`-adic and mod-` Galois Representations.
Alejandro Argáez-Garcı́a
A.Argaez-Garcia@warwick.ac.uk
06/11/2014
The aim of this note is to describe how to find a stable lattice for a Galois
representation ρE,` : GK → GL2 (Q` ) attached to an elliptic curve E. In this
way we will obtain an integral Galois representation we will be able to define
a residual mod-` Galois representation ρE,` : GK → GL2 (F` ).
Contents
1 Lattices on Q2` and the Bruhat-Tits Tree.
1.1 Bruhat-Tits Tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
3
2 Stable lattices and Galois representations.
2.1 Stable lattices and isogeny classes. . . . . . . . . . . . . . . . . . . . . . .
4
4
3 Surjectivity of Galois Representations.
3.1 Dickson’s Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Surjectivity for ` ≥ 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Non-surjectivity for ` = 2, 3. . . . . . . . . . . . . . . . . . . . . . . . . . .
6
7
7
8
Bibliography
9
Definitions and Notation.
• K is a number field.
• S is a finite set of primes of K.
• E is an elliptic curve over K.
• GK = Gal(K/K).
• V = Q2` (or any other 2-dimensional vector space over Q` ).
1
• L, L0 , . . . , are lattices, i.e., rank 2 Z` -submodules of V .
• Λ, Λ0 , . . . , are the classes of L, L0 , . . . , with respect to the equivalence relation
given by homothety.
• ρ : GK → GL(V ) ∼
= GL2 (Q` ) is a continuous Galois representation, unramified
outside S.
1 Lattices on Q2` and the Bruhat-Tits Tree.
Definition 1.1. We call a subset L ⊆ Q2` a lattice if L is a rank 2 free Z` -module of
Q2` . Equivalently L is a lattice of Q2` if there exist two independent vectors v1 , v2 ∈ Q2`
such that
L = Z` v1 + Z` v2 = {xv1 + yv2 |x, y ∈ Z` }.
For example we have that
(a) L0 = Z2 = Z` (1, 0) + Z` (0, 1), and
(b) La,b = Z` (`a , 0) + Z` (0, `b ) with a, b ∈ Z,
are examples of lattices of Q2` .
Now, fix a lattice L = Z` v1 + Z` v2 . We want to characterize all the lattices L0 such
that L ⊇ L0 ⊇ `L.
Define
φ : L → F2`
xv1 + yv2 7→ (x, y),
where we denote by x 7→ x the reduction map Z` → F` = Z` /`Z` . We see that ker(φ) =
`L and so we have L/`L ∼
= F2` . Hence any L0 such that L ⊇ L0 ⊇ `L, L0 /`L is a subspace
2
of F` . The extreme cases are when
(
0
iff L0 = `L
0
L /`L =
F2` iff L0 = L.
The other cases are when L0 /`L is isomorphic to a one dimensional subspace of F2` .
There are ` + 1 such subspaces and they are generated by (a, 1) with a ∈ F` and (1, 0).
(This is an exercise from the exam of modular forms.)
1.1 Bruhat-Tits Tree.
We want to construct a graph whose vertices are the equivalence classes of the lattices
of Q2` . We define an equivalence relation on the set of lattices of Q` as follows
L ∼ L0 ⇔ L0 = λL for some λ ∈ Q∗` .
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Definition 1.2. The Bruhat-Tits tree is the graph T, with:
1. vertices Λ := [L], where Λ is the equivalent class of some lattice L of Q2` .
2. There is an edge between two vertices v1 and v2 of T if and only if
∃L such that v1 = Λ
∃L0 such that v2 = Λ0
and
L ⊃ L0 ⊃ `L.
Since L ⊃ L0 ⊃ `L implies L0 ⊃ `L ⊃ `L0 , the graph T is an undirected graph.
For every pair of vertices Λ, Λ0 there are representative lattices L, L0 with L ⊃ L0 and
L/L0 cyclic of order `d . Now there is a unique sequence of intermediate lattices each of
index ` in the previous one linking L to L0 : we define the distance between Λ and Λ0 to
be d. Hence the graph is connected and there is a unique path between any two vertices
so it is a tree.
Example 1. There are eight 2-isogeny classes for the elliptic curves with conductor 15.
2 Stable lattices and Galois representations.
The following chapter was sent to me by Professor John E. Cremona as part of a joint
work.
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2.1 Stable lattices and isogeny classes.
Let ρ be an `-adic representation of GK , i.e., ρ : GK → GL2 (Q` ).
Definition 2.1. A lattice L is GK -stable (with respect to ρ) if ρ(GK )(L) ⊆ L. This
property only depends on the homothety class Λ of L.
Proposition 2.2. Every representation ρ has at least one stable lattice.
Proof. Let L be any lattice; the subgroup H of GK which leaves L stable is open and
hence of finite index since GK is compact. Hence L has only finitely many images under
GK , and the sum of these is stable.
Given a stable lattice L and using a Z` -basis for L as a basis for V we obtain an
integral representation ρL : GK → GL2 (Z` ). We will be interested in the collection of
these, for fixed ρ and varying stable lattice L.
Definition 2.3. Two integral representations ρj : GK → GL2 (Z` ) are isogenous if they
are conjugate as representations into GL2 (Q` ), i.e., if there exists U ∈ GL2 (Q` ) such
that ρ2 (σ) = U ρ1 (σ)U −1 for all σ ∈ GK .
Definition 2.4. Let ρ : GK → GL2 (Z` ) be an integral representation. The residual
representation associated to ρ is the map ρ : GK → GL2 (F` ) obtained by composing ρ
with the reduction modulo `.
GK
ρ
/ GL2 (Z` )
mod `
ρ
GL2 (F` )
If ρ is any representation and L a stable lattice for ρ, then the associated residual
representation ρL is the induced representation on L/`L ∼
= F2` .
Note that isogenous representations need not in general have the same (or isomorphic)
residual representations. If a rational representation (into GL2 (Q` )) has more than one
stable lattice L (up to homothety) then it will have more than one residual representation
ρL and these need not be isomorphic.
Example 2. Let E1 and E2 be elliptic curves defined over K with a K-rational 2-isogeny
from E1 → E2 . For each curve we obtain an integral representation into GL2 (Z2 ) by
letting GK act on the 2-adic Tate module of each curve (with respect to a fixed Z2 -basis
for each). The residual representations have image which is either of order 1 (if Ej (K)[2]
has order 4) or 2 (if Ej (K)[2] has order 2). Both can occur in the same isogeny class.
In fact there must be a curve in the class with non-trivial residual image by the result
known as “Ribet’s wrench” [5].
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In Example 1 we saw the Bruhat-Tits tree of the 2-isogeny classes for all the elliptic
curves of conductor 15. In particular, if we take
E15.a1 : y 2 + xy + y = x3 + x2 − 2160x − 39540
E15.a2 : y 2 + xy + y = x3 + x2 − 135x − 660
we will get that ρE15.a1 ,2 (GQ ) ∼
= C2 and ρE15.a2 ,2 (GQ ) ∼
= C1 .
We will mainly be interested in irreducible representations, such as those attached
to elliptic curves without CM. For these, the number of stable lattices is finite (up to
homothety). The following result must be well-known, but I do not know any reference.
Proposition 2.5. The number of stable lattices (up to homothety) is finite if and only
if ρ is irreducible.
Proof. If ρ is reducible, let L be a stable lattice and hwi a stable line for some w ∈ V .
We may scale w by a power of ` so that w ∈ L but `−1 w ∈
/ L, and then there exists
v ∈ L such that L = hv, wi. Set Ln = h`n v, wi for n ≥ 0. Then every Ln is stable and
no two are homothetic. Notice that the stable line hwi is the limit of the Ln as n → ∞.
Conversely suppose that there are infinitely many pairwise non-homothetic stable
lattices. These determine infinitely many stable vertices in the Bruhat-Tits tree. Note
that if Λ1 , Λ2 are both stable and distance d apart, then all of the d − 1 lattices between
them are also stable. To see this, we may represent the classes Λj by lattices Lj such
that L1 ⊃ L2 and L1 /L2 is cyclic of order `d . Now GK acts on L1 /L2 and leaves every
subgroup invariant, since (being cyclic) it has only one subgroup of each order `k for
0 ≤ k ≤ d. These subgroups have the form L/L2 where L1 ⊇ L ⊇ L2 , and the class of
L is a vertex between Λ1 and Λ2 , which is therefore stable.
Now any infinite subtree of the Bruhat-Tits tree is unbounded and contains an infinite
half line, so there is an infinite sequence of stable lattices Ln for n ≥ 0 such that
n
Ln ⊃ L
Tn+1 with index ` and L0 /Ln is cyclic of order ` for all n. The intersection
L∞ = n≥0 Ln is a stable Z` -module of rank at most 1 (since it has infinite index in
L0 ), and to complete the proof we show that it has rank exactly 1 (a line).
Let L0 = hv, wi. Each Ln is determined by an element (cn : dn ) ∈ P1 (Z/`n Z) such
that
Ln = {xv + yw | x, y ∈ Z` , cn x + dn y ≡ 0 (mod `)}.
Without loss of generality, (c1 : d1 ) = (1 : 0) and L1 = h`v, wi. Since Ln+1 ⊂ Ln we have
(cn+1 : dn+1 ) ≡ (cn : dn ) (mod `n ) and in particular cn ∈ Z∗` , so again without loss of
generality we may take cn = 1 and then dn+1 ≡ dn (mod `n ) for all n. This implies that
d = limn dn exists in Z` (in fact in `Z` ). Hence L∞ = {xv + yw | x = −dy} = hw − dvi,
which is a stable line as required.
Proposition 2.6. Let ρ be an integral representation. The number of stable lattices (up
to homothety) is 1 if and only if the residual representation ρ is irreducible.
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Proof. (Note: this is Exercise 1.4 in [6].) Let L be any stable lattice for L and let ρ = ρL
be the induced representation on L/`L. Suppose that there is another stable lattice L0 ,
not homothetic to L; without loss of generality, we may take L0 to have homothety class
adjacent to that of L in the Bruhat-Tits tree, and hence (replacing L0 by a homothetic
lattice if necessary) be contained in L with index `. Now GK leaves stable the line L0 /`L
in L/`L, so ρ is reducible.
Conversely, if ρ is reducible, then it leaves stable a line in L/`L which must have the
form L0 /`L where L0 has index ` in L and is GK -stable, so the class of L0 is stable and
distinct from that of L.
The preceding propositions say that the isogeny class of a representation ρ is finite if
and only if ρ is irreducible, and is a singleton if and only if the residual representation
ρ (with respect to any stable lattice) is irreducible.
Example 3. As we saw in Example 2, the representations attached to those elliptic
curves are reducible so we get a Bruhat-Tits tree with more than one vertex (see Example
1).
By Propositions 2.5 and 2.6 we can see that
1. For the isogeny class 44a we have
E44.a1 :y 2 = x3 + x2 − 77x − 289
E44.a2 :y 2 = x3 + x2 + 3x − 1
and its Bruhat-Tits tree looks like
2. For E245.a1 : y 2 + y = x3 − 7x + 12 we have that its Bruhat-Tits tree looks like
3 Surjectivity of Galois Representations.
In this section we will show two important results about Galois Representations.
3.1 Dickson’s Theorem.
In the following theorem, due to Dickson, see [1] and [4], are listed all finite subgroups
of PGL2 (F` ), for ` ≥ 3, up to conjugation:
Theorem 3.1 (Dickson’s Theorem.). Let ` ≥ 3 be a prime and H a finite subgroup of
PGL2 (F` ). Then a conjugate of H is one of the following groups:
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1. a finite subgroup of the upper triangular matrices (Borel subgroup),
2. PSL2 (F`r ) or PGL2 (F`r ) for r ∈ Z>0 ,
3. a dihedral group D2n with n ∈ Z>1 and (`, n) = 1,
4. a subgroup isomorphic to either A4 , S4 or A5 .
Let
π : GL2 (F` ) → PGL2 (F` )
G 7→ H
where H = π(G). Then taking G = π −1 (H) we will have
H
Borel
PSL2 (F`r )
PGL2 (F`r )
Dihedral
A4 , S4 , A5
G
Borel
(F∗`s )2 · SL2 (F`r )
F∗`s · GL2 (F`r )
Normalizer of
split or non-split
Cartan
F∗`s · π −1 (H)
3.2 Surjectivity for ` ≥ 5.
Let E be an elliptic curve over Q and ρE,` its respective `-adic (integral) Galois representation. A natural question about ρE,` is whether the representation is surjective or
not-surjective.
Lemma 3.2 ([6], IV, 3.4, Lemma 3). Let X be a closed subgroup of SL2 (Z` ) whose
image under the reduction mod ` map in SL2 (Z/`Z) is surjective. Assume ` ≥ 5. Then
X = SL2 (Z` ).
For E and σ ∈ Gal(Q(E[`n ])/Q), we know that det(ρE,` ) = χ(σ), where χ : Gal(Q(E[`n ])/Q) →
Z∗` is the `-adic cyclotomic character. Also, because of the Weil-paring we have that
µ`n ⊆ Q(E[`n ]) and Q does not contain any elements of µ`n apart from 1 or ±1 if `
is odd or even respectively. Hence χ is surjective mod `n and the determinant map is
surjective. If the image of ρE,` mod `n contains SL2 (Z/`n Z) and an element of every
determinant mod `n , then it must contain all the elements of GL2 (Z/`n Z). Therefore
we conclude that if ρE,` : Gal(Q(E[`n ])/Q) → GL2 (Z/`n Z) mod `n is surjective on
SL2 (Z/`n Z), then it is surjective on GL2 (Z/`n Z).
Now if ρE,` is surjective it contains SL2 (Z/`Z). Then Lemma 3.2 implies that for
` ≥ 5, ρE,` is surjective on SL2 (Z` ) and by previous observation, ρE,` is surjective on
GL2 (Z` ). There we have shown that for ` ≥ 5 if ρE,` is surjective then ρE,` is also
surjective. However, for ` < 5, the above phenomenon no longer occurs.
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3.3 Non-surjectivity for ` = 2, 3.
The mod-` Galois representation over Q for ` ∈ {2, 3} have been explicitly explored by
[2] and [3]. For ` = 2 the following theorem describes when ρ is surjective mod 2 but
not mod 4 and mod 4 but not mod 8.
Theorem 3.3 ([2], Theorem). Let E : y 2 = x3 + ax + b be an elliptic curve over Q with
discriminant ∆ = −16(4a3 + 27b2 ) and j-invariant j = −1728(4a)3 /∆. Then
(1) ρ2 is surjective ⇔ x3 + ax + b is irreducible and ∆ 6∈ (Q∗ )2 .
(2) ρ4 is surjective ⇔ ρ2 is surjective, ∆ 6∈ −1 · (Q∗ )2 and j 6= −4t3 (t + 8) for any t ∈ Q.
(3) ρ8 is surjective ⇔ ρ4 is surjective and ∆ 6∈ ±2 · (Q∗ )2 .
For ` = 3, the elliptic curves that are surjective mod 3 but not mod 9 have been
classified by [3]; the j-invariant is of the form
j=−
37(t2 − 1)3 (t6 + 3t5 + 6t4 + t3 − 3t2 + 12t + 16)3 (2t3 + 3t2 − 3t − 5)
,
(t3 − 3t − 1)9
t ∈ Q.
References
[1] Leonard Eugene Dickson. Linear groups: With an exposition of the Galois field
theory. with an introduction by W. Magnus. Dover Publications, Inc., New York,
1958. 3.1
[2] Tim Dokchitser and Vladimir Dokchitser. Surjectivity of mod 2n representations of
elliptic curves. Math. Z., 272(3-4):961–964, 2012. 3.3, 3.3
[3] Noam D. Elkies. Elliptic curves with 3-adic galois representation surjective mod 3
but not mod 9, 2006. 3.3, 3.3
[4] Serge Lang. Introduction to modular forms, volume 222 of Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
Springer-Verlag, Berlin, 1995. With appendixes by D. Zagier and Walter Feit, Corrected reprint of the 1976 original. 3.1
[5] Barry Mazur. How can we construct abelian Galois extensions of basic number fields?
Bull. Amer. Math. Soc. (N.S.), 48(2):155–209, 2011. 2
[6] Jean-Pierre Serre. Abelian l-adic representations and elliptic curves. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute.
W. A. Benjamin, Inc., New York-Amsterdam, 1968. 2.1, 3.2
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