Algebraic curves through Fernando Torres' lens

Algebraic curves through Fernando Torres’ lens

Herivelto Borges

Universidade de Sao Paulo

Carleton Finite Fields eSeminarA tribute to Fernando Torres

Dec 2020

Picture by Cıcero Carvalho 1 / 28

Fernando Torres: in the begginng

Fernando Torres was born in Tarma-Peru in June 23, 1961.

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Fernando Torres: The student

Fernando went to Elementary and Middle school in Tarma, at

the school Nino Jesus en Praga and San Vicente de Paul.

He went to high school in Lima, at

the Santo Tomas de Aquino school.

After finishing high school, Fernando went to the PontificiaUniversidad Catolica del Peru - PUCP, where he got his

Undergraduate degree in Mathematics in 1985.

Masters degree in Mathematics in 1988.

In August of 1988, Fernando came to Brazil, and started hisPh.D at IMPA.

He finished his Ph.D in 1993.

Arnaldo Garcia was his Ph.D. advisor.

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Fernando’s Carrer

Fernando remained at IMPA in 1994, on a postdoctoralfellowship.

In 1995 and 1996 he was in the math department at ICTP,in Trieste.

In 1997 he went to Essen, Germany, and this was his lastyear as a postdoc.

In 1998 he became a faculty in the University of Campinas(Unicamp).

During his time at Unicamp he went abroad several times,especially to Italy and Spain, where he made many friendsand collaborated with many mathematicians.

Fernando became a full Professor in 2015.

In May 28th, 2020, Fernando passed away.

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Fernando’s Carrer

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Fernando’s Carrer

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Fernando’s Carrer

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Fernando’s Carrer

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Fernando’s Carrer

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Fernando’s Carrer

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Fernando’s Carrer:

Fernando published over 50 articles, and co-authored,together with G. Korchmaros and J.W.P. Hirschfeld, thefamous book

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Fernando supervised 12 Ph.D. students:

Stefani Concolato Vieira, 2020+ε

Matheus Bernardini de Souza, 2017.

Wanderson Tenorio, 2017.

Steve Vicentim, 2016.

Paulo Cesar Cavalcante de Oliveira 2016.

Arnoldo Rafael Teheran Herrera, 2014.

Beatriz Casulari da Motta Ribeiro, 2011.

Rafael Peixoto, 2011.

Alonso Sepulveda Castellanos, 2008.

Guilherme Chaud Tizziotti, 2008.

Juan Elmer Villanueva Zevallos, 2008.

Ercılio Carvalho da Silva, 2004

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Fernando’s Ph.D. thesis

Weierstrass points and double cover of curves (1993)

It is well known that if Y is a curve of genus g ≥ 2, then thefollowing are equivalent:

(i) Y is a hyperelliptic curve;

(ii) there exists a point P ∈ Y such that H(P ) is hyperelliptic(i.e. 2 ∈ H(P ))

(iii) there exists a point P ∈ Y such that w(P ) =

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Fernando’s thesis

Let X be a curve (algebraic, projective, non-singular,irreducible) defined over an algebraically closed field ofcharacteristic zero.Fernando’s definition: Let γ be a non-negative integer. Acurve X is called γ -hyperelitic curve when X is a doublecover of a genus γ curve.

A numerical semigroup H ⊂ N0 is called γ -hyperelitic ifthe following holds.

(i) H has exactly γ even elements in the interval [2, 4γ].(ii) 4γ + 2 ∈ H

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Fernando’s thesis

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Fernando’s thesis

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Fernando’s thesis

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Fernando’s thesis

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Fernando’s Thesis

In his thesis, Fernando generalized these equivalences for thecase of the γ -hypertelitic curves,

Theorem (Torres 1995)

Let X be a curve of genus g ≥ 6γ + 4, for some γ ∈ Z≥0. Thenthe following are equivalent.

(i) X is γ-hyperelliptic

(ii) There exists P ∈ X such that H(P ) is γ-hyperelliptic

(iii) There exists a base-point-free complete linear series of Xof projective dimension 2γ + 1 and degree 6γ + 2.

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Fernando’s Thesis

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Fernando’s Thesis

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Fernando’s Thesis

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Fernando’s Thesis

Let X be a curve of genus g, and let γ ∈ Z≥1 be such that

{30, if γ = 1(12γ−6

)+ 1, if γ ≥ 2.

Then the following are equivalent.

(ii) There exists P ∈ X such that(g−2γ2

)≤ w(P ) ≤

(g−2γ2

)+ 2γ2

(iii) There exists P ∈ X such that(g−2γ2

)≤ w(P ) ≤

(g−2γ+2

)10 / 28

Fernando’s Thesis

{30, if γ = 1(12γ−6

)+ 1, if γ ≥ 2.

)≤ w(P ) ≤

(g−2γ2

)+ 2γ2

)≤ w(P ) ≤

(g−2γ+2

)10 / 28

Fernando’s Thesis

{30, if γ = 1(12γ−6

)+ 1, if γ ≥ 2.

)≤ w(P ) ≤

(g−2γ2

)+ 2γ2

)≤ w(P ) ≤

(g−2γ+2

)10 / 28

Fernando’s Thesis

{30, if γ = 1(12γ−6

)+ 1, if γ ≥ 2.

)≤ w(P ) ≤

(g−2γ2

)+ 2γ2

)≤ w(P ) ≤

(g−2γ+2

)10 / 28

A very influential paper in Fernando’s career

Seminal paper: Stohr, K.O. and Voloch, J.F.: Weierstrasspoints and curves over finite fields. Proc LondonMath. Soc .52, 1− 19(1986)

It gives a new proof to the Riemann-Hypothesis for curves overfinite fields, and in several circumstances, it improves thefamous Hasse-Well bound:

N ≤ 1 + q + 2g√q. (1)

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N ≤ 1 + q + 2g√q. (1)

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N ≤ 1 + q + 2g√q. (1)

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N ≤ 1 + q + 2g√q. (1)

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A flavor of the Stohr-Voloch theory

Let X be a curve of genus g, and let D = grd be a base-point-freeFq-linear series on X . Associated to a point P ∈ X we have theHermitian P -invariants

j0(P ) = 0 < j1(P ) < . . . < jr(P ) ≤ d

of D, also called the (D, P )-orders.

This sequence is the same, namely,

ε0 < ε1 < · · · < εr,

for all but finitely many points P ∈ X .

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j0(P ) = 0 < j1(P ) < . . . < jr(P ) ≤ d

ε0 < ε1 < · · · < εr,

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j0(P ) = 0 < j1(P ) < . . . < jr(P ) ≤ d

ε0 < ε1 < · · · < εr,

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The Stohr-Voloch divisor

In addition to the order-sequence ε0, ε1, · · · , εr, there existsanother sequence of nonnegative integers

ν0 < ν1 < · · · < νr−1,

called Frobenius orders (of D), which is such that

{ν0, ..., νr−1} = {ε0, ..., εr}\{εI},

for some suitable I ∈ {1, . . . , r}.

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The Stohr-Voloch theorem

Theorem

Let X be an irreducible, nonsingular, projective, algebraic curveof genus g defined over Fq, equipped with a base-point-free linearseries D = grd over Fq. If D has order-sequence (ε0, ..., εr), andFq-Frobenius order-sequence (ν0, ..., νr−1), then the number N ofFq-rational points of X satisfies

N ≤ (ν1 + · · ·+ νr−1)(2g − 2) + (q + r)dr∑i=1

(εi − νi−1). (2)

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Maximal Curves

Fernando’s main contributions to the theory of Maximal curvesare related to the following important problems:

Determination of the possible genera of maximal curvesover Fq2Determination of explicit equations for maximal curvesover Fq2 .

Classification of maximal curves over Fq2 of a given genus.

The main ingredient of Fernando’s approach was the systematicstudy of the Frobenius linear series

D = |(q + 1)P0|,

attached to an Fq2-maximal curve, form the Stohr-Volochtheory viewpoint.

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Maximal Curves

D = |(q + 1)P0|,

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Maximal Curves

D = |(q + 1)P0|,

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Maximal Curves

D = |(q + 1)P0|,

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Maximal Curves

D = |(q + 1)P0|,

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Genera of Maximal Curves

Let us consider the set

:= {g ∈ N : g is the genus of an Fq2 -maximal curve }

The full description of the above set is a hard problem even forsmall values of q.The two first important results regarding M

1 (Ihara-1981) M(q2)⊆ [0, q(q − 1)/2].

2 (Ruck and Stichtenoth-1994) Up to isomorphism, theHermitian curve

H : yq + y = xq+1

over Fq2 is the unique Fq2-maximal of genus g = q(q−1)2 .

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1 (Ihara-1981) M(q2)⊆ [0, q(q − 1)/2].

H : yq + y = xq+1

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1 (Ihara-1981) M(q2)⊆ [0, q(q − 1)/2].

H : yq + y = xq+1

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In 1995, results by Stichtenoth and Xing gave rise to thefollowing question

Is is true that M(q2)⊆ [0, (q − 1)2/4] ∪ {q(q − 1)/2}?

Theorem (Fuhrmann-Torres (1996))

If the curve X is an Fq2-maximal of genus g, then either

g = q(q−1)2 or

g ≤ (q − 1)2

The important ingredients of their proof was the study of thelinear system

D = gr+1q+1 := |(q + 1)P0|

via Stohr-Voloch theoory and Castelnuovo’s genus bound forcurves in projective spaces.

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g = q(q−1)2 or

g ≤ (q − 1)2

D = gr+1q+1 := |(q + 1)P0|

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g = q(q−1)2 or

g ≤ (q − 1)2

D = gr+1q+1 := |(q + 1)P0|

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g = q(q−1)2 or

g ≤ (q − 1)2

D = gr+1q+1 := |(q + 1)P0|

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g = q(q−1)2 or

g ≤ (q − 1)2

D = gr+1q+1 := |(q + 1)P0|

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g = q(q−1)2 or

g ≤ (q − 1)2

D = gr+1q+1 := |(q + 1)P0|

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Theorem (Fuhrmann-Garcia-Torres (1997))

If q is odd, then any Fq2-maximal of genus g = (q − 1)2/4 isFq2-isomorphic to the curve

yq + y = x(q+1)/2

Theorem (Abdon-Torres 1998)

If q is even, and X is an Fq2-maximal curve of genus

g =⌊(q−1)2

⌋, then X is Fq2-isomorphic to

yq/2 + yq/4 + · · ·+ y = xq+1

provided that q/2 is a Weierstrass non-gap at some point of theX .

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yq + y = x(q+1)/2

g =⌊(q−1)2

yq/2 + yq/4 + · · ·+ y = xq+1

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yq + y = x(q+1)/2

g =⌊(q−1)2

yq/2 + yq/4 + · · ·+ y = xq+1

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Theorem (Korchmaros-Torres (2002))

The genus g of an Fq2-maximal curve satisfies either

g ≤⌊(q2 − q + 4

)/6⌋

or g =

⌊(q − 1)2

⌋or g = (q − 1)q/2.

To finish this part, I want to point out that one of Fernando’smost recent results regarding the set M

can be found in

(Arakelian-Tafazolian-Torres) On the spectrum for thegenera of maximal curves over small Fields. Adv. Math.Commun. 12 (2018), 143–149.

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g ≤⌊(q2 − q + 4

)/6⌋

or g =

⌊(q − 1)2

⌋or g = (q − 1)q/2.

can be found in

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g ≤⌊(q2 − q + 4

)/6⌋

or g =

⌊(q − 1)2

⌋or g = (q − 1)q/2.

can be found in

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Natural Embedding Theorem

Every Fq2-maximal curve is Fq2-isomorphic to a curve of degreeq + 1 lying on a non-degenerate Hermitian variety Hm,q ⊆ Pm.

There are many other results which are very important in thetheory of Maximal curves and which a due to Fernando.

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Algebraic Geometry Codes

Let X be a non-singular, projective, geometrically irreducible,algebraic curve of genus g over Fq. Let D = P1 + · · ·+ Pn andG be Fq-rational divisors on X , where P1, . . . , Pn are pairwisedistinct rational points on X , not in the support of G. Let L(G)be the Riemann-Roch space associated to G; The Fq-vectorspace

CL(D,G) := {(f (P1) , . . . , f (Pn)) | f ∈ L(G)} ⊆ Fnq

is an [n, k, d]-code with parameters

k = `(G)− `(G−D) and d ≥ n− degG

Some challenging problems are

Computation of the parameters [k, d].

Construction of codes with large minimum distance d.

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Fernando’s contributions in AG Codes

Some of the most important results by Fernando and hiscollaborators in coding theory are:

Carvalho, C.; Munuera, C.; da Silva, E.; Torres, F..Nearorders and codes. IEEE Trans. Inform. Theory53(5)(2007), 1919− 1924

Munuera, C.; Torres, F.. A note on the order boundon the minimum distance of AG codes and acutesemigroups. Adv. Math. Commun. 2(2)(2008), 175− 181

Geil, O.; Munuera, C.; Ruano, D.; Torres, F.. On theorder bounds for one-point AG codes. Adv. Math.Commun. 5(3)(2011), 489− 504.

Munuera, C.; Sepulveda, A.; Torres, F.. Castle curvesand codes. Adv. Math. Commun. 3(2009), no.4, 399− 408

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Munuera, C.; Tenorio, W.; Torres, F.. Quantumerror-correcting codes of Castle type. Quantum Inf.Process. 15 (2016), no. 10, 4071− 4088

Munuera, C.: Tenorio, W.; Torres, F.. Locallyrecoverable codes from algebraic curves withseparated variables. Adv. in Mathematics ofCommunications 14(2), 2020, 265− 27

Farran, J. I.; Munuera, C.; Tizziotti, G.; Torres, F..Grobner basis for norm-trace codes. J. SymbolicComput. 48(2013), 54− 63.

Munuera, C.; Sepulveda, A.; Torres, F. GeneralizedHermitian codes. Des. Codes Cryptogr. 69(2013), no.1, 123− 130.

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Fernando’s contributions Finite Geometry

An (n, d)-arc K is a set of n points of the projective planePG(2, q) over Fq such that no line meets K in more than dpoints and there exists a line that meets K in exactly d points.

The arc is called complete if it is not contained in a(n+ 1, d)-arc; that is to say, if for every point P of PG(2, q) \ Kthere is a line through P meeting K in d points.

FACT: (n, d)-arcs give rise linear codes with parameters[n, 3, n− d]

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A few facts:

Plane curves of degree d in PG(2, q) are natural sources of(n, d)-arcs

The Hermitian curve yq + y = xq+1 gives rise to a complete(q3 + 1, q + 1)-arc in PG(2, q2).

Non-singular Fq2-maximal curves give rise to complete(1 + q2 + 2q, 3)-arc in PG(2, q2).

Question: Which curves of degree d and with n rationalpoints give rise to a complete (n, d)-arc in PG(2, q) ?

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A few facts:

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A few facts:

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A few facts:

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M. Giulietti, F. Pambianco, F. Torres and E. Ughi, Oncomplete arcs arising from plane curves, Des. CodesCryptogr. 25(2002), 237− 246.

H. Borges, B. Motta, F. Torres, Complete arcs arisingfrom a generalization of the Hermitian curve, ActaArith., 164(2014)101− 118

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M. Giulietti, F. Pambianco, F. Torres and E. Ughi, Oncomplete arcs arising from plane curves, Des. CodesCryptogr. 25(2002), 237− 246.

H. Borges, B. Motta, F. Torres, Complete arcs arisingfrom a generalization of the Hermitian curve, ActaArith., 164(2014)101− 118

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Other important results by Fernando

There are many other important results by Fernando thatwere not reported here.

The same is true for his endless list of importantcollaborators: A. Cossidente, R. Pellikaan, M. Bernadini, J.Villanueva, D. Bartoli, M. Montanucci, A. Kazemifard, J.J.Moyano-Fernandez, ...

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Other important results by Fernando

There are many other important results by Fernando thatwere not reported here.

The same is true for his endless list of importantcollaborators: A. Cossidente, R. Pellikaan, M. Bernadini, J.Villanueva, D. Bartoli, M. Montanucci, A. Kazemifard, J.J.Moyano-Fernandez, ...

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One of his many happy moments:

We express our profound gratitude for each moment that wewere privileged to live and work with you, Fernando.

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Algebraic curves through Fernando Torres' lens

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