Optimal curves of genus $1$, $2$ and $3$

by

Christophe Ritzenthaler

Abstrat. In this survey, we disuss the problem of the maximum numberof points of

urvesofgenus

1, 2

^and

3

^{overniteelds.}

Résumé (Courbes optimales de genre

1, 2

^et

3

^{). Nous examinons la question du}

nombremaximumdepointspourlesourbesdegenre

1, 2

^et

3

^{surlesorpsnis.}

1. Introdution

The foundations of the theory of equations over nite elds were laid, among others, by

mathematiianslike Fermat,Euler, GaussandJaobi(see [Di66℄). Subsequently,therewas

little ativityinthe eld at leastuntil theend of the19thentury andthe studyof thezeta

funtion of a urve. Initiated by Dedekind, Weber, Artin and Shmidt, this work led to an

analogue ofthe Riemann hypothesiswhihwasprovedbyHasseintheaseofellipti urves

andthen byWeil ingeneral in1948 (see[Wei48 ℄). Thethird,modern, periodstarts in1980

with the work of Goppa [Gop77, Gop88℄. His onstrution of error-orreting odes with

good parameters fromurvesovernite eldsrenewed theinterest inthis theory.

Withthis appliationinmind,thetheoryhasfousedonthemaximumnumberofpointsofa

(projetive,geometriallyirreduible,nonsingular)urveofgenus

g

^overaniteeld

k = F q

^,

denoted

N _q (g)

^{. Asymptoti results, i.e. values of}

N _q (g)/g

^when

g

^{goes to innity and}

q

^is

xed,drewattentionrst,butSerre,inhisleturesatHarvard[Ser85℄,gaveequaltreatment

to the `dual'ase, i.e. values of

N _q (g)

^when

g

^{is xedand}

q

^{varies. It quiklyappearedthat}

determining

N _q (g)

^{was a hard problem and as soon as}

g ≥ 3

^{, only sparse values are known}

(see forinstane the web page www.manypoints.orgfor thebestestimates when

q

^issmall).

Inthissurvey,we aregoingtodesribe themainideas thathave been developedtodealwith

2000 MathematisSubjetClassiation. Primary11G20,11G10Seondary14K25,14H45.

Key wordsand phrases. Optimalurve,isogenylass,indeomposablepolarization,hermitianmodule,

Serre'sobstrution,planequarti, Siegelmodularform,Hasse-Weil-Serrebound.

TheauthoraknowledgespartiallysupportedbygrantMTM2006-11391fromtheSpanishMECandbygrant

ANR-09-BLAN-0020-01fromtheFrenhANR..

theases

1 ≤ g ≤ 3

^{. Itis}interestingtonotethatforeahvalueof

g

^{,wewillbeonfrontednot}

onlytoharderomputationsbutalsotoaompletelynewkindofissue. Inordertoemphasize

thisprogression,wewillnotonsidertheatualvalueof

N _q (g)

^{butthefollowing}sub-problem.

As we shall reall in Setion 2.1 ,

N q (g) ≤ 1 + q + g ⌊ 2 √ q ⌋

^{and we an wonder when}

N q (g)

reahes this bound. If it does, a urve with this number of points is alled optimal and we

aregoingto askfor whih valuesof

q

^{suh urvesexist.}

When

g ≤ 3

^{,thelassial gameto prove or disprove the existeneof optimalurvesis}

1. to prove theexistene(ornot)ofan abelian variety

A/k

^{witha`good'Weil polynomial}

(Setion 2). Thisisgoingto ontrol thenumberofpointson apossibleurve

C/k

^suh

that

Jac C ≃ A

^.

2. to put a good polarization

a

^{on A suh that}

(A, a)/k

^is geometrially (i.e. over

¯ k

^{) the}

Jaobian ofa urve

C ¯

^{withits anonial} polarization (Setion3).

3. to see if

C ¯

^{admitsa model}

C/k

^{suh that}

(Jac C, j) ≃ (A, a)

^(where

j

^{is the anonial}

polarizationof

C

^{). Wewillseethatif}

C ¯

^isnonhyperellipti,thereanbeanobstrution to this desent (see Setion 4) and for

g = 3

^{,we will propose solutions to address the}

omputation ofthe obstrution (see Setion 5).

Mostideas we aregoing to present here are alreadyontained in[Ser85℄but our proofsfor

g = 1

^and

2

^{aresometimes dierent fromthe original onesandtake advantageof subsequent}

simpliationsof thetheory.

Conventions and notation. In the following

g ≥ 1

^{is an integer and}

q = p ⁿ

^with

p

^a

primeand

n > 0

^{an integer. The letter}

k

^{denotes the niteeld}

F q

^and

K

^{any perfet eld.}

When we speak about a genus

g

^{urve we mean that the urve is projetive,} geometrially irreduible and non-singular. If

A

^and

B

^{are varieties over a eld}

K

^{, when we speak of a}

morphismfrom

A

^to

B

^{wealways meanamorphism dened over}

K

^{. So,forinstane}

End(A)

isthering ofendomorphisms denedover

K

^,

A ∼ B

^means

A

^{isogenous to}

B

^over

K

^{,et. If}

(A, a)

^and

(B, b)

^{arepolarizedabelian varieties,byan}isomorphismbetween them,wealways mean`aspolarizedabelian varieties'.

Aknowledgements. I would like to thank Christian Maire for suggesting me to write

this survey. This is part of my `habilitation' thesis [Rit09℄ whih was defended during the

workshopTheory of Numbers and Appliations whih was organizedbyKarim Belabas and

Christian Maire in Luminy in Deember 2009. I am really grateful to Detlev Homann for

thereferenes of Remark3.5and to the Number TheoryList ommunityand partiularly to

SamirSiksek for helping mewithRemark 3.10 .

2. Control of the isogeny lass

2.1. Bounds. Let

C/k

^beagenus

g

^{urve. We reallthatitsWeilpolynomial}

χ _C

^isthe

Weilpolynomialof

Jac C/k

^{,i.e. the}harateristipolynomialoftheationofthe

k

^-Frobenius

endomorphismon an

ℓ

^{-adiTatemoduleforanyprime}

ℓ 6 = p

^{. It iswellknownthatitanbe}

written

χ C = Y g i=1

(X ² + x i X + q) ∈ Z [X]

with

x _i ∈ R

^and

| x _i | ≤ 2 √ q

^{. Sine}

#C(k) = q + 1 + X g

i=1

x _i ,

it islear that

#C(k) ≤ 1 + q + ⌊ 2g √ q ⌋

^{,whih is known asHasse-Weil bound [Wei48 ℄ and}

so

N _q (g)

^{is less than this bound too. It is possible to improve this bound as the following}

lemmashows.

Lemma 2.1 (Hasse-Weil-Serre bound [Ser83b℄). Let

m = ⌊ 2 √ q ⌋

^{. Then}

N _q (g) ≤ 1 + q + gm.

Proof. It isenough to usethearithmeti-geometri meaninequality:

1 g

X g i=1

(m + 1 − x _i ) ≥ Y g i=1

(m + 1 − x _i )

! 1/g

≥ 1,

thelast inequalityoming from the fatthattheprodutis anon-zero integer.

Thismotivates us to give the following denition.

Denition 2.2. We saythata genus

g

^urve

C/ F q

^{isoptimal if}

#C( F q ) = q + 1 + gm.

Inthat ase

N q (g) = q + 1 + gm

^.

Note that the previous denition is not universally aepted. Some authors all maximal

(or

F ^q

^{-maximal) what we all optimal by referene to the historial ases with}

n

^{even and}

N _q (g) = q + 1 + gm

^{. Wepreferto keepthewordmaximal forurveswhihnumbersof points}

isequal to

N _q (g)

^andour terminology is oherent withthehistorial one aswell.

Remark 2.3. If

g ≥ (q − √ q)/2

^{, the bound an be improved, thanks to the expliit}

methods of Oesterlé (and is known as Oesterlé bound [Ser83b℄). It uses the fat that the

number of plaes of eah degree on the urve is non negative. As we will mainly deal with

small values of

g

^{ompared to}

q

^,theHasse-Weil-Serrebound will be our referene.

2.2. Existene of the isogeny lass. Equality in the arithmeti-geometri mean in-

equality is equivalent to the fat that all terms in the sum are equal and so

x _i = m

^{for all}

1 ≤ i ≤ g

^{. Hene, ifanoptimal urve}

C

^{exists, itsWeilpolynomialhasthepartiular simple}

expression

χ _C = (X ² + mX + q) ^g .

Honda-Tatetheory asexplained in[Tat66℄, [Hon68℄,[Wat69℄, [MW71 ℄ or [Tat71 ℄shows

that if

p ∤ m

^(resp.

n

^{is even) then}

Jac C

^{is isogenous to}

E ^g

^where

E

^{is an ordinary (resp.}

supersingular)elliptiurvewithtrae

− m

^{. However,if}

p | m

^and

n

^{isodd,this might not be}

true(seetheproofofProposition2.5below)andthereisfor instaneasimpleabelianvariety

ofdimension

9

^over

F 5 ⁹

^withsuhWeil polynomial. Ifwe restrit to

g ≤ 3

^{,it an be proved}

thatthisneverhappens (seefor instane the proofof Corollary4.2 of[NR10℄).

Lemma 2.4. If

C/ F q

^{is an optimal urve of genus}

g ≤ 3

^then

Jac C

^{is isogenous to}

E ^g

where

E

^{is an ellipti urve of trae}

− m

^.

Therstneessaryondition isthento seewhether suh anellipti urve existsor not.

Proposition 2.5 (Deuring [Deu41℄). There does not exist an ellipti urve with trae

− m

^{if and onlyif}

n ≥ 3

^,

n

^isoddand

p | m

^.

Proof. Let

F = X ² + mX + q

^{. Sine}

m < 2 √ q

^{if and only if}

q

^{is not a square,}

F

^is

irreduibleover

Q

^when

n

^isoddand

F = (X + √ q) ²

^when

n

^iseven.

If

n

^{is odd, by [Wat69 , p.527℄, the minimal}

e

^{for whih}

χ = F ^e

^{is the Weil polynomial of}

an abelian variety of dimension

e

^over

k

^{is the least ommon} denominator of

v _p (F _ν (0))/n

where

F _ν

^{denotes thefatorsof}

F

ⁱⁿ

Q p [t]

^and

v _p

^the

p

^{-adi valuationof}

Q p

^{. Hene}

F

^isthe

Weilpolynomialofanelliptiurve ifand onlyif

n | v _p (F _ν (0))

^{forallfators. Thisisofourse}

satisedif

n = 1

^{. LookingattheNewtonpolygonof}

F

^,weseethatif

p ∤ m

^then

v p (F ν (0)) = n

or

0

^{, so}

e = 1

^{. Withthe same tehnique, if}

n > 1

^{odd and}

p | m

^{, then}

v _p (F _ν (0)) < n

^{and so}

e > 1

^.

If

n

^{is even,we apply the previous arguments to}

F = X + √ q

^{. Sine}

v p ( √ q)/n = 1/2

^,

e = 2

so

F ² = X ² + mX + q

^{isthe Weil polynomialof anellipti urve.}

Atually,theonly valuesof

q = p

^{for whih}

p | m

^are

q = 2

^or

q = 3

^.

Remark 2.6. For any value of

− m ≤ t ≤ m

^{, one knows if an ellipti urve with trae}

t

^{exists (see [Wat69, Th.4.1℄). Also, thepossible Weil} polynomials of the isogeny lasses of abelian surfaes(resp. threefolds) an be found in [MN02 , Lem.2.1,Th.2.9℄ (resp. [Xin96,

Hal10℄).

3. Existene of an indeomposable prinipal polarization

The Jaobian of a genus

g

^urve

C/K

^{is naturally equipped witha prinipal} polarization

j

induedbytheintersetionpairingontheurve

C

^{. Sinethethetadivisor}

Sym ^g−1 C ֒ → Jac C

assoiated to

j

^is geometrially irreduible,

(Jac C, j)

^is geometrially indeomposable, i.e.

theredoesnot existsanabelian subvariety

B ⊂ Jac C

^{dened over}

K ¯

^{suh that}

j

^{indues on}

B

^{a prinipal}polarization. Conversely,starting with

A = E ^g

^where

E

^{is an elliptiurve,it}

islearthat

A

^{alwaysadmitsaprinipal}polarization

a 0

^{given bytheprodutoftheprinipal}

polarizations on eah fator. As

a ₀

^is deomposable,

(A, a ₀ )

^{is not (even} geometrially) a Jaobian. Hene `good' prinipal polarizations on

A

^{(or on abelian varieties in the isogeny}

lassof

A

^{) have to be moresubtle. Lukily,} equivalenesofategory have been developed to translatetheexisteneofanindeomposablepolarizationintotheexisteneofpurelyalgebrai

objets. As far asI know several points of view o-existand it is not lear to seehowto go

fromone to the other. IshalluseSerre's one andmentionothers inremark.

Remark 3.1. Howe [How95℄,[How96 ℄hasdevelopedapowerfulmahineryto provethe

existene of a prinipally polarized abelian variety inthe isogenylass of an abelian variety

A/k

^{. Butonlywhen}

A

^{issimple,itiseasyto seethatthe}polarizationisindeomposable (see [Ryb08℄forthease

E × B

^where

E

^{isanelliptiurveand}

B

^ageometriallysimpleabelian surfae).

3.1. The equivalenes. Let us start with

E

^{ordinary. Let}

E/k

^{be anordinary ellipti}

urve with trae

t

^{. If}

π

^{denotes the}

F q

^-Frobenius endomorphism of the urve

E

^{, then the}

ring

R := Z [X]/(X ² − tX + q)

^{is isomorphi to}

Z [π] ⊂ End(E)

^{. Serre [Ser85, Se.50-53℄,}

[Lau02 , Appendix℄ denes an equivalene of ategory

T

^{between the ategory of abelian}

varietieswhih are isogenous to a power of

E

^and

R

^{-modules of nite type without torsion.}

Thefuntor

T

^{mapsan objet}

A

^{to the}

R

^-module

L = Hom(E, A)

^{. Obviously,therank of}

L

isequalto thedimensionof

A

^{. Thisfuntoralsobehavesnielywithrespettoduality: ifwe}

denote

L ˆ

^{thering of}anti-linear homomorphism

f : L → R

^(i.e.

f(rx) = ¯ rf (x)

^{for all}

r ∈ R

and

x ∈ L

^)then

T ( ˆ A) = ˆ L

^{. Thus amorphism}

a : A → A ˆ

^{denesa morphism}

h : L → L ˆ

^and

heneanhermitian form

H : L × L → R

^{. Serreprovesthat}

a

^isapolarizationifandonlyif

H

is positive denite, that

a

^{is prinipal if}

(L, H )

^{is unimodular (i.e.}

h(L) = ˆ L

^{) and moreover}

(geometrially)indeomposableifandonlyif

(L, H)

^isindeomposable,i.e. annotbewritten asasum of orthogonalsub-modules. Theouple

(L, H )

^{isalleda hermitian module.}

Remark 3.2. This equivalene is inspired by the lassial theory over

C

^{, whih is not}

surprisingsineordinaryabelianvarietiesanbeliftedanoniallyandthisisusedin[Del69℄.

When

A = E ^g

^and

End(E) ≃ R

^{,a more expliit point of viewan be onsidered looking at}

thehermitian matrix

M := a ⁻ ₀ ¹ a ∈ End(A) = M g (End(E)) ≃ M g (R)

^{(see [Rit10℄,[Lan06 ℄).}

For

g = 2

^{, Kani's onstrution [Kan97℄ also gives neessary and suient onditions for}

`gluing' twoelliptiurvesalongtheir

n

^-torsionfor

n > 1

^{. Bothpointsofviewarerelatedby}

theCholewskydeomposition of

M

^.

A lassiation of rank

2

^and

3

^{hermitian modules wasahieved in [Hof91,} Th.8.1,8.2℄ (see also [Sh98℄for further omputations) andtranslates into the following result.

Proposition 3.3. Let

E

^{be an ordinary ellipti urve with trae}

t

^{. There is no abelian}

surfae (resp. threefold)withageometrially indeomposableprinipalpolarizationinthelass

of

E ²

^(resp.

E ³

^{) ifand onlyif}

t ² − 4q ∈ {− 3, − 4, − 7 }

^(resp.

t ² − 4q ∈ {− 3, − 4, − 8, − 11 }

^).

Remark 3.4. For

g = 2

^{,theresultanbetraedbakto[HN65,p.14℄,wheretheauthors}

prove the existene of genus

2

^{urves whih Jaobian is isomorphi to}

E ²

^{by onstruting}

free indeomposable hermitian modules (in [Hay68 ℄, the preise number of isomorphism

lasses of suh urves is omputed). For

g = 2

^or

3

^{, it ould also be dedued from the}

massformulae(i.e. numberof weightedlasses bytheorder oftheirautomorphismgroup)of

[HK86,HK89℄ (although, aordingto Homann (lo. it. p.400) thereis aminor mistake

inthese omputations).

Remark 3.5. For

g > 3

^{, there have been several partial answers on the existene of}

indeomposable unimodular positive denite hermitian modules of rank

g

^{over the ring of}

integers of an imaginary quadrati eld

Q ( √

− d)

^{. It seems that in[Zhu97℄ and [WL01 ℄ a}

omplete answer is given: there always exists one, exept when

d = 1

^and

g = 5

^or

d = 3

and

g = 4, 5, 7

^{. Oneshould be areful sine, aordingto the Mathsinet reviewof [Zhu97℄}

by Homann, the proofs ontain several mistakes. Also, I do not know if the ase of non

maximalorders hasbeen onsidered.

Assumenowthat

E

^issupersingular. Morepreisely,let

E/ F p

^{beanelliptiurvewithtrae}

0

^,sothat

E

^issupersingular,all the geometri automorphisms of

E

^{aredened over}

F p ²

^and

Tr(E/ F p ² ) = − 2p = − m

^{. Onesaysthatanabelianvariety}

A

^{(resp. aurve}

C

^{)issuperspeial}

if

A

^(resp.

Jac C

^)isgeometriallyisomorphito aprodutofsupersingularelliptiurves. A resultofDeligne (see[Shi79,Th.3.5℄)shows thatwhen

g > 1

^,asuperspeialabelian variety ofdimension

g

^isgeometriallyisomorphito

E ^g

^(whereasfor

g = 1

^therearenon-isomorphi supersingularelliptiurvesassoonas

p > 7

^{). However,thedesriptionoftheisogenylassis}

mademoreompliated thaninthe ordinary asebytheexisteneof `ontinuous' familiesof

isogenies. For instane, already when

g = 2

^{(see[Oor75℄), a} supersingular abelian surfae is eithergeometriallyisomorphito

E ²

^(andsosuperspeial)oroftheform

E ² /α _p

^where

α _p

^is

theunique loal-loal groupshemeover

F ^p

^{,theinjetionof}

α p

ⁱⁿ

E ²

^beingparametrizedby

P ¹ (¯ F p ) \ P ¹ ( F p ² )

^{. Inthe latter,itanbeshownthat}

A

^isnotsuperspeialandthedesription ofthepolarizations on this objetis moreevolved. For this reason, we will onentrate only

onexisteneresults and limit ourselvesto thesuperspeialase.

Remark 3.6. Note that it is still possible to obtain a omplete desription for

g = 2

ⁱⁿ

the non-superspeial ase like in [IKO86℄ or [HNR09, Part.2℄ where the mass formula of

[Ibu89℄wereused.

As in Remark 3.2, we desribe the polarizations on

A = E ^g

^{by matries}

M := a ⁻ ₀ ¹ a

ⁱⁿ

End(A) = M g (End(E))

^{. Now,}

End(E)

^{is a quaternion algebra, so we need results on the}

number

n _g

^{of positive denite quaternion hermitian forms. Then, to obtain the number of}

(geometrially)indeomposablepolarizationson

E ^g

^{,theideaistosubtratto}

n _g

^thenumber

ofpolarizationsomingfromombinationsoflowerdimensionalabelianvarieties. Inthisway,

one gets

Proposition 3.7 ([Eke87 ,Prop.7.5℄). There is no geometrially indeomposable prini-

pal polarization on

E ^g

^{ifand onlyif}

g = 2

^and

p = 2

^or

3

^{, or}

g = 3

^and

p = 2

^.

Remark 3.8. Morepreisely,Ekedahlgivesin[Eke87 ,Prop.7.2℄themassofindeompos-

able prinipal polarizations on

E ^g

^{. However, Brok[Bro93,Th3.10.℄ orrets a mistake in}

thease

g = 3

^{. He alsoompletes and reoversseveral resultsobtained in[HI83,I℄, [KO87℄}

for

g = 2

^{and in [Has83 ℄, [Oor91℄ for}

g = 3

^{. For instane, in [Bro93,} Th.3.14,Th.3.15℄, he gives the number of genus

2

^{and genus}

3

superspeial urves for eah possible group of automorphisms.

3.2. Appliation. We annowanswerthequestion oftheexisteneofa goodpolariza-

tionwhen

g ≤ 3

^.

Theorem 3.9. Let

E

^{be an ellipti urve with trae}

− m

^{. There is no abelian surfae}

(resp. threefold) with a geometrially indeomposable prinipal polarization in the isogeny

lass of

E ²

^(resp.

E ³

^{) if and only if}

q = 4

^or

9

^or

m ² − 4q ∈ {− 3, − 4, − 7 }

^(resp.

q = 4

^or

16

^or

m ² − 4q ∈ {− 3, − 4, − 8, − 11 }

^).

Proof. When

p ∤ m

^,

E

^{is ordinarysowe an use}Proposition3.3 .

When

n

^isoddand

p | m

^,

E

^{existsifandonly if}

q = 2

^or

3

^{,whih leavesthese twoases tobe}

treated apart(for instane by extensive omputer researh of urvesusing Theorem 4 or by

Remark3.10 ).

When

n

^{iseven then}

p | m

^{. We} distinguish several ases.

When

p > 3

^and

g = 2

^(resp.

p > 2

^{and g}

= 3

^), Proposition 3.7 shows that there is alwaysanindeomposableprinipalpolarizationon

E ²

^(resp.

E ³

^{). Notethanwhen}

4 | n

^,

thepresent

E

^{isthe quadratitwist oftheellipti urve in}Proposition3.7.

When

p = 2

^and

g = 2

^,expliitonstrutionsasin[MN07 ℄ora`gluing'argumentasin [Ser85,Se.32℄,[Sha01,Prop.30℄showsthatonean getaurve

C/ F 2 ⁿ

^suhthat

Jac C

isisogenous to

E ²

^assoonas

n > 2

^.

When

p = 2

^,

g = 3

^and

n > 4

^{, An expliit non} hyperellipti urve

C/ F ^{2 n}

^{suh that}

Jac C ∼ E ³

^{an be onstruted. (see [Rit09,} Lem.2.3.8℄). Note that in [NR08℄, the moregeneralquestionoftheexisteneofaJaobianintheisogenylassofasupersingular

abelianthreefold inharateristi

2

^{is addressed.}

Finally when

p = 3

^and

g = 2

^{, one an nd an expliit onstrution in [Kuh88℄ (see}

also[Sha01,Cor.37℄ whereamistakeisorreted)assoonas

n > 2

^{. Thisworkwasalso}

generalized to all supersingular abelian surfaesinharateristi

3

^in[How08℄.

Remark 3.10. Theases

m ² − 4q ∈ {− 3, − 4 }

^{analsobeexludedthankstoaproofdue}

to Beauville [Sha01,Th.16℄, [Ser85,Se.13℄ without any hypothesis onthe

p

^-rankof

E

^(and

then

q = 2, 3

^areovered).

Asonjeturally, there is innitely many

p

^{in theforms}

p = x ² + 1

^and

p = x ² + x + 1

^the

equations

m ² − 4p ⁿ ∈ {− 3, − 4 }

^{have innitely many solutions with}

n = 1

^{. For}

n > 1

^odd,

one knows that theset of solutions is nite. For instane, in[Ser83a ℄, we nd that there is

onlyone solution tothe equation

q = x ² + x + 1

^namely

q = 7 ³

^{and noneto}

q = x ² + 1

^.

Similarly,theaseofdisriminant

− 7

^{orrespondstotheequation}

q = x ² + x + 2

^withunique

solutions

q ∈ { 2 ³ , 2 ⁵ , 2 ¹³ }

^when

n > 1

^isodd.

Thease ofdisriminant

− 8

^orrespondsto

q = x ² + 2

^,whih,when

n > 1

^isodd,has

q = 3 ³

forunique solution. Thisis proved usingthe same argumentsasthelast asebelow.

Finally, the ase of disriminant

− 11

^{orresponds to}

q = x ² + x + 3

^{. When}

n > 1

^{is odd,}

q = 3 ⁵

^{is the unique solution thanks to the following argument due to Samir Siksek. The}

equationan be rewritten

(2x + 1) ² + 11 = 4p ⁿ

^{and fatorsin}

K = Q ( √

− 11)

^as

2x + 1 + √

− 11 2

2x + 1 − √

− 11 2

= p ⁿ .

Sine

n

^{is odd and}

O K

^{is a prinipal domain, there exists}

α = (a + b √

− 11)/2 ∈ O K

^suh

that

α ⁿ = (2x + 1 + √

− 11)/2

^and

αβ = p

^with

β = ¯ α

^{. Now,notethat}

α ⁿ − β ⁿ = √

− 11

^and

sine

(α ⁿ − β ⁿ )/(α − β) = 1/b ∈ O K

^{,we seethat}

b = ± 1

^{. Hene, ifwe x}

n

^{,we an ndthe}

niteset of integer solutions of this polynomial equationin

a

^{. However, to solve itfor all}

n

we have to invoke the muh deepertheorem from [BHV01℄ whih tells us that ifthere is a

solutionthen

n < 4

^,

n = 5

^or

n = 12

^{. Indeed, withthe} terminology andnotation of lo. it., one seesthat

u _n = (α ⁿ − β ⁿ )/(α − β) = ± 1

^{is a Luasnumber withoutprimitive divisor, so}

theLuas pair

(α, β)

^is

n

^-defetive.

4. Optimal urves

Aswehave seeninSetion 3,thestrategy we have applied sofar worksinanydimension. If

wenowhavetorestritourselvesto thedimensionslessthanorequalto

3

^{isbeause,inthese}

ases,theondition`has anindeomposable prinipal polarization'is geometrially suient

to be theJaobian of a urve. Thisis not truewhen thedimension is bigger, asitis proved

simplybynoting that the dimension of the moduli spaeof urvesof genus

g

^,

3g − 3

^{,is less}

thanthedimensionofthe modulispaeofprinipally polarizedabelianvarietiesofdimension

g

^,

g(g + 1)/2

^{. However,}

Proposition 4.1 ([OU73℄). For

g ≤ 3

^{,any geometrially}indeomposable prinipally po- larized abelian variety

(A, a)/K

^{is theJaobian of a urve}

C ¯

^over

K ¯

^.

So,given

(A, a)/K

^asin Proposition 4.1 , the question boilsdown to know whether one an desend the urve

C ¯

^{to a urve}

C

^over

K

^{suh that}

(Jac C, j) ≃ (A, a)

^. Surprisingly the answeris `notalways'.

Theorem 4.2 (Arithmeti Torelli theorem). Thereisauniquemodel

C/K

^of

C ¯

^suh

that:

1. If

C ¯

^ishyperellipti, there is an isomorphism

(Jac C, j) −−−−→ ^∼ (A, a).

2. If

C ¯

^{is not} hyperellipti, there is a unique quadrati harater

ε

^of

Gal( ¯ K/K )

^{, and an}

isomorphism

(Jac C, j) −−−−→ ^∼ (A, a) _ε

where

(A, a) _ε

^{is the quadrati twist of}

A

^by

ε

^.

Remark 4.3. Itistrikyto ndtheright origin ofthepreviousresult. In[Ser85,Se.69℄,

Gouvéa indiates `Oort +...' as a referene. One an indeed nd in [Oor91, Lem.5.7℄ a

similarresult (butthisis 1991). Sekiguhialsoworked on thisquestion but aftertwo errata,

he gives in[Sek86℄ only the existene of the model

C/K

^{but doesnot speak about}

ε

^{. One}

an alsond this resultin[Maz86 ,p.236℄.

The notation

(A, a) ε = (A ε , a ε )

^{should be understood asfollows. The variety}

A ǫ

^isuniquely

denedupto isomorphismbythefollowingproperty: there existsaquadratiextension

L/K

and an isomorphism

φ : A → A _ǫ

^{dened over}

L

^{suh that for all}

σ ∈ Gal( ¯ K/K)

^{one has}

φ ^σ = ε(σ)φ

^{. The}polarization

a ε

^{isthepull-bak of}

a

^by

φ ^{− 1}

^.

This result is a onsequene of Weil's desent as explained in [Ser68, 4.20℄ and of Torelli

theorem [Mat58 , p.790-792℄. The shism whih appears between the hyperellipti and non

hyperellipti aseisdue to thefatthat

Aut(Jac C, j) ≃

( Aut(C)

^if

C

^ishyperellipti,

Aut(C) × {± 1 }

^if

C

^isnon hyperellipti.

Denition 4.4. The harater

ε

^{(or the} disriminant of the extension

L/K

^{) is alled}

Serre's obstrution. Byextension, inthehyperellipti ase, we saythat

ε

^istrivial.

Letus emphasizewhythisobstrutionisanissueinourstrategy. Sofarwehavebeenableto

prove inertain ases the existene of a geometrially indeomposable prinipally polarized

abelian variety

(A, a)/k

^{withWeil polynomial}

(X ² + mX + q) ^g

^{. Thanks to} Proposition 4.1, we knowthatitis geometriallytheJaobian ofa urve

C ¯

^{. Iftheobstrutionis trivial,then}

C ¯

^{desendsto aurve}

C/k

^suhthat

Jac C ≃ A

^{and so}

C

^{isoptimal. Ontheontrary, ifthe}

obstrution is not trivial, then

C ¯

^{desendsto a urve}

C/k

^{suh that}

Jac C

^{is isomorphi to}

the(unique)quadratitwistof

A

^{andsoitsWeilpolynomialis}

(X ² − mX + q) ^g

^{. Inpartiular}

#C(k) = q + 1 − gm

^and

C

^{isnotoptimal(andatually}

C

^{hastheminimumnumberofpoints}

agenus

g

^{urve over}

k

^{an have).}

4.1. The end of the genus

1

^and

2

^{ases. Sine a genus}

1

^{urve over a nite eld}

alwayshasarationalpoint,itisanelliptiurveandProposition2.5tellsuswhenanoptimal

genus

1

^{urve exists(forthevalueof}

N _q (1)

^{see[Deu41℄ or [Ser83a℄).}

When

g = 2

^{, all genus}

2

^urvesare hyperellipti sotheobstrution isalways trivialand the resultis similarto Theorem 3.9,namely

Theorem 4.5 (Serre). There isno optimalurve of genus

2

^over

F q

^ifandonlyif

q = 4

or

9

^or

m ² − 4q ∈ {− 3, − 4, − 7 }

^.

In[Ser83a℄, a losed formulafor the value of

N q (2)

^{is given. Morereently,ompleting the}

work started by many authors, we obtained in [HNR09℄ the omplete piture for abelian

surfaes, i.e. we determined whih isogeny lasses ontain theJaobian of a genus

2

^urves

interms oftheoeients ofthe Weilpolynomial.

5. The genus

3

^ase

As there exist non hyperellipti genus

3

^{urves (the} non-singular plane quartis), Serre's obstrution may not be trivial. One hope is that, for eah

q

^{, there would be an optimal}

hyperellipti urve but this possibilityhasto bedisarded: for instane, theredoesnot exist

anyoptimalhyperelliptigenus

3

^urvesover

F 2 ⁿ

^with

n

^evensinesupersingularhyperellipti urvesdonot existsinharateristi

2

^{[Oor91℄. Other} ounterexamples an be foundinodd harateristi aswell asit will be apparent in Proposition 5.6. Therefore,it is important to

beabletoomputeSerre'sobstrution. Currently,thereisnoperfetsolutiontothisproblem

butwewill summarizesome of the ideas andpartial answers whihhave beenobtained.

5.1. Speial families. The key-idea is to use some families of urves with non trivial

automorphisms,suhthattheir Jaobian isaprodutofelliptiurvesexpliitlyobtained as

quotientbyertainautomorphismsubgroups. Thenonetriestoreversetheproessandseeif

oneangluegivenelliptiurvestogethertogetaurveinthefamily. Thepossiblequadrati

extension one has to make during the onstrution is Serre's obstrution. Let us illustrate

thisproedure withanexample.

Example 5.1. The following family represents genus

3

^non hyperellipti urves in har- ateristi

2

^withautomorphism groupontaining

( Z /2 Z ) ²

C : (a(x ² + y ² ) + cz ² + xy + ez(x + y)) ² = xyz(x + y + z), ac(a + c + e) 6 = 0.

Theinvolutions are

(x : y : z)

^{maps to}

(y : x : z), (x + z : y + z : z)

^or

(y + z : x + z : z)

^{. To}

gettheequation ofthe urve

E ₁ = C/ h (x : y : z) 7→ (y : x : z) i

^{, one introduesthe invariant}

funtions

X = x + y, Y = xy

^andnds

E ₁ : (aX ² + c + Y + eX ) ² = Y (X + 1).

Doing similarly with the other involutions and rewriting the equations of theellipti urves

(see[NR10℄ for details) one getsthat

Jac C ∼ E ₁ × E ₂ × E ₃

^where

E 1 : y ² + xy = x ³ + ex ² + a ² (a + c + e) ² ,

E ₂ : y ² + xy = x ³ + ex ² + c ² (a + c + e) ² ,

E ₃ : y ² + xy = x ³ + ex ² + c ² a ² .

Conversely,wenowwant toglueordinaryelliptiurves

E _i

^with

j

^-invariant

j _i 6 = 0

^{. Theyan}

always be written

E _i : y ² + xy = x ³ + ex ² + 1/j _i

^where,if

q > 2

^,

Tr _{F q / F 2} (e) = 0

^{ifandonly if}

Tr(E _i ) ≡ 1 (mod 4)

^{. Let}

s ⁴ _i = 1/j _i

^,then

a = s ₁ s ₃ s 2

, c = s ₂ s ₃

s ₁ , e = s 1 s 3

s ₂ + s 2 s 3

s ₁ + s 1 s 2

s ₃ .

(1)

Now, for instane, assume that

m ≡ − 1 (mod 8)

^{. This happens for}

n = 35, 37, 63, . . .

^{. We}

hoose

E = E ₁ = E ₂ = E ₃

^{an ordinary ellipti urve with trae}

− m

^and

j

^-invariant

j

⁽

E

existssine

2 ∤ m

^{). Sine we an assume that}

q > 4

^{,the urve}

E

^hasan

8

^{-torsion point and}

itis not diultto hekthatthis implies(atuallyis equivalent to)

Tr _{F q / F 2} 1/j = 0

^{. Hene}

Tr _{F q / F 2}

s ₁ s ₃ s 2

+ s ₂ s ₃ s 1

+ s ₁ s ₂ s 3

= Tr _{F q / F 2} (1/j) = 0.

On theother hand,sine

Tr(E) ≡ 1 (mod 4)

^{, we have}

Tr _{F q / F 2} (e) = 0

^{aswell, sothere is no}

obstrution to (1) . Atually,we getan expliitequation

C : (j ^−1/4 (x ² + y ² + z ² + xz + yz) + xy) ² = xyz(x + y + z)

for theoptimalurve.

Exploitingother familiesof urvesinharateristi

2

^{,we getthefollowing result.}

Theorem 5.2 ([NR08 , NR10℄). If

n

^{is even, there existsan optimal urve over}

F 2 ⁿ

^if

andonly if

n ≥ 6

^.

If

n

^{isodd and}

m ≡ 1, 5, 7 (mod 8)

^{, there isan optimalurve over}

F 2 ⁿ

^.

When

n > 1

^isoddand

m

^{iseven,thereisofoursenooptimal urvesinethereisnoellipti}

urve with trae

− m

^{. So only the ase}

m ≡ 3 (mod 8)

^{ismissing to get a omplete answer}

when

p = 2

^.

Morereently,Mestre[Mes10 ℄hasworked withafamily ofurveswithautomorphismgroup

S ₃

^{and showed that if}

p = 3

^(resp.

p = 7

^),

3 ∤ m

^(resp.

3 | m

^{) and}

− m

^{is a non-zero square}

modulo

7

^(resp.

n ≥ 7

^{),thenthere existsanoptimal urve over}

F ^{3 n}

^(resp.

F ^{7 n}

^).

To onlude on this approah, let us point out that one ould use the family with auto-

morphism group

( Z /2 Z ) ²

^{(alled Ciani quartis) also in} harateristi greater than

2

^{, sine}

Serre's obstrution hasbeen worked out in[HLP00℄. Unfortunately,one doesnot see when

this obstrution is trivial knowing only

m

^{(one needs theequations of theellipti fators to}

deide).

5.2. Serre's analyti strategy. InspiredbyresultsofKlein[Kle90 ,Eq.118,p.462℄ and

Igusa [Igu67, Lem.10,11℄, in a 2003 letter to Jaap Top [LR08℄, Serre stated a strategy to

ompute the obstrution when the harateristi is dierent from

2

^{. Roughly speaking, his}

ideawasthataertainSiegelmodularformevaluatedata`modulipoint'

(A, a)/K

^isasquare

in

K

^{ifand only if the obstrution is trivial. In a series of three papers, it was shown that}

thisisaurate(rst forCianiquartis,theningeneral)andhowtoompute theobstrution

inthe ase of thepowerof a CMellipti urve. Let us state the general result without any

omments on the proof whih would lead us to far from our initial purpose (see however

[Rit09,Chap.4℄ fordetails).

Theorem 5.3 ([LRZ10 ℄). Let

A = (A, a)/K

^{be a prinipally polarized abelian threefold}

denedoveraeld

K

^with

char K 6 = 2

^{. Assumethat}

a

^isgeometriallyindeomposable. There existsa unique primitive geometriSiegel modular form of weight

18

^{dened over}

Z

^{, denoted}

χ ₁₈

^{, suh that}

i)

(A, a)

^isa hyperellipti Jaobian if and onlyif

χ ₁₈ (A, a) = 0

^.

ii)

(A, a)

^{isa non} hyperellipti Jaobian if andonlyif

χ 18 (A, a)

^{isa non-zero square.}

Moreover, if

K ⊂ C

^{, let}

(ω ₁ , ω ₂ , ω ₃ )

^{be a basis of regular} dierentials on

A

^;

γ ₁ , . . . γ ₆

^{be a sympleti basis (for}

a

^)of

H ₁ (A, Z )

^;

Ω a := [Ω 1 Ω 2 ] = [ R

γ j ω i ]

^{be a period matrix with}

τ a := Ω ⁻ ₂ ¹ Ω 1 ∈ H ³

^{a Riemann matrix.}

Then

(A, a)

^{isa Jaobian if and onlyif}

(2)

χ ₁₈ ((A, a), ω ₁ ∧ ω ₂ ∧ ω ₃ ) := (2π) ⁵⁴ 2 ²⁸ ·

Q

[ε] θ[ε](τ _a ) det(Ω 2 ) ¹⁸

isa square in

K

^.

Letus reallthatthe Thetanullwerte

θ[ε](τ )

^arethe

36

^{onstants suh that}

[ε] =

ǫ ₁ ǫ ₂

∈ { 0, 1 } ³ ⊕ { 0, 1 } ³ ,

with

ǫ ₁ ^t ǫ ₂ ≡ 0 (mod 2)

^andfor

τ ∈ H 3

θ ǫ ₁

ǫ ₂

(τ ) = X

n∈Z ³

exp(iπ(n + ǫ ₁ /2)τ ^t (n + ǫ ₁ /2) + iπ(n + ǫ ₁ /2) ^t ǫ ₂ ).

Remark 5.4. For adierentapproah onthis result, see[Mea08℄.

The initial aim of Serre's letter was of ourse the existene of optimal urves of genus

3

^.

However, one does not know how to ompute diretly the value of

χ ₁₈

^{over nite elds.}

Therefore, as Serre suggested, when

A

^{is ordinary, we lift}

(A, a)

^{anonially over a number}

eld and there, we use formula (2). Doing the omputation with enough preision, we an

reognizethis value asan algebrai number. Finallywe redue itto theinitial nite eldto

seeifitisa square.

As the Jaobian of an optimal urve is isogenous to the power of an ellipti urve

E

^{, in}

[Rit10℄,we worked out this proedure expliitinthe partiular ase

A = E ³

^{. Let}

a ₀

^{be the}

produt prinipal polarization on

E ³

^and

M = a ⁻ ₀ ¹ a ∈ M ₃ (End(E))

^{. When}

End(E)

^{is an}

order in an imaginary quadrati eld, it is well known that

M

^{is the matrix of a prinipal}

polarizationon

E ³

^{ifandonly if}

M

^{isa positive denite hermitian matrix with} determinant

1

^{(see Remark3.2 and [Mum08,p.209℄). Moreover, when}

E

^{is dened overa number eld,}

weshowhowto translatethe data

(E ³ , a 0 M)

^{intoaperiodmatrixofthe}orrespondingtorus inorder to ompute theanalyti expressionof

χ ₁₈

^{. Letus illustrate thisproedure withthe}

following example.

Example 5.5. Does there exist an optimal urve

C

^{of genus}

3

^over

k = F 47

^{? If so,}

by Lemma 2.4 we know that

Jac C

^{is isogenous to}

E ³

^where

E

^{is an ellipti urve with}

trae

−⌊ 2 √

47 ⌋ = − 13

^{. The urve}

E

^{is then an ordinary ellipti urve and}

End(E)

^on-

tains

Z [π] ≃ Z [(13 + √

13 ² − 4 · 47)/2] = Z [τ ]

^(where

π

^{is the}

k

^-Frobeniusendomorphismand

τ = (1 + √

− 19)/2

^{). Hene}

End(E) = Z [π]

^{istheringofintegers}

O L

^of

L = Q ( √

− 19)

^{. Sine}

O L

^isprinipal,

E

^isuniqueuptoisomorphism. Usingtheworkof[Sh98℄,one anseethat, up to automorphism, there is a unique positive denite hermitian matrix

M ∈ M ₃ ( O L )

^of

determinant

1

^whihisindeomposable. InthelanguageofSetion 3.1,thismeansthatthere exists a unique positive denite unimodular indeomposable rank

3

^hermitian

O L

^-module.

The abelian threefold

(E ³ , a ₀ M )

^{isthenthe unique prinipally polarized} geometrially inde- omposable abelian threefoldwithWeil polynomial

(X ² + 13X + 47) ³

^,upto isomorphism.

Lifting

E

^{anonially over}

Q

^as

E ¯ : y ² = x ³ − 152x − 722

^{we an onsider the prinipally}

abelianthreefold

( ¯ E ³ , a ₀ M)

^sine

End( ¯ E) = O L

^{aswell. Let}

[w ₁ w ₂ ]

^{beaperiodmatrix of}

E

withrespetto the anonialregulardierential

dx/(2y)

^{. Ifwe let}

Ω ₀ =









w ₁ 0 0 0 w ₁ 0 0 0 w 1









w ₂ 0 0 0 w ₂ 0 0 0 w 2







 ,

C ³ /Ω 0 Z ⁶ ≃ E ¯ ³ ( C )

^{withthe produt}polarization

a 0

^{. Wethenneed tond asympletibasis}

of

Ω ₀ Z ⁶

^forthepolarization

a ₀ M

^{. ItisnotdiulttoprovethattherstChernlassof}

a ₀ M

with respet to the pull-bak

ω _i

^{of the} dierentials

dx/(2y)

^{on eah urve is} represented by thematrix

H = 1 w 1 w 2

t M .

The alternated form

T

^{lassially assoiated to}

H

^{on thelattie}

Ω ₀ Z ⁶

^is

T = Im( ^t Ω ₀ HΩ ₀ )

^.

Onethennds a matrix

B ∈ GL ₆ ( Z )

^{suh that}

BT ^t B =

0 I 3

− I ₃ 0

and

Ω = Ω 0 t B

^{isaperiodmatrixfor the}polarization

a 0 M

^{. Finally,one omputesanapprox-}

imation of

χ = χ ₁₈ (( ¯ E ³ , a ₀ M ), ω ₁ ∧ ω ₂ ∧ ω ₃ )

thankstotheanalytiformula(2 )andwereognizeitasanelementof

L

^{. Wendinourase}

χ = (2 ¹⁹ · 19 ⁷ ) ² .

Thevalue

χ

^{isanon-zero squareover}

F 47

^{sobyTheorem5.3(ii )Serre'sobstrutionis trivial}

andthere isa nonhyperellipti optimal urve ofgenus

3

^over

k

^.

Similaromputationsshowthatthereisanoptimalurveover

F q

^for

q = 61, 137, 277

^butnot

for

q = 311

^{. Notethatthisresultfor}

q = 47

^and

q = 61

^{hasalreadybeen obtainedin[Top03℄}

usingexpliitmodelsand the othershave been onrmed by[AAMZ09℄. In [Rit10℄,tables

of values of

χ

^{as the one from Example 5.5 are given for}

( ¯ E ³ , a ₀ M)

^where

E ¯

^{is an ellipti}

urvewithlassnumber

1

^and

M

^{istakenfrom[Sh98℄. Fromthem,weangetforinstane:}

Proposition 5.6. Assume that

q = p ⁿ

^{is suh that}

4q = m ² + d

^with

d = 7

^(resp.

19

^).

Thenthere exists an (expliit) genus

3

^{optimal urve over}

F ^q

^{if and onlyif}

m ≡ 1, 2

^or

4 (mod 7)

^(resp.

m

19

− 2 p

= 1

⁾

.

Moreover if this urve exists,it isnon hyperellipti.

Assume that

q = p ⁿ

^{is suh that}

4q = m ² + 43

^{. In partiular}

43

^{is a square in}

F ^p

^{, let say}

43 = r ²

^with

r ∈ F ^p

^{. Thenthere existsa genus}

3

^{optimalurve over}

F ^q

^{if and onlyif}

m 43

α p

= 1

where

α

^iseither

− 2 · 3 · 7, − 487, − 47 · 79 · 107 · 173

^or

− 15156 ± 8214r

^{. Moreover ifthis urve}

exists,it isnon hyperellipti.

Remark 5.7. The term `expliit' inProposition 5.6omes from thefatthat for ertain

( ¯ E ³ , a ₀ M )

^{of [Rit10℄ we were able to give the equation of a urve}

C ¯

^{suh that}

Jac ¯ C

^is

isomorphito

( ¯ E ³ , a ₀ M)

^{using [Guà09℄. Henefor these ases,we have a} `universal' family ofexpliitequations for theoptimal urve.

Thefatthatthepreviousstatementisembarrassedlyumbersomerevealseithertheintrinsi

diultyoftheproblemorawrongattakangle. Moreover,thelimitsofthisstrategyalready

appearintheexample: theomputationoftheanoniallift,ofthematries

M

^andofaperiod

matrix make it algorithmi in nature. Worse, the omputation of an approximation of

χ

^is

time-onsuming sine one has to reognize it as an algebrai number (atually for a good

hoie of themodel

E ¯

^,

χ

^{isan algebrai integer). Therefore large values of the}disriminant of

End(E)

^{seemout of reah.}

Itmight thenbe interesting totry tounderstand theprimedeomposition of

χ

algebraially.

Klein's formula linking

χ ₁₈

^{to the squareof the} disriminant of plane quartis (see [Kle90 , Eq.118,p.462℄ and [LRZ10 , Th.2.23℄) makesus think about an analogue of the Néron-Ogg-

Shafarevih formulafor ellipti urve [Sil92,Appendix C,16℄. We shalltheninterpret

p | χ

ⁱⁿ

termsofthenatureof

(A, a) := ( ¯ E ³ , a ₀ M ) (mod p)

^{. For instane, using[Ih95,p.1059℄,}

p | χ

ifandonlyif

(A, a)

^isgeometriallydeomposable orahyperelliptiJaobian. Unfortunately, wedonot knowhowto detetalgebraiallythis lastpossibility(seethedisussionin[Rit09,

Se.4.5.1℄).

Remark 5.8. We have not spoken yetabout thease

q

^squarewhen

p > 2

^{. First, when}

p ≡ 3 (mod 4)

^{, one knows [Ibu93, p.2℄ that there exists an optimal genus}

3

^{urve. This}

urveiseven hyperellipti [Oor91℄butnot expliit(seehowever [KTW09℄for someexpliit

sub-ases). Also Fermat urve

x ⁴ + y ⁴ + z ⁴ = 0

^{is optimal if}

n ≡ 2 (mod 4)

^{. Then, when}

p ≡ 1 (mod 4)

^and

n ≡ 2 (mod 4)

^{, Ibukiyama (lo. it.) shows that there is an optimal}

urve. Ibukiyama'sstrategy uses a massformulaon quaternion hermitian forms to showthe

desentofanindeomposableprinipal polarizationonamodelover

F p

^of

E ³

^where

E/ F p ²

^is

an ellipti urve withtrae

− 2p = − m

^{. The abelian threefoldand its quadrati twist being}

isomorphiover

F p ²

^{,he avoids theissue ofomputing Serre's}obstrution.

5.3. The geometri approah. Following aonstrution of Reillas[Re74℄,we were

able to give in [BR10℄ a geometri haraterization of Serre's obstrution. For the sake of

simpliity,let usassume that

char k 6 = 2

^{and that}

(A, a)/k

^is geometrially theJaobian ofa nonhyperellipti genus

3

^{urve. Sine}

k

^{isaniteeld,thereexistsasymmetrithetadivisor}

Θ

^(forthepolarization

a

^{)dened over}

k

^{. Let}

Σ

^betheunionof

2 _∗ Θ

^{andoftheuniquedivisor}

in

| 2Θ |

^withmultipliity greaterthan or equal to

4

^at

0

^.

Proposition 5.9. Let

α ∈ A(¯ k) \ { 0 }

^{. The urve}

X ˜ _α = Θ ∩ (Θ + α)

^{is smooth and}

onneted if and onlyif

α ∈ A(¯ k) \ Σ

^.

Hene, thedivisor

Σ

^{representsabad lousthatneedsto beavoidedinthesequel. Assuming}

that

α / ∈ Σ

^{,theinvolution}

(z 7→ α − z)

^of

X ˜ _α

^{isxedpoint freeandso}

X _α = ˜ X _α /(z 7→ α − z)

isa smooth genus

4

^urve.

Proposition 5.10. The urve

X _α

^{is non} hyperellipti and its anonial model in

P ³

^lies

on a quadri

Q _α

^{whihis smooth.}

To gofurther,weneed to assumethat

α

^{isrational. When}

k

^{isbigenough,suh an}

α

^always

exists. Wethenobtain the following result.

Theorem 5.11. Assume there exists

α ∈ A(k) \ Σ

^{. Then}

(A, a)

^{isa Jaobian ifand only}

if

δ = Disc Q _α

^{isa square in}

k ^∗

^.

Let us sketh the proof. A non hyperellipti genus

4

^urve

X

^{lies anonially in}

P ³

^{on the}

intersetion of a unique quadri

Q

^{and a ubi surfae}

E

^{. If we assume that}

Q

^{is smooth,}

then

X

^hastwo

g ₃ ¹

^{oming fromthetwo rulingsof}

Q

^byintersetingthemwith

E

^{. Moreover,}

an easy omputation shows that

Disc Q

^{is a square if and only if these two}

g ¹ ₃

^{are dened}

over

k

^{. NowReillas'} onstrution,whihanbeusedwhen

(A, a)

^{istheJaobianof aurve,}

shows that

X α

^{hastwo (ratherexpliit) rational}

g ¹ ₃

^{. To onlude, itis then enough to show}

thata quadrati twist of

(A, a)

^{(whih isno morea Jaobian)leads to two onjugate}

g ₃ ¹

^.

The advantage of this approah is that it stays over the nite eld

k

^{and is ompletely}

algebrai. Unfortunately, sofar, we do not seehow to ompute

δ

^for

A = E ³

^and

a = a ₀ M