Constructing hyperelliptic curves with surjective Galois representations

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by  Samuele Anni

CONSTRUCTING HYPERELLIPTIC CURVES WITH
SURJECTIVE GALOIS REPRESENTATIONS
7
1
SAMUELEANNIANDVLADIMIRDOKCHITSER
0
2
Abstract. In this paper we show how to explicitly write down equations of
n
hyperelliptic curves over Q such that for all odd primes ℓ the image of the
a
J modℓGaloisrepresentation isthegeneral symplecticgroup. Theproofrelies
onunderstandingtheactionofinertiagroupsontheℓ-torsionoftheJacobian,
0
including at primes where the Jacobian has non-semistable reduction. We
2
alsogiveaframeworkforsystematicallydealingwithprimitivityofsymplectic

modℓGaloisrepresentations.
]
T Themainresultofthepaperisthefollowing. Supposen=2g+2isaneven
integerthatcanbewrittenasasumoftwoprimesintwodifferentways,with
N
noneoftheprimesbeingthelargestprimeslessthann(thishypothesisappears
. toholdforallg6=0,1,2,3,4,5,7and13). ThenthereisanexplicitN ∈Zand
h
t anexplicitmonicpolynomialf0(x)∈Z[x]ofdegreen,suchthattheJacobian
a J of every curve of the form y2 =f(x) has Gal(Q(J[ℓ])/Q)∼= GSp2g(Fℓ) for
m alloddprimesℓandGal(Q(J[2])/Q)∼=S2g+2,whenever f(x)∈Z[x]ismonic
[ with f(x)≡f0(x)modN and withno roots of multiplicitygreater than 2 in
Fp foranyp∤N.
1
v Contents
5
1 1. Introduction 2
9 1.1. Notation, t-Eisenstein polynomials and type t q ,...,q 4
1 k
5 2. Inertia action on J[ℓ] −{ } 5
0
2.1. J[ℓ] when ℓ=p: clusters 5
. 6
1 2.2. J[ℓ] when ℓ=p: fundamental characters 8
0
2.3. Creating a transvection 8
7
3. Irreducibility 9
1
3.1. Local representations 9
:
v
3.2. Global representations 10
i
X 4. Primitivity 11
r 4.1. Quasi-unramified representations 11
a 4.2. Criteria for being quasi-unramified 11
4.3. Admissible polynomials 12
4.4. p-admissible polynomials 13
4.5. Miscellaneous linear algebra 14
5. Surjectivity 14
5.1. Generating symplectic groups 14
5.2. Symplectic representations and abelian varieties 15
6. Maximal Galois images over Q 16
7. Congruence conditions 19
7.1. Main theorem: explicit curves 19
2010 Mathematics Subject Classification. Primary11F80,Secondary 12F12,11G10, 11G30.
Key words and phrases. Galoisrepresentations,abelianvarieties,hyperellipticcurves,inverse
Galoisproblem,Goldbach’s conjecture.
1
2 SAMUELEANNIANDVLADIMIRDOKCHITSER
7.2. Congruences and type t q ,...,q 20
1 k
−{ }
7.3. Semistability at odd primes 21
7.4. Good reduction at p=2 21
8. An example 22
References 23
1. Introduction
AclassicalmethodforshowingthatthegroupGL (F )isaGaloisgroupoverQis
2 ℓ
byrealisingitastheGaloisgroupofthefieldgeneratedbytheℓ-torsiononanelliptic
curve. OnecansimilarlytrytoconstructthegeneralsymplecticgroupGSp (F )as
2g ℓ
theGaloisgroupassociatedtotheℓ-torsionofag-dimensionalabelianvariety. The
maindifficulty isthatitismuchhardertowritedownexplicitabelianvarietiesand
then verify that the Galois group obtained is not a proper subgroup of GSp (F ).
2g ℓ
This approach has been successfully used in a number of works, including [Zar79],
[SZ05],[Hal08], [Zar10],[Hal11], [AdRV11], [AdRK13], [AdRAK+15], whichrealise
GSp (F ) for every (odd) prime ℓ using Jacobians of hyperelliptic curves, and
2g ℓ
show that one curve often realises GSp (F ) for all sufficiently large ℓ. More
2g ℓ
recently [LSTX16] gave a non-constructive proof that many hyperelliptic curves
realiseGSp (F )foralloddprimesℓ,and[Die02],[Zyw15],[ALS16]whoexhibited
2g ℓ
explicit curves of genus 2 and 3 with this property. There has also been numerical
work[AdRAK+16],[ALS16]investigatingtheGaloisimagesofJacobiansofcurves.
The main contribution of the present article to this topic is an explicit con-
struction of hyperelliptic curves, such that for every prime ℓ their Jacobian has
maximal mod ℓ Galois image; in other words hyperelliptic curves whose Jacobian
J has Gal(Q(J[ℓ])/Q) = GSp (F ) for every odd prime ℓ, and isomorphic to the
∼ 2g ℓ
symmetric group S for ℓ=2.
2g+2
The key new tool is a way of controlling the action of inertia groups on J[ℓ]
at places where J has non-semistable reduction. Our approach does, however,
require a rather unorthodox constraint on the dimension g: we need the even in-
teger 2g+2 to satisfy Goldbach’s conjecture! In fact, we require it to satisfy the
following somewhat stronger statement, which appears to hold1 for every genus
g =0,1,2,3,4,5,7,13:
6
Conjecture (Double Goldbach). Every positive even integer n can be written as a
sum of two primes in two different ways with none of the primes being the largest
prime less than n, except for n=0,2,4,6,8,10,12,16,28.
TheresultonGaloisimagesofhyperellipticcurvesthatweobtainisthefollowing:
Theorem 1.1. Let g be a positive integer such that 2g+2 = q +q = q +q
1 2 4 5
for some primes q , with q ,q = q ,q , and that there is a further prime q
i 1 2 4 5 3
{ } 6 { }
with q ,q ,q ,q <q <2g+2. Then there exist an explicit N Z and an explicit
1 2 4 5 3
∈
f Z[x] monic of degree 2g+2 such that if
0
∈
(1) f(x) f modN, and
0
≡
(2) f(x)modp has no roots of multiplicity greater than 2 for all primes p∤N,
GSp (F ) for all primes ℓ=2,
then Gal(Q(J[ℓ])/Q)= 2g ℓ 6
∼
(S2g+2 for ℓ=2.
1Wehavemadeaquicknumericalcheck: thepropertyholdsforallotherg<107.
CONSTRUCTING HYPERELLIPTIC CURVES 3
See Theorem 7.1 for the explicit description of N and f (x), and Remark 7.2 for
0
anexplanationonhow to find explicitcurves satisfyinghypothesis (2). An explicit
example for g =6 is given in Section 8.
Fortheexceptionalgenerag =1,2,3,4,5,7,13ourmethodstillmakesitpossible
to construct hyperelliptic curves with maximal image at all but a small number of
primes, e.g. all primes except ℓ=5,11,13 when g =7 (see Remark 6.6).
Itis worthnotingthathypotheses(1)and(2)inthe abovetheoremaresatisfied
by a positive density of monic polynomials f(x). In particular,this shows that the
hyperelliptic curves of genus g with maximalmod ℓ Galois image for everyprime ℓ
have a positive (lower) density among all hyperelliptic curves of genus g.
Throughout the paper we work with hyperelliptic curves
C :y2 =f(x)
and write
J =Jac(C)
for their Jacobian. The layout is as follows.
In Section 2 we examine the Galois representations H1(C,Q ) and J[ℓ] as rep-
e´t ℓ
resentations of local Galois groups. For the “ℓ = p” theory we use the method of
6
clusters, recently introduced in [DDMM17]. For “ℓ = p” we restrict our attention
to primes p of semistable reduction, and use the description given by the theory of
fundamental characters. We also give a simple criterion guaranteeing that a local
Galois group contains a transvection in its action on J[ℓ].
In Section 3 we develop criteria for the representations H1(C,Q ) and J[ℓ] to
e´t ℓ
be globally irreducible. This is the place where the “double Goldbach” hypothesis
enters. The reason for it is that we cannot always guarantee that the above repre-
sentationsarelocally irreducible. Infact, this appearsto be agenuine obstruction:
for example, there is no 17-dimensional abelian variety A/Q and primes p > 35
p
and ℓ = p, such that T (A) Q is an irreducible Gal(Q /Q )-module2. However,
6 ℓ ⊗ ℓ p p
when 2g+2 is a sum of two primes (other than ℓ), we are able to force the local
representation to have at most two irreducible constituents. To guarantee global
irreducibilitywethen useconditions ofthis kindatseveralplaces. The reasonthat
we require “double Goldbach” rather than the classical Goldbach’s conjecture, is
so that we can treat the case when ℓ is one of the Goldbach prime summands of
2g+2 by using the other pair of primes.
In Section 4 we develop criteria for the representations H1(C,Q ) and J[ℓ] to
e´t ℓ
be primitive. The basic method essentially follows that of Serre (see e.g. [Maz78,
Theorem 4] or [AS15, 1]) or, more recently [ALS16]: we ensure that every inertia
§
group acts trivially on every possible Galois stable partition of the representation
and then invoke the Hermite-Minkowski theorem to deduce that no such partition
exists. We formalise this approach by introducing “quasi-unramified” representa-
tions, and develop the necessary conditions on f(x) that make the argument work
(“admissible”and“p-admissible”polynomials). Unlike[ALS16],itisimportantfor
us to allow for curves with non-semistable reduction.
2Thehypotheses ensurethatAhaspotentiallygoodreductionandtheinertiagroupatpacts
tamely and semisimply on Tℓ(A), through a cyclic quotient of order k, say. Any element of the
inertiagroupmusthaverationaltraceonTℓ(A)(asthetraceisindependentofℓ),soirreducibility
forcestheeigenvaluesofageneratoroftheimageofinertiatobepreciselythesetofk-throotsof
unity. Hence34=dimTℓ(A)⊗Qℓ=ϕ(k),whichisimpossible.
4 SAMUELEANNIANDVLADIMIRDOKCHITSER
InSection5werecallthe classificationofsubgroupsofGSp (F )of[Hal08]and
2g ℓ
[ADW16]andrephraseitasacriterionforJ to haveGal(Q(J[ℓ])/Q)=GSp (F ).
∼ 2g ℓ
As inmany previousworks,the basic rule is thatif the actionis irreducible,primi-
tiveandcontainsatransvection,thentheGaloisrepresentationhasmaximalimage
(see Theorem 5.1).
In Section 6 we tie everything together to give a list of (essentially local) con-
straints that guarantee that Gal(Q(J[ℓ])/Q) = GSp (F ) for every odd prime ℓ,
∼ 2g ℓ
and that Gal(Q(J[2])/Q)=S ; see Theorems 6.2 and 6.5.
∼ 2g+2
In Section 7 we give explicit congruence conditions on the coefficients of f(x)
that ensure that the above list of constraints is satisfied, and prove the precise
version of Theorem 1.1 (see Theorem 7.1).
WeendinSection8byworkingthroughtheconditionsinTheorem7.1forg =6,
and constructing a hyperelliptic curve satisfying all the hypotheses.
Acknowledgments. We would like to thank Adam Morgan and Tim Dokchitser
forseveralusefuldiscussionsrelatedtothiswork. Thefirstauthorwassupportedby
EPSRCProgrammeGrant‘LMF:L-FunctionsandModularForms’EP/K034383/1
during his position at the University of Warwick. The second author is supported
byaRoyalSocietyUniversityResearchFellowship. Bothauthorswouldalsoliketo
thank the Warwick Mathematics Institute where most ofthis researchwas carried.
1.1. Notation, t-Eisenstein polynomials and type t q ,...,q .
1 k
−{ }
Local setting. For a finite extension F of Q we write:
p
π for a fixed uniformizer of F;
F
•
for the ring of integers of F;
F
• O
e for the ramification degree of F;
F
•
F for a fixed algebraic closure of F;
•
Fnr for the maximal unramified extension of F;
•
v for a valuation on F normalized such that v(π )=1;
F
•
F for the residue field of F;
•
G for the absolute Galois group Gal(F/F);
F
•
I for the inertia subgroup of G ;
F F
•
g(x) and α for the reduction modulo π of every g(x) [x] and α .
F F
• ∈O ∈O
Definition 1.2 (t-Eisenstein polynomials). Let t 1 be an integer. We say that
≥
a polynomial with -coefficients
F
O
f(x)=xm+a xm−1+ +a
m−1 0
···
is t-Eisenstein if v(a ) t for all i>0 and v(a )=t.
i 0
≥
Definition 1.3 (Polynomialsoftypet q ,...,q ). Letq ,...,q beprimenum-
1 k 1 k
−{ }
bers and let t 1 be an integer. Let f(x) [x] be a monic squarefree polyno-
F
≥ ∈ O
mial. We say that f(x) is of type t q ,...,q if it can be factored as
1 k
−{ }
k
f(x)=h(x) g (x α )
i i
−
i=1
Y
over [x],forsomeα withα =α foralli=j,whereg (x)isat-Eisenstein
F i F i j i
O ∈O 6 6
polynomial of degree q and h(x) is separable with h(α )=0 for all i.
i i
6
CONSTRUCTING HYPERELLIPTIC CURVES 5
In other words, the monic polynomial f(x) is a product of shifted t-Eisenstein
polynomialsofdegreesq ,...q andlinearpolynomials,suchthatthesepolynomials
i k
have no common roots in the residue field. See Section 8 for explicit examples.
Global setting. For a number field K we write:
for the ring of integers of K;
K
• O
F for the residue field of K at a prime p of K;
p
•
K for a fixed algebraic closure of K;
•
G for the absolute Galois group Gal(K/K);
K
•
I =I for the inertia subgroup at p.
• p Kp
Definition 1.4. Let q ,...,q be prime numbers. Let f(x) [x] be a monic
1 k K
∈ O
squarefreepolynomial. Wesaythatf isoftype t q ,...,q atpiff(x) [x]
−{ 1 k} ∈OKp
is of type t q ,...,q .
1 k
−{ }
Roots of unity. Let q be a positive integer, we will denote by ζ a primitive q-th
q
root of unity. Throughout this article we will choose primitive roots of unity to
form compatible systems, i.e. if ζ is a primitive q-th root of unity and q = q′q′′
q
thenζqq′′ =ζq′. Incharacteristicℓdividingq,wehaveζq =ζm whereℓrm=q,with
(ℓ,m)=1.
2. Inertia action on J[ℓ]
Theconstructionofhyperellipticcurvespresentedinthisarticlewillcruciallyrely
on understanding the action of the inertia groups on the ℓ-torsion of the Jacobian
for every prime number ℓ. In this section, F will be a local field of odd residue
characteristic p. Let
C :y2 =f(x)
be a genus g hyperelliptic curve over F with f(x) [x] monic and squarefree,
F
∈ O
and let J =Jac(C). We will describe the inertia action on J[ℓ] in terms of f(x).
2.1. J[ℓ] when ℓ=p: clusters.
6
In this section we describe the inertia action on J[ℓ] when ℓ=p. In particular, we
6
will prove the following theorem:
Theorem 2.1. Suppose that f(x) [x] has type t q ,...,q for odd primes
F 1 k
∈O −{ }
q = p. Then for every ℓ = p, the inertia group I acts tamely on H1(C/F,Q )
i 6 6 F e´t ℓ
and on J[ℓ] through a quotient of order dividing 2 q . Moreover, the non-trivial
i i
eigenvalues (with multiplicity) of any generator τ of tame inertia are either
Q
ζ , ζ2 ,..., ζq1−1,..., ζ , ζ2 ,..., ζqk−1 if t is odd,
− q1 − q1 − q1 − qk − qk − qk
or
ζ ,ζ2 ,...,ζq1−1,...,ζ ,ζ2 ,...,ζqk−1 if t is even.
q1 q1 q1 qk qk qk
The main ingredient of the proof of Theorem 2.1 is the theory of clusters devel-
oped in [DDMM17].
Definition 2.2. Let f(x) [x] be a squarefree monic polynomial and let R be
F
∈O
its set of roots in F. A cluster s R is a non-empty set of roots of f(x) of the
⊆
form s=R for a disc F with respect to the p-adic topology.
∩D D⊆
6 SAMUELEANNIANDVLADIMIRDOKCHITSER
For a cluster s with s 2 define:
| |≥
d =min v(r r′):r,r′ s ;
s
• { − ∈ }
s to be the set of maximal subclusters of s of odd size;
0
•
I =Stab (s);
• s IF
µ = v(r r ) for any r s;
• s r∈R\s − 0 0 ∈
λ = 1(µ +d s );
• s P2 s s| 0|
trivial character 1 of I if s is even and ord µ 1
s 2 s
| | ≥
ǫ = order two character of I if s is even and ord µ <1
s  s 2 s
• | |
zero representation of Is if s is odd;
| |
γ : I C∗ any character of order equal to the prime-to-p-part of the
s s
• denomin→ator of λ (with γ =1 if λ =0);
s s s
V =γ (C[s ]⊖1)⊖ǫ , an I -representation.
s s 0 s s
• ⊗
Theorem 2.3 ([DDMM17]). Let ℓ be a prime different from p, then
H1(C/F,Q )=H1 (H1 Sp(2))
e´t ℓ ∼ ab⊕ t ⊗
as I -representations, with
s
H1 = IndIF V , H1 = (IndIF ǫ )⊖ǫ ,
ab Is s t Is s R
s∈MX/IF s∈MX/IF
whereX isthesetofclustersthatareneithersingletonsnor(proper)disjointunions
of even size clusters, and where Sp(2) denotes the 2-dimensional special ℓ-adic
representation.3
Remark 2.4. When J is semistable we will refer to the dimension of H1 as the
t
toric dimension of J. If the dimension of H1 is equal to the genus g, we say that
t
the reduction is totally toric.
Lemma 2.5. Let f(x) [x] be a t-Eisenstein polynomial of degree n, with
F
∈ O
(n,tp)=1. Then I acts tamely on the roots of f(x) and permutes them cyclically
F
and transitively. Moreover, v(r r′)= t for any two roots r=r′ of f(x).
− n 6
Proof. Let r be a root of f(x). The Newton polygon of f(x) has a unique slope
equal to t, so all the roots of f(x) have valuation t. In particular, f(x) is
−n n
irreducible and the field Fnr(r) is a tamely ramified extension of degree n of Fnr.
By uniqueness, it is Galois and its Galois group is C . As f(x) is irreducible over
n
Fnr, the cyclic group C acts transitively on the roots of f(x).
n
Since the extension Fnr(r)/Fnr is tame, the standard homomorphism
Gal(Fnr(r)/Fnr) F∗; σ σ(r)
→ 7→ r
is injective. In particular if σ is non-trivial then σ(r) = 0,1 in F, and hence
r 6
v(σ(r) r) = v(r) = t. As I acts transitively on the roots, this shows that
− n F
v(r r′)= t for any distinct pair of roots r,r′ of f(x). (cid:3)
− n
3 Let ℓ and p be distinct prime numbers. The special representation Sp(2) over a local field
F/Qp isthe(tame) 2-dimensionalℓ-adicrepresentationgivenby:
Sp(2)(τ)=(cid:18)10 tℓ(1τ)(cid:19), Sp(2)(FrobF/F)= 10 01q!,
where τ ∈ IF, the character tℓ(τ) is an ℓ-adic tame character, FrobF/F is a fixed Frobenius
elementandq=#F.
CONSTRUCTING HYPERELLIPTIC CURVES 7
Lemma 2.6. Suppose that f(x) [x] has type t q ,...,q with (q ,tp)=1
F 1 k i
∈O −{ }
for all i and let f(x)=h(x) k g (x α )be thecorresponding factorisation as in
i=1 i − i
Definition 1.3. Let β ,...,β be the roots of g (x α ) and let β ,...,β
i,1 Qi,qi i − i 0,1 0,degh
be the roots of h(x).
(i) If i=j, then v(β β )=0 for all a,b.
i,a j,b
6 −
(ii) If i=0, then v(β β )= t for all a=b.
6 i,a− i,b qi 6
(iii) v(β β )=0 for all a,b.
0,a 0,b
−
(iv) Theclustersoff(x)arethewholesetofrootsR,sets β ,...,β for every
{ i,1 i,qi}
i=0 and singleton roots.
6
(v) For s=R, the inertia subgroup I acts trivially on C[s ] and γ =ǫ =1.
F 0 s s
(vi) For s = β ,...,β with i = 0, the inertia subgroup I acts tamely on
{ i,1 i,qi} 6 F
C[s ] and the eigenvalues of a generator τ of tame inertia are precisely the
0
q -th roots of unity. Moreover, γ and ǫ are also tame and
i s s
1 if t is even 0 if q =2
i
γ (τ)= ǫ (τ)= 6
s s
( 1 if t is odd, (1 if qi =2.
−
Proof. (i) The roots of g (x) all reduce to 0 in F, so those of g (x α ) all reduce
i i i
−
to α . The result follows as α α for i=j and α β for all i,j.
i i 6≡ j 6 i 6≡ 0,j
(ii) This follows from Lemma 2.5.
(iii) The statement follows from the definition of type t q ,...,q and of h(x).
1 k
−{ }
(iv) Clear from (i), (ii) and (iii).
(v) I acts trivially on the roots of h(x) and on α ,...,α , and hence trivially
F 1 k
{ }
on C[s ]. Here µ =d =λ =0, so γ =ǫ =1.
0 s s s s s
(vi) The result for C[s ]=C[s] follows from Lemma 2.5. By (ii) d = t , µ =0 so
0 s qi s
λ = 1(µ +d s ) = t. Therefore γ is either trivial or it has order 2 depending
s 2 s s| 0| 2 s
on the parity of t. Since µ =0, ǫ is tame and
s s
0 if q =2
i
ǫ (τ)= 6
s
(1 if qi =2.
(cid:3)
Proof of Theorem 2.1. By Lemma 2.6 (iv) the set X of clusters of f(x) which are
not singletons nor unions of even clusters consists of the whole set of roots R and
β ,...,β for every i=0, with β as in Lemma 2.6.
{ i,1 i,qi} 6 i,j
The inertia group I does not permute the clusters so by Theorem 2.3 we have
F
He´1t(C/F,Qℓ)∼=Ha1b⊕(Ht1⊗Sp(2)) with Ha1b = s∈XVs and Ht1 =( s∈Xǫs)⊖
ǫ =0, where the last equality follows from Lemma 2.6 (vi) since q =2 for all i.
R i
L 6 L
By Lemma 2.6 (v) and (vi), V is tame for each cluster s.
s
For s=R, by Lemma 2.6 (v) inertia acts trivially on V .
R
For s = β ,...,β , by Lemma 2.6 (vi) the eigenvalues of τ on V are
{ i,1 i,qi} s
ζqi,ζq2i,...,ζqqii−1or−ζqi,−ζq2i,...,−ζqqii−1dependingonwhethertisevenoroddre-
spectively. Inparticular,τ actssemisimplyonH1(C/F,Q )byanelementoforder
e´t ℓ
dividing2 q . ThisprovestheclaimabouttheactionofinertiaonH1(C/F,Q ).
i i e´t ℓ
As ℓ = p, H1(C/F,Q ) is the dual of T (J) Q as an I -representation. In
6 Q e´t ℓ ℓ ⊗ ℓ F
particular, the action of I on T (J) factors through the same tame quotient and
F ℓ
τ has the same set of eigenvalues. The result for J[ℓ] follows by reducing the
characteristic polynomial of τ modulo ℓ. (cid:3)
8 SAMUELEANNIANDVLADIMIRDOKCHITSER
2.2. J[ℓ] when ℓ=p: fundamental characters.
GivenanabelianvarietyA/F withsemistablereduction,aresultduetoRaynaud
allowsus to recoverthe eigenvalues of a generatorof the tame inertia groupacting
on A[p]. Recall that for an integer d coprime to p, we write ζ for a primitive d-th
d
root of unity, chosen such that for all divisors d′ of d we have ζd′ =ζ .
d dd′
Theorem 2.7. Let A/F be an abelian variety with semistable reduction. Then the
eigenvalues of a generator of the tame inertia group on A[p] are all of the form
Pni=−01aipi
ζ
pn−1
for 1 n 2dim(A) and 0 a e , and where the ζ form some compatible
i F d
≤ ≤ ≤ ≤
system of roots of unity.
For ease of reading, we will recall briefly the theory of fundamental characters.
For further details see [Ser72, 1].
§
Let I denote the tame inertia quotient of I .
t F
A surjective homomorphism ψ :I F× defined by
n t → pn
σ(π )
ψ (σ)= n modπ where π = pn−√1π ,
n n n F
π
n
is a fundamental character of level n. The set of fundamental characters of level n
isthesetofthencharactersψ ,ψp,...,ψpn−1;thissetisindependentofthechoice
n n n
of pn−√1πF. The fundamental characters of level n satisfy compatibility relations
with fundamental characters of level m for any integer m dividing n:
pn−1
ψ (τ)pm−1 =ψ (τ), for allτ I .
n m t
∈
Proof of Theorem 2.7. Let V be a Jordan-Ho¨lder factor of dimension n over F
p
of the I -module A[p]. Then, by [Ray74, Corollaire 3.4.4], V has the structure
F
of a 1-dimensional F -vector space with the action of I given by a character
pn F
̟ : I F× , where
F → pn
̟ =ψPni=−01aipi
n
with 0 a e .
i F
Let τ≤be a≤fixed generatorof tame inertia. Then ψ (τ)=ζ F× and̟(τ)
n pn−1 ∈ pn
acts as multiplication by ζpPn−ni=−101aipi on Fpn.
Let Φ be the minimal polynomial of ζpn−1 over Fp. Since Fp[ζpn−1] ∼= Fpn, the
minimal polynomial Φ has degree n and hence its roots are
ζpn−1,ζppn−1,ζppn2−1,...,ζppnn−−11.
Therefore, by the Cayley-Hamilton theorem, these are precisely the eigenvalues of
multiplication by ζ on V.
pn−1
Hence, the eigenvalues of ̟(τ) are ζPni=−01aipi. (cid:3)
pn−1
2.3. Creating a transvection.
Finally, we will need a criterion to ensure that some element of Gal(Q(J[ℓ])/Q)
acts as a transvection on J[ℓ]. We again use inertia groups for achieving this.
CONSTRUCTING HYPERELLIPTIC CURVES 9
Definition 2.8. Recall that a transvection in GSp (F ) is a unipotent element σ
2g ℓ
such σ Id has rank 1.
2g×2g
−
Lemma 2.9. Suppose that p ∤ 2ℓ and f(x) [x] has type 1 2 . Then some
F
∈ O −{ }
element of I acts as a transvection on J[ℓ].
F
Proof. The model of the curve consisting of the chart y2 = f(x) and the usual
chart at infinity is a regular proper semistable model of C. The dual graph of the
special fibre is a vertex with a loop. The homology group of the dual graph is Z
with intersection pairing (1), so the Tamagawa number of the Jacobian over Knr
is c(J/Knr)=det(1)=1 (see e.g. [Pap13, Theorem 3.5 and Theorem 3.8]).
On the other hand, for a principally polarised g-dimensional semistable abelian
variety A of toric dimension d, the inertia group acts on T (A) by block matrices
ℓ
of the form
Id 0 t (σ)N
d ℓ
σ 0 Id 0
2g−2d
7→ 
0 0 Id
d
where t is the ℓ-adic tame character and N is a d dsymmetric integer-valued
ℓ
×
matrix that satisfies c(A/Knr)= coker(N) (it is the matrix induced by the mon-
| |
odromy pairing composed with the principal polarisation on A); see e.g. [GR72,
9, 10] or the summary in [DD09, 3.5.1]. In our case d=1 and c(J/Knr)=1,
§ § §
soN is a 1 1 matrix with entry 1. In particular,picking σ appropriatelygivesan
×
element of the inertia group that acts on J[ℓ] as a transvection. (cid:3)
3. Irreducibility
The aim of this section is to provide explicit criteria on f(x) that force irre-
ducibilityofthemodℓGaloisrepresentation. Thekeyideaistoensurethatimages
oflocalGaloisgroupsaresufficientlylargeandcanbepatchedtogethertoguarantee
global irreducibility.
3.1. Local representations.
Proposition 3.1. Let C : y2 = f(x) be a hyperelliptic curve over a local field
F of odd residue characteristic p, with f(x) [x] monic and squarefree, and
F
∈ O
let J = Jac(C). Suppose that f(x) has type t q ,...,q where q ,...,q are
1 k 1 k
−{ }
odd primes, coprime to t. Suppose moreover that the size of the residue field #F
is a primitive root modulo each of the q . Then for every prime ℓ = p,q ,...,q ,
i 1 k
6
the semisimple representation (J[ℓ] F ) decomposes as a direct sum of one
⊗Fℓ ℓ ss
(q 1)-dimensional,one(q 1)-dimensional,...,one(q 1)-dimensionalirreduci-
1 2 k
− − −
bleG -subrepresentation,andallotherirreducibleconstituentsbeing1-dimensional.
F
Proof. By Theorem 2.1, G acts tamely on J[ℓ] and the non-trivial eigenvalues of
F
a generator of tame inertia are ζri,j for r =1,...,q 1, where each sign is + if
± qi i,j i−
t is even and if t is odd. We claim that the conclusion of the proposition holds
−
for any semisimple F -representationV with this property.
ℓ
The action on V factors through a finite group G= τ,φ , where τ ⊳G is the
h i h i
(tame) inertia subgroup and φ is any lift of Frobenius.
Write V for the z-eigenspace of τ on V. Since φτφ−1 = τ#F, and hence
z
τφ−1 = φ−1τ#F, it follows that φ−1 maps Vz to Vz#F. In particular, φ−(qi−1)
is an endomorphism of V .
±ζqi
10 SAMUELEANNIANDVLADIMIRDOKCHITSER
Pick v ∈ V±ζqi which is an eigenvector for the action of φ−(qi−1) on V±ζqi and
considerthe subspaceW = v,φ−1v,...,φ−(qi−2)v . Since W is closedunderτ and
h i
φ−1, it is a G -submodule of V.
F
Moreover,as #F is a primitive root modulo q , it follows that the eigenvalues of
i
τ on v ,..., φ−(qi−2)v are precisely the non-trivial qi-th roots of unity, or their
h i h i
negatives. Inparticular,astheseτ-eigenvaluesaredistinct,anyG-submoduleofW
must be a direct sum of some of the φjv ’s. As φ−1 permutes these τ-eigenspaces
h i
transitively, it therefore follows that W is irreducible.
Now the result follows by substituting V by V/W and then proceeding by in-
duction on the dimension. (cid:3)
3.2. Global representations.
Lemma 3.2. Let C : y2 = f(x) be a hyperelliptic curve over a number field K,
where f(x) [x] is a monic squarefree polynomial of degree 2g +2. Suppose
K
∈ O
that f(x) has type t q ,q at p and type t′ q at p , where:
1 2 2 3 3
−{ } −{ }
q , q and q are primes with q q <q <2g+2 and q +q =2g+2;
1 2 3 1 2 3 1 2
• ≤
p ,p ∤2;
2 3
•
t is coprime to q q , and t′ is coprime to q ;
1 2 3
•
#F is a primitive root modulo q and q ;
• p2 1 2
#F is a primitive root modulo q .
• p3 3
Then for every prime ℓ ∤ q ,q ,q ,#F ,#F , the G -module J[ℓ] is absolutely
1 2 3 p2 p3 K
irreducible, where J is the Jacobian of C.
Proof. By Proposition 3.1, the restriction of J[ℓ] F to G contains an irre-
⊗Fℓ ℓ Kp3
ducible (q 1)-dimensional subquotient. Also its restriction to G has exactly
3− Kp2
two Jordan-Holder factors and these have dimension (q 1) and (q 1). It
1 2
− −
follows that, on the one hand, J[ℓ] F can have at most two Jordan-Holder
⊗Fℓ ℓ
factors,inwhichcasethey havedimensions (q 1)and(q 1),and,onthe other
1 2
− −
hand, J[ℓ] F has a Jordan-Holder factor of dimension at least (q 1). Hence
⊗Fℓ ℓ 3−
J[ℓ] F is irreducible. (cid:3)
⊗Fℓ ℓ
Theorem 3.3. Let C : y2 = f(x) be a hyperelliptic curve over a number field K,
where f(x) [x] is a monic squarefree polynomial of degree 2g +2. Suppose
K
∈ O
that f(x) has type t q ,q at p , type t′ q ,q at p′, type t′′ q at p ,
−{ 1 2} 2 −{ 4 5} 2 −{ 3} 3
and type t′′′ q at p′, where:
−{ 5} 3
q ,q ,q ,q and q are primes such that:
1 2 3 4 5
•
2g+2=q +q =q +q q <q q <q <q <2g+2.
1 2 4 5 4 1 2 5 3
≤
p ,p′,p ,p′ ∤2 q and they have distinct residue characteristics;
• 2 2 3 3 i i
t is coprime to q q ; t′ is coprime to q q ; t′′ is coprime to q and t′′′ is
1 2 4 5 3
• Q
coprime to q ;
5
#F is a primitive root modulo q and q ;
• p2 1 2
#Fp′ is a primitive root modulo q4 and q5;
• 2
#F is a primitive root modulo q ;
• p3 3
#Fp′ is a primitive root modulo q5.
• 3
Then for every prime ℓ, the G -module J[ℓ] is absolutely irreducible, where J is
K
the Jacobian of C.