arXiv subjects

Saeed Tafazolian

Publications and source records attributed to Saeed Tafazolian.

At least 19 recordsLinked to original sources

arXiv2026

Fermat quotients with CM Jacobians and trivial automorphism group

Let $F_n$ be the Fermat curve of degree $n$ over $\mathbb Q$, and let $D_n=F_n/S_3$ be its quotient by coordinate permutations. We prove that $D_n$ has trivial geometric automorphism group for every $n\ge 11$ with $\gcd(n,6)=1$. Its Jacobian has complex multiplication. For $n=11$, this Jacobian is absolutely simple of dimension five. We construct an isogeny over $\mathbb Q$ of degree $11^5$ from $\operatorname{Jac}(D_{11})$ to the Jacobian of $y^{11}=x(1-x)^2$. For primes $p\equiv 1\pmod 3$, we also consider the metacyclic quotients $C_p$ of $F_p$ over $\mathbb Q$. By a theorem of Carocca, Lange and Rodríguez, their Jacobians are absolutely simple and have complex multiplication. We prove that $C_p$ has trivial geometric automorphism group if and only if $p\ge 19$. Write $p=r^2+rs+s^2$ with $r,s$ positive integers. The natural degree-three cover $Y_p\to C_p$ satisfies $\operatorname{gon}(Y_p)=r+s$, so $\operatorname{gon}(C_p)\ge \lceil(r+s)/3\rceil$. Since $g(C_p)=(p-1)/6$, both the genus and the gonality of $C_p$ tend to infinity with $p$.

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arXiv2026

Projective symmetries of Chebyshev polynomials

Let $k$ be an algebraically closed field of odd characteristic $p$, and let $φ_d$ be the monic Chebyshev polynomial defined by $φ_d(t+t^{-1})=t^d+t^{-d}$. For $d>4$ with $p\nmid d$, let $H_d\le\text{PGL}_2(k)$ be the stabilizer of the zero set of $φ_d$. We show that \[ H_d\cong \begin{cases} D_d,&\text{if }2d=p^r+1\;\text{ for some integer }\;r\geq 1,\\ S_3,&\text{if }4d=p^r+1\;\text{ for some integer }\; r\geq 1,\\ C_2,&\text{otherwise}. \end{cases} \]

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arXiv2026

Automorphisms of Plane Curves defined from Chebychev polynomials

For any $d\geq 4$ we determine the automorphism groups of the algebraic curves \[ \mathcal{C}_d : y^d = φ_d(x), \] where $φ_d(x)$ denotes the Chebyshev polynomial of degree $d$, defined over a field $k$ with $p:=\operatorname{char}(k) \nmid 2d$. For $d>4$, it is $\mathbb{Z}/d\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$, except when $2d=p^r+1$ or $4d=p^r+1$ for some $r\geq 1$. In these cases it is, respectively, $(\mathbb{Z}/d\mathbb{Z}\times\mathbb{Z}/d\mathbb{Z})\rtimes S_3$ and $\mathbb{Z}/d\mathbb{Z}\times S_3$. Two ingredients used here are a description of all total inflection points on $\cC_d$, and knowledge of the group of fractional linear transformations that preserve the set of zeros of $φ_d$.

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arXiv2026

Transitive automorphism groups of maximal curves

Let $q$ be an odd prime power and let $X$ be a maximal curve of genus at least two over $\F_{q^2}$. We prove that the automorphism group of $X$ acts transitively on its rational points if and only if $X$ is the Hermitian curve, or $q=5$ and $X$ is the unique $\F_{25}$-maximal curve of genus $3$, namely the maximal Klein quartic. Together with the result of Giulietti and Korchmáros in characteristic two, this gives the classification for every prime power $q$. In particular, the Hermitian curve is the only maximal curve of genus at least two whose automorphism group is doubly transitive on its rational points.

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arXiv2026

A classification of zero $p$-rank curves with large automorphism groups in odd characteristic

Let $\cX$ be a curve of genus $g\ge2$ and $p$-rank zero over an algebraically closed field of odd characteristic $p$, and let $G\le\Aut(\cX)$ satisfy $|G|>24g(g-1)$. We prove that $G$ fixes a point unless $(\cX,G)$ belongs to an explicit list. For $g\ge4$, the curve is a cyclic quotient of a Hermitian curve, of a Giulietti--Korchmáros curve, or of Skabelund's cover of a Ree curve. We determine the equations and all the groups $G$ that occur, including their central kernels. The exceptional $A_7$ action in characteristic $5$ is on the Hermitian curve of degree $6$; genera $2$ and $3$ are treated separately. The equations are recovered from the group action by two rigidity lemmas for equivariant one-point $p$-group covers of the affine line. If $G$ has no common fixed point and its Sylow $p$-subgroups are noncyclic, then, unless $G$ induces ${}^2G_2(3)$ on the orbit of Sylow fixed points, the bound $|G|>\max\{84(g-1),2g^2\}$ gives a rational quotient with exactly one wild and one tame short orbit, and a rational Sylow quotient.

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arXiv2026

$a$-numbers of covers in characteristic two: semigroups and Riemann-Roch spaces

We compute the $a$-numbers of the Garcia-Güneri-Stichtenoth curves, the Beelen-Montanucci curves, and the Skabelund covers of Suzuki curves in characteristic two. The proofs use the identity $a(X)=\ell(\tfrac12\operatorname{div}(dt))$ and reduce the calculations to Riemann-Roch spaces on the base curves. We obtain transfer formulas for odd-degree radical covers, with one or several points at infinity, and evaluate them using Weierstrass semigroups and plane sections of the Hermitian curve. For the first two families we also give cyclic models over the projective line and relate the formulas to Elkin's rank formula. We then treat two classes of Artin-Schreier fibre products. A supersingular genus-six family has a-number $3$ at the zero parameter and $2$ elsewhere, although the genera and $a$-numbers of its quadratic quotients remain fixed. We determine its Ekedahl--Oort types as $(0,0,0,1,2,3)$ and $(0,1,1,2,3,4)$,

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arXiv2026

$a$-Numbers, higher Cartier ranks and maximality of van der Geer-van der Vlugt curves

Let $p$ be an odd prime and let $C_R$ be the van der Geer-van der Vlugt curve $y^p-y=xR(x)$ over an algebraically closed field of characteristic $p$, where $R(x)=\sum_{i=0}^{h}a_ix^{p^i}$ is an additive polynomial of degree $p^h$. We determine the image and the rank of every power of the Cartier operator $\Cart$ on $H^0(C_R,\Omg)$. For $1\le r\le h-1$ the image of $\Cart^r$ is spanned by an explicit set of monomial differentials that does not depend on $R$, and $\rank(\Cart^r)=p^{h-r-1}\binom{p+r}{r+2}$. The last iterate detects the linear coefficient: $\rank(\Cart^h)=0$ if $a_0=0$ and $\rank(\Cart^h)=\binom{(p-1)/2+h}{h+1}$ if $a_0\neq0$, so the nilpotency index of $\Cart$ is $h$ or $h+1$. Hence, for $h\ge2$, the loci $a_0=0$ and $a_0\ne0$ have the same $a$-number, which is the largest $a$-number of a one-point Artin-Schreier curve with pole order $p^h+1$, but they share no Ekedahl-Oort type. Over finite fields we deduce that a curve over $\F_{p^{2n}}$ that is geometrically isomorphic to $C_R$ can be $\F_{p^{2n}}$-maximal or $\F_{p^{2n}}$-minimal only if $n\ge h$, and only if $n\ge h+1$ when $a_0\neq0$. Explicit examples show that the second bound is attained for $(p,h)=(3,1)$, $(3,2)$ and $(5,2)$.

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arXiv2026

Artin-Schreier curves of prescribed p-rank without extra automorphisms

Let p be an odd prime. We study the Artin-Schreier analogue of Poonen's problem of constructing curves without extra automorphisms. An Artin-Schreier curve carries an unavoidable deck group Cp, so the smallest possible full geometric automorphism group inside the Artin-Schreier locus is Cp. We determine the exact numerical range in which this minimum is attained, for every p-rank. More precisely, for integers r >= 0 and N >= 1, there exists an Artin-Schreier curve X/Fp with g(X) = (p - 1)N/2, gamma(X) = r(p - 1), Aut_Fp(X) is isomorphic to Cp if and only if either r = 0, p does not divide N + 1, N >= 3 and (p,N) not equal to (3, 3), or r >= 1, N >= 2r and (r,N) not equal to (1, 2). All models are defined over the prime field by explicit equations together with elementary choices of auxiliary polynomials, and every excluded admissible case is shown to carry an extra geometric automorphism. The main structural input is the theorem of Valentini and Madan on the uniqueness of the rational subfield of index p outside three exceptional families. We exclude these families by intrinsic arguments: the genus and p-rank in most cases, Nakajima's bound on p-subgroups of automorphism groups of ordinary curves for the ordinary family, and the Lehr-Matignon normal form for isomorphisms of one-point covers for the monomial family.

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arXiv2026

Explicit LCP of MDS Codes and LCD Codes on Hyperelliptic Curves via Mumford Representation

We study algebraic geometry codes on hyperelliptic curves of genus $g \geq 2$ equipped with a rational Weierstrass point, with a focus on complementarity properties. Our first contribution is a characterization of non-special divisors of degree $g$ and $g-1$ via the polynomial degrees of their reduced Mumford representation, reducing a classical geometric problem to a degree test on univariate polynomials. Using this, we construct Linear Complementary Pairs (LCP) of codes via polynomial arithmetic on the Jacobian and provide a criterion in terms of Mumford degrees for the resulting codes to be Maximum Distance Separable (MDS). Under a $2$-torsion condition in the Jacobian, equivalently a divisibility condition on the Mumford polynomials, we obtain explicit multipliers that turn these pairs into Linear Complementary Dual (LCD) codes. We give an MDS LCP with parameters $[8,4,5]_{16}$ in characteristic two, an MDS LCD code with parameters $[10,5,6]_{25}$ on the maximal hyperelliptic curve $y^2=x^5+x$, and an LCD code with parameters $[28,14]_{49}$ on $y^2=x^7+x$. Motivated by the explicit $q=5$ computation, we conjecture for odd prime powers $q\ge5$ the existence of MDS LCD codes with parameters $[2q,q,q+1]_{q^2}$ arising from this construction.

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arXiv2026

Sharp bounds and gap phenomena for large $p$-subgroups of automorphism groups of curves

Let $X$ be a curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\le G_P$ satisfies $|S|>\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, or Ree curves. We determine the sharp boundary and the first gap below it. At equality, besides the classical curves, exactly three types occur: the ordinary curves $y^p-y=x+c/x$, the zero-$p$-rank curves $y^3=x^p-x$ for $p\equiv2\pmod3$, and, in characteristic $3$, the ordinary genus-six family $y^3-y=c\left(x+\frac1{x^3-x}\right)$. As a key ingredient, we classify zero-$p$-rank curves whose full automorphism group fixes no point and which admit a $p$-subgroup of order larger than the genus. We then prove that, when $|S|>p$, the boundary classification already follows from the weaker inequality $|S|>p(g-p+1)/(p-1)$. For $|S|\ge p^3$, the forbidden interval above $m=(p-1)|S|/p$ has width at least $p(p-1)/2$, and below $m+(p^2-1)/2$ at most one further genus can occur, with a rigid ramification structure. Finally, we classify the first admissible layer: it occurs only for $|S|=p^2$, and the resulting ordinary curves admit a complete description in terms of two-dimensional Artin--Schreier spaces.

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arXiv2026

A Quadratic Bound for Automorphism Groups of Curves of Positive $p$-Rank in Odd Characteristic

Let $X$ be a nonsingular projective curve of genus $g\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchmáros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\ge2$, positive $p$-rank implies $|\Aut(X)|<900g^2$. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse--Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.

math.AG↗

arXiv2026

Cyclic automorphisms beyond the canonical bound

Let X be a smooth projective curve of genus g >= 2 over an algebraically closed field K, and suppose that Aut(X) contains a cyclic subgroup G of order N > 2g - 2. The cases N >= 2g + 1 are known. We treat the two boundary values N = 2g and N = 2g - 1. In the tame case we obtain an explicit finite list of ramification signatures. In odd characteristic we prove that the Sylow p-subgroup of G has order p and determine all wild boundary cases by a direct ramification analysis; the resulting curves are Artin-Schreier-Kummer fiber products. In characteristic two we prove that the 2-part has order at most four and classify all wild boundary cases, including Artin-Schreier-Witt normal forms for the C4 and C12 cases.

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arXiv2026

A characterization of the Ree curve

Let $q=3q_0^2$, where $q_0=3^s$ and $s\ge3$. We prove that a smooth, projective, geometrically irreducible curve over $\F_q$ with $q^3+1$ rational points and genus \[ \frac32q_0(q-1)(q+q_0+1) \] is $\F_q$-isomorphic to the Ree curve.

math.AG↗

arXiv2026

Nonexistence of maximal curves of genus five over $\F_{64}$

We show that there is no maximal curve of genus five over $\F_{64}$. As a consequence, $N_{64}(5)=140$, and the genus spectrum of maximal curves over $\F_{64}$ is determined. The proof uses the vanishing of the third iterate of the Cartier operator. We prove that a nonhyperelliptic curve of genus five in characteristic two satisfying this condition is nontrigonal. Its canonical theta characteristic defines a separable cover of degree four with one geometric branch value. The two possible ramification types give either a rational subcanonical point or a point bound obtained from the cubic resolvent. Both cases exclude maximality over $\F_{64}$.

math.AG↗

arXiv2026

On the generalized Ree curve

Let p be an odd prime, let q_0 := p^s with s >= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the classical Ree curve. In analogy with the generalized Suzuki curve, we call R_{p,s} the generalized Ree curve associated with p and s. For p > 3 our principal geometric result determines the full geometric automorphism group: Aut_{F_q}(R_{p,s}) is isomorphic to the semidirect product of U and F_q^*, with |U| = q^3. The natural p-group U gives a big action; we determine the complete ramification filtration of the elementary abelian cover R_{p,s} -> P^1_x and prove that (R_{p,s}, P_infinity) is Castle. In characteristic 3 the same subgroup is only the stabilizer of P_infinity in the full Ree group, so the automorphism structure exhibits a sharp characteristic-three dichotomy. On the arithmetic side, writing Y_{p,s} : y^q - y = x^{q_0}(x^q - x), Z_{p,s} : z^q - z = x^{2q_0}(x^q - x), we prove the F_q-isogeny Jac(R_{p,s}) ~ Jac(Y_{p,s}) x Jac(Z_{p,s})^q. The first factor is supersingular and all three curves have p-rank zero. For p > 3 the first Newton slope of Z_{p,s} satisfies 1/(pq_0 + 2) <= lambda_min(Z_{p,s}) <= 1/3, so Z_{p,s} and R_{p,s} are not supersingular. For fixed p > 3 and s -> infinity, the order of the full automorphism group is asymptotic to 2^(8/5) p^(-4/5) q^(8/5).

math.AG↗

arXiv2026

Nakajima-extremal Artin--Schreier covers of ordinary elliptic curves in characteristic $2$

Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which \[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \] These curves realize case \textup{(ib)} in the classification of Giulietti--Korchmáros and give an infinite family answering a problem posed by Korchmáros. More generally, we prove that every curve in case \textup{(ib)} arises from the same construction. Case \textup{(ib)} occurs exactly in genera $g=2^h+1$ with $h\ge2$. For $g\ge9$ we determine the full automorphism group; in genus $5$ we determine its Sylow $2$-subgroup but do not claim the full automorphism group. For every fixed $q\ge8$, the isomorphism classes in case \textup{(ib)} of genus $q+1$ are parametrized bijectively by $(k^\times)^2$. The construction is described in terms of an ordinary elliptic curve and an invariant differential. We determine the short orbits and ramification, the unramified cyclic quotients, and the quotients by the central involutions. For $q=8$ an explicit plane model over $\F_2$ is given.

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arXiv2026

On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves

Let $\Sm_q$ and $\Rm_q$ denote the Suzuki and Ree curves. Motivated by the Giulietti--Korchmáros curve, Skabelund constructed cyclic covers $\tSm_q$ and $\tRm_q$ of these curves and proved that they are maximal over $\F_{q^4}$ and $\F_{q^6}$, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class field covers $\Sm_{\rm rcf}$ and $\Rm_{\rm rcf}$, and showed that there are towers \[ \Sm_{\rm rcf}\longrightarrow\tSm_q\longrightarrow\Sm_q, \qquad \Rm_{\rm rcf}\longrightarrow\tRm_q\longrightarrow\Rm_q . \] Computations for small values of $q$ suggested that the first map in each tower is always an isomorphism, and the general case was left open. We prove that \[ \Sm_{\rm rcf}=\tSm_q,\qquad \Rm_{\rm rcf}=\tRm_q \] for every admissible $q$, the comparison being made over $\F_{q^4}$ in the Suzuki case and over $\F_{q^6}$ in the Ree case. The proof combines a Kummer normal form of the ray class extension, allowing a constant twist, with the centrality of its Galois group among lifted automorphisms, the standard involution of the base curve, and the first positive non-gap at the rational point at infinity.

math.AG↗

arXiv2026

Maximal and minimal curves of the form $y^3=x^{(q^2+1)/2}+x$

Let $p\ge 5$ be a prime with $p\equiv -1\pmod 3$, let $q=p^r$, and consider \[ \cC:\qquad y^3=x^{(q^2+1)/2}+x \] over $\F_{q^6}$. We prove the exact formula \[ \#\cC(\F_{q^6})=q^6+1+(-1)^{r+1}(q^2-1)q^3. \] Since $g(\cC)=(q^2-1)/2$, the curve is maximal when $r$ is odd and minimal when $r$ is even. The proof uses a birational Kummer model and an explicit Jacobi-sum point count. A congruence together with Frobenius invariance reduces the relevant Jacobi sums to cubic Gauss sums, whose sign is determined from the Fermat cubic. In particular, the maximality of $y^3=x^{13}+x$ over $\F_{5^6}$ appears as the first case of an infinite family.

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