𝔽 p 2 -maximal curves with many automorphisms are Galois-covered by the Hermitian curve

2021 β—½  
Vol 0 (0) β—½  
Author(s):  
Daniele Bartoli β—½  
Maria Montanucci β—½  
Fernando Torres
Keyword(s):  
Finite Field β—½  
Prime Numbers β—½  
Hermitian Curve β—½  
Maximal Curve β—½  
Maximal Curves

Abstract Let 𝔽 be the finite field of order q 2. It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curve H q + 1 : y q + 1 = x q + x ${{\mathcal{H}}_{q+1}}:{{y}^{q+1}}={{x }^{q}}+x$ is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x $\mathcal{x}$ of genus g β‰₯ 2 with | Aut(𝒳) | > 84(g βˆ’ 1) is Galois-covered by H q + 1 . ${{\mathcal{H}}_{q+1}}.$ The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x $\mathcal{x}$ for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 βˆ’ 1) which is not Galois-covered by the Hermitian curve H 72 . ${{\mathcal{H}}_{72}}.$

Magic squares of squares over a finite field

2021 β—½  
pp. 111-122
Author(s):  
Stewart Hengeveld β—½  
Giancarlo Labruna β—½  
Aihua Li
Keyword(s):  
Finite Field β—½  
Prime Numbers β—½  
Similar Question β—½  
Magic Squares β—½  
Content Type β—½  
Magic Square β—½  
Twin Primes β—½  
Quadratic Residues β—½  
Perfect Squares β—½  
Open Question

A magic square M M over an integral domain D D is a 3 Γ— 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: β€œIs there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≑ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

scholarly journals Automaticity of primitive words and irreducible polynomials

10.46298/dmtcs.632 β—½  
2013 β—½  
Vol Vol. 15 no. 1 (Automata, Logic and Semantics) β—½  
Author(s):  
Anne Lacroix β—½  
Narad Rampersad
Keyword(s):  
Finite Field β—½  
Finite Automaton β—½  
Prime Numbers β—½  
Deterministic Finite Automaton β—½  
Irreducible Polynomials β—½  
Letter Alphabet β—½  
International Audience β—½  
Primitive Words

Automata, Logic and Semantics International audience If L is a language, the automaticity function A_L(n) (resp. N_L(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.

scholarly journals On space maximal curves

2019 β—½  
Vol 53 (supl) β—½  
pp. 223-235
Author(s):  
Paulo César Oliveira β—½  
Fernando Torres
Keyword(s):  
Rational Points β—½  
Positive Element β—½  
Weierstrass Semigroup β—½  
Cubic Surface β—½  
Maximal Curve β—½  
Maximal Curves

Any maximal curveΒ X is equipped with an intrinsic embedding Ο€: X β†’ Pr which reveal outstanding properties of the curve. By dealing with the contact divisors of the curve Ο€(X) and tangent lines, in this paper we investigate the first positive element that the Weierstrass semigroup at rational points can have whenever r = 3 and Ο€(X) is contained in a cubic surface.

scholarly journals On maximal curves which are not Galois subcovers of the Hermitian curve

2012 β—½  
Vol 43 (3) β—½  
pp. 453-465 β—½  
Author(s):  
Iwan Duursma β—½  
Kit-Ho Mak
Keyword(s):  
Hermitian Curve β—½  
Maximal Curves

An 𝔽 p2 -maximal Wiman sextic and its automorphisms

2021 β—½  
Vol 21 (4) β—½  
pp. 451-461
Author(s):  
Massimo Giulietti β—½  
Motoko Kawakita β—½  
Stefano Lia β—½  
Maria Montanucci
Keyword(s):  
Riemann Surface β—½  
Automorphism Group β—½  
Finite Field β—½  
Symmetric Group β—½  
Closed Field β—½  
Complex Field β—½  
Hermitian Curve β—½  
Full Automorphism Group β—½  
Algebraically Closed Field

Abstract In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field β„‚ defined by the equation X 6+Y 6+ℨ 6+(X 2+Y 2+ℨ 2)(X 4+Y 4+ℨ 4)βˆ’12X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p β‰₯ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) β‰… PGL(2,𝕂). We also show that Wiman’s 𝔽192 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192 .

scholarly journals Weierstrass semigroup at $$m+1$$ rational points in maximal curves which cannot be covered by the Hermitian curve

2020 β—½  
Vol 88 (8) β—½  
pp. 1595-1616 β—½  
Author(s):  
Alonso Sepúlveda Castellanos β—½  
Maria Bras-Amorós
Keyword(s):  
Rational Points β—½  
Weierstrass Semigroup β—½  
Hermitian Curve β—½  
Maximal Curves

A QUOTIENT CURVE OF THE FERMAT CURVE OF DEGREE TWELVE ATTAINING THE SERRE BOUND

2005 β—½  
Vol 04 (02) β—½  
pp. 173-178 β—½  
Author(s):  
MOTOKO QIU KAWAKITA
Keyword(s):  
Finite Field β—½  
Prime Number β—½  
Prime Numbers β—½  
Fermat Curve β—½  
Prime Fields

We find a new curve of genus four attaining the Serre bound over prime fields. It is defined by the equation y12=x4(1-x), which attains the bound over the finite field [Formula: see text] if and only if the prime number p satisfies p ≑ 1 mod 12, [Formula: see text] and [Formula: see text] with an integer n. Furthermore, we show that if a standard conjecture of prime numbers is true then infinitely many prime numbers satisfy these conditions.

scholarly journals A new family of maximal curves over a finite field

2008 β—½  
Vol 343 (1) β—½  
pp. 229-245 β—½  
Author(s):  
Massimo Giulietti β—½  
Gábor Korchmáros
Keyword(s):  
Finite Field β—½  
New Family β—½  
Maximal Curves

scholarly journals SMALLEST IRREDUCIBLE OF THE FORM x2-dy2

2009 β—½  
Vol 05 (03) β—½  
pp. 449-456
Author(s):  
SHANSHAN DING
Keyword(s):  
Field Theory β—½  
Finite Field β—½  
Polynomial Ring β—½  
Function Field β—½  
Classical Result β—½  
Prime Numbers β—½  
Density Theorem β—½  
Chebotarev Density Theorem β—½  
Effective Version β—½  
Infinite Set

It is a classical result that prime numbers of the form x2 + ny2 can be characterized via class field theory for an infinite set of n. In this paper, we derive the function field analogue of the classical result. Then, we apply an effective version of the Chebotarev density theorem to bound the degree of the smallest irreducible of the form x2 - dy2, where x, y, and d are elements of a polynomial ring over a finite field.

A maximal curve which is not a Galois subcover of the Hermitian curve

2006 β—½  
Vol 37 (1) β—½  
pp. 139-152 β—½  
Author(s):  
Arnaldo Garcia* β—½  
Henning Stichtenoth
Keyword(s):  
Hermitian Curve β—½  
Maximal Curve
Sign in / Sign up

Export Citation Format

Share Document