)=v_{2}(n)$. ∎

Example 4.2.3**.**

Suppose that $\ell_{1},\ell_{2}\equiv 1\bmod 4$ are primes, with the property that maximality of ${\mathcal{H}}_{\ell_{j}}$ in characteristic $p\equiv 1\bmod 4$ only occurs when $p$ is a primitive root modulo $\ell_{j},j=1,2$. This condition is satisfied, for example, if the $\ell_{j}$ are as described in Cor. 3.4.5. As remarked at the end of Section 3, it also holds for $\ell_{1},\ell_{2}\leq 101$. Combining Lemma 4.2.2 and Thm. 3.4 yields that under these conditions, a necessary condition for maximality of ${\mathcal{H}}_{\ell_{1}\ell_{2}}$ in characteristic $p\equiv 1\bmod 4$ (besides $p$ being a primitive root modulo $\ell_{1}$ and $\ell_{2}$) is that $v_{2}(\ell_{1}-1)=v_{2}(\ell_{2}-1)$.

For all pairs $\ell_{1}\neq\ell_{2}$ satisfying these conditions and $\ell_{1},\ell_{2}\leq 101$, we checked using Magma that the condition mentioned in (b) above does not hold for any $p\equiv 1\bmod 4$ which is a primitive root modulo $\ell_{1},\ell_{2}$. In other words, we did not find any ${\mathcal{H}}_{d}$ with $d$ divisible by at least two distinct prime numbers, such that ${\mathcal{H}}_{d}$ attains maximality in some characteristic $p\equiv 1\bmod 4$.

Question 4.2.4**.**

Is it true that a necessary condition for maximality of ${\mathcal{H}}_{d}$ for $d>2$ in some characteristic $p\equiv 1\bmod 4$, is that $d$ is a power of a prime $\equiv 1\bmod 4$?

Next, curves ${\mathcal{H}}_{\ell^{n}}$ are discussed. An analogue of Lemma 4.2.1 is as follows.

Lemma 4.2.5**.**

Let $\ell$ be an odd prime number and $n\in\mathbb{Z}_{\geq 2}$. The map ${\mathcal{H}}_{\ell^{n}}\to{\mathcal{H}}_{\ell^{n-1}}$ coming from $\varphi_{\ell^{n}}(x)=\varphi_{\ell^{n-1}}(\varphi_{\ell}(x))$ leads to a $\mathbb{Q}$-isogeny

[TABLE]

for some abelian variety $B/\mathbb{Q}$ of dimension $\ell^{n-1}(\ell-1)/2$.

Write $d=\ell^{n}$. In characteristic [math], this $B$ has complex multiplication with endomorphism algebra $\mathbb{Q}(i,\zeta_{d}+\zeta_{d}^{-1})$. Identifying $\mbox{\rm{Gal}}(\mathbb{Q}(i,\zeta_{d}+\zeta_{d}^{-1})/\mathbb{Q})$ with $(\mathbb{Z}/4\mathbb{Z})^{\times}\times(\mathbb{Z}/d\mathbb{Z})^{\times}/(\pm 1)$, the corresponding CM type $\Phi$ is described by the following subset of this Galois group:

[TABLE]

Proof.

With notations as in Lemma 4.2.1 and its proof, one has the diagram

In $H^{0}({\mathcal{X}}_{\ell^{n}},\Omega^{1})$ the $\sigma$-invariant differentials are those coming from ${\mathcal{H}}_{\ell^{n}}$ and they have as basis

[TABLE]

The ones coming from ${\mathcal{H}}_{\ell^{n-1}}$ are spanned by $\{\omega_{n}\colon\ell^{n-1}|n\}$. The remainder of the proof is similar to that of Lemma 4.2.1. ∎

Note that the case $n=1$ (although excluded in the statement of the lemma) is in fact the result of [16, Prop. 4]. A difference between the current case and the one described in Lemma 4.2.1, is that one can identify cases such that $B$ has supersingular reduction:

Proposition 4.2.6**.**

With notations as in Lemma 4.2.5, if the prime $p\equiv 1\bmod 4$ generates $(\mathbb{Z}/\ell^{n}\mathbb{Z})^{\times}$ then $B$ has supersingular reduction at $p$.

Proof.

Put $d=\ell^{n}$. By assumption, in $\textrm{Gal}(\mathbb{Q}(i,\zeta_{d}+\zeta_{d}^{-1})/\mathbb{Q})\cong(\mathbb{Z}/4\mathbb{Z})^{\times}\times(\mathbb{Z}/d\mathbb{Z})^{\times}/(\pm 1)$ the decomposition group $D_{p}$ at $p$ equals $\{1\}\times(\mathbb{Z}/d\mathbb{Z})^{\times}/(\pm 1)$. As a consequence, $1/2$ is the only slope of Frobenius at $p$, proving the claim. ∎

Remark 4.2.7**.**

A test with Magma (considering all primes $\ell\equiv 1\bmod 4$ upto $101$ and $n=2$, as well as $\ell\leq 29$ and $n=3$) showed that in these cases, the only primes $p\equiv 1\bmod 4$ such that ${\mathcal{H}}_{\ell}$ attains maximality in characteristic $p$ and moreover $B$ has supersingular reduction at $p$, are the ones for which $p$ generates $(\mathbb{Z}/\ell^{n}\mathbb{Z})^{\times}$.

If this holds in general, then it would provide a necessary condition for maximality of ${\mathcal{H}}_{\ell^{n}}$ in characteristic $p\equiv 1\bmod 4$.

Example 4.2.8**.**

For $\ell=5$ and $n=2$ and every $p\in\{13,17,37,53,73,97,233,277\}$ (primes $1\bmod 4$ ranging over the generators modulo $25$), Magma verifies that the numerator of the zeta-function of ${\mathcal{H}}_{25}/{\mathbb{F}}_{p}$ equals $(p^{2}x^{4}+1)(p^{10}x^{20}+1)$. Hence here ${\mathcal{H}}_{25}/{\mathbb{F}}_{p^{20}}$ is maximal.

Taking $\ell=5$ and $n=3$, the prime $p=13$ generates the units modulo $125$. According to Magma, the numerator of the zeta-function of ${\mathcal{H}}_{125}/{\mathbb{F}}_{13}$ is $(p^{2}x^{4}+1)(p^{10}x^{20}+1)(p^{50}x^{100}+1)$. As a consequence, ${\mathcal{H}}_{125}/{\mathbb{F}}_{13^{100}}$ is maximal! Magma code used for these cases is presented on p. Appendix: some Magma code. Although additional examples of this kind seem beyond the power of Magma, the ones presented here motivates the following, generalizing Questions 3.4.8 and 4.2.4.

Question 4.2.9**.**

Let $\ell\equiv 1\bmod 4$ be prime and $n\in\mathbb{Z}_{\geq 1}$ and $p\equiv 1\bmod 4$ a prime generating $(\mathbb{Z}/\ell^{n}\mathbb{Z})^{\times}$; write $m=\ell^{n-1}(\ell-1)$ for the order of this group. Is it true that the curve ${\mathcal{H}}_{\ell^{n}}$ is maximal over ${\mathbb{F}}_{p^{m}}$ (and hence over all odd degree extensions of the latter field)?

Moreover, suppose ${\mathcal{H}}_{d}$ is maximal over some field of characteristic $p\nmid 2d$ (and $d\geq 3$), with $p\equiv 1\bmod 4$. Is it true that $d=\ell^{n}$ for some prime $\ell\equiv 1\bmod 4$ and $n\in\mathbb{Z}_{\geq 1}$?

The results and described experiments in this text are consistent with positive answers to both questions. As an example, in the case $n=1$, Thm. 3.4.1 provides a positive answer to the first question posed here.

Appendix: some Magma code

In this section some of the used Magma code (see [2] for the language) is presented.

The following code computes the curve ${\mathcal{H}}_{d}$ and checks maximality over ${\mathbb{F}}_{q}$ (provided $d,q$ are reasonably small).

⬇

1PZ<x>:=PolynomialRing(Integers());

2cheb := function(d)

3 ch1:=x; ch2:=x^2-2;

4 for m in [3..d] do

5 ch:=x*ch2-ch1; ch1:=ch2; ch2:=ch;

6 end for;

7 return ch;

8end function;

9

10F:=GF(13^20); PF<x>:=PolynomialRing(F);

11H:=HyperellipticCurve(PF!cheb(25));

12SerreBound(H) eq #H;

It turns out that some “larger” examples can be verified using the numerator of the zeta function of ${\mathcal{H}}_{d}/{\mathbb{F}}_{p}$. Here is an example.

⬇

1p:=13; F:=GF(p); PF<x>:=PolynomialRing(F);

2H:=HyperellipticCurve(PF!cheb(125));

3Numerator(ZetaFunction(H));

Executing the lines above outputs $(p^{2}x^{4}+1)(p^{10}x^{20}+1)(p^{50}x^{100}+1)$ in case $d=125,p=13$. As a consequence, ${\mathcal{H}}_{125}$ is maximal over ${\mathbb{F}}_{13^{n}}$ if and only if $n=100m$ with $m$ odd.

The next code computes the slopes $\frac{\#\left(\Phi_{\ell}\cap D_{p}\cdot\tau\right)}{\#D_{p}}$ used in Section 3.

⬇

1mg:=function(m,n) return Min(m mod n,n-(m mod n));end function;

2slopes:=function(ell, p)

3 Phi:={<(-1)^(n+1), mg(n,ell)> : n in [1..(ell-1)div 2]};

4 ord:=Modorder(p,ell); sl:={};

5 Dp:={<(-1)^(((p-1)div 2)*n), mg(p^n,ell)> : n in [1..ord]};

6 for m in [1..(ell-1)div 2]

7 do Dpmp:= { <x[1], mg(m*x[2],ell)> : x in Dp};

8 Include(~sl, #{x: x in Dpmp | x in Phi}/#Dp);

9 Dpmm:={ <-x[1], mg(m*x[2],ell)> : x in Dp};

10 Include(~sl, #{x: x in Dpmm | x in Phi}/#Dp);

11 end for;

12 return sl;

13end function;

The next code verifies that the only characteristics $p\equiv 1\bmod 4$ such that ${\mathcal{H}}_{89}$ attains maximality over extensions of ${\mathbb{F}}_{p}$, are the ones for which $p$ is a primitive root modulo $89$:

⬇

1ell:=89;

2classes:={p : p in [1..4ell-1] | Gcd(p,4ell) eq 1};

3for p in classes do

4 if slopes(ell,p) eq {1/2} then

5 if p mod 4 eq 1 then <p , Modorder(p,ell)>; end if;

6 end if;

7end for;

Below is code computing the slopes of Frobenius at $p$ for the CM abelian varieties described in Lemma’s 4.2.1 and 4.2.5.

⬇

1slopes2 := function(d, p)

2 enns:= {n : n in [1..(d-1)div 2] | Gcd(n,d) eq 1};

3 Phi:= {<(-1)^(n+1), mg(n,d)> : n in enns};

4 ord:= Modorder(p,d); sl:={};

5 Dp:= {<(-1)^(((p-1)div 2)*n), mg(p^n,d)> : n in [1..ord]};

6 for m in [1..(d-1)div 2] do

7 if Gcd(m,d) eq 1 then

8 Dpmp:= { <x[1], mg(m*x[2],d)> : x in Dp};

9 Include(~sl, #{x: x in Dpmp | x in Phi}/#Dp);

10 Dpmm:= { <-x[1], mg(m*x[2],d)> : x in Dp};

11 Include(~sl, #{x: x in Dpmm | x in Phi}/#Dp);

12 end if;

13 end for;

14 return sl;

15end function;

Finally, we present code used to check whether a given ${\mathcal{H}}_{\ell_{1}\ell_{2}}$ might attain maximality in some characteristic $p\equiv 1\bmod 4$. Following the discussion in §3.4 it is assumed (verified) that a necessary condition is that $p$ is a primitive root modulo both $\ell_{j}$’s. For primes $p$ in the congruence classes satisfying this, the code checks whether the abelian variety $A$ described in Lemma 4.2.1 has supersingular reduction at $p$ (so, set of slopes of Frobenius equal to $\{1/2\}$).

⬇

1primroots := function(p)

2 a:=PrimitiveRoot(p);

3 return {a^m mod p : m in [1..p-2] | Gcd(m,p-1) eq 1};

4end function;

5check := function(ell1, ell2)

6 s1:=primroots(ell1); s2:=primroots(ell2); ss:={};

7 cls:={ CRT([a,b,1], [ell1,ell2,4]) : a in s1, b in s2 };

8 for p in cls do

9 if slopes2(ell1*ell2,p)eq{1/2} then Include(~ss,p); end if;

10 end for;

11 return ss;

12end function;

Acknowledgment. We thank J.D. Top for making the two diagrams used in §4.2. The first author was partially supported by FAPESP grant No. 2023/08271-5.