We prove that the density of ramified primes in semisimple p-adic
representations of Galois groups of number fields is 0. Ravi Ramakrishna has
produced examples of such representations that are infinitely ramified

Khare, Chandrashekhar

Rajan, C. S.

English

arXiv

We show that the set of primes which ramify in a semisimple p-adic representation of the absolute Galois group of a number field is of upper density zero

Indian Academy of Sciences

The density of Ramified primes in semisimple p-adic Galois representations

Publications of the IAS Fellows

IMRN International Mathematics Research Notices
2001, No. 12
The Density of Ramified Primes in Semisimple
p-adic Galois Representations
Chandrashekhar Khare and C. S. Rajan
1 Introduction
Let L be a number field. Consider a continuous, semisimple p-adic Galois representation
ρ : GL −→ GLm(K)
of the absolute Galois groupGL of L and with K a finite extension ofQp. In [R] in the case
when n = 2 and L = Q examples of such representations were constructed which were
ramified at infinitely many primes, which had open image, and which had determinant
ε, the p-adic cyclotomic character (see also the last section of [KR]). We say that such
representations are infinitely ramified. One may ask if in these examples of [R] the set
of ramified primes is of small density.
Theorem 1. Let ρ : GL → GLm(K) be a continuous, semisimple representation. Then the
set of primes Sρ that ramify in ρ is of density zero. 
The semisimplicity assumption is crucial as, using Kummer theory (see the ex-
ercise of [S2 , III-12]), one can construct examples of continuous, reducible, indecompos-
able representations ρ : GL → GL2(Qp) that are ramified at all primes. Note that as in
[R] infinitely ramified representations, though notmotivic themselves, arise as limits of
motivic p-adic representations.
After Theorem 1, we know that the set of primes that are unramified in a con-
tinuous, semisimple representation ρ : GL → GLm(K) is of density 1. Hence many of the
results (e.g., the strong multiplicity 1 results of [Ra]), which are available in the classi-
Received 11 October 2000.
Communicated by Barry Mazur.
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602 Khare and Rajan
cal case when ρ is assumed to be finitely ramified, extend to this more general situation.
After Theorem 1, it also makes good sense to talk of compatible systems of continuous,
semisimple Galois representations in the sense of [S2] without imposing the condition
that these be finitely ramified. We raise the following question.
Question 1. Given two compatible continuous, semisimple representations ρ : GL →
GLm(Q) and ρ
′ : GL → GLm(Q ′) with  =  ′, is the set of primes at which either ρ or ρ ′
ramifies finite?
2 Proof of theorem
2.1
Let ρ be as inTheorem 1. As ρ is continuous, we can regard ρ as taking values in GLm(O),
where O is the ring of integers of K. We denote the maximal ideal of OK by m, and we
denote by ρn the reduction modmn of ρ.
Wedefine Sρ,n to be the set of primesq of Lwhich satisfy the following conditions:
(1) q is of degree 1 overQ (this assumption is merely for notational convenience;
we denote abusively the prime of Q lying below it by q);
(2) q is unramified in ρ1 and not equal to p;
(3) ρn|Dq is unramified, but there exists a “lift” of ρn|Dq , with Dq the decompo-
sition group at q corresponding to a place above q in Q, to a representation ρ˜q of Dq
to GLm(K) that is ramified at q; by a lift we mean some conjugate of ρ˜q which reduces
modmn to ρn|Dq . Note that by (2) any such lift ρ˜q factors through Gq, the quotient of
Dq which is the Galois group of the maximal tamely ramified extension of Qq.
Let cρ,n be the upper density of the set Sρ,n.
Proposition 1. Given any ε > 0, there is an integer Nε such that cρ,n < ε for n > Nε. 
We claim that the proposition implies Theorem 1. To see this, first observe that
the primes of L which do not lie above primes of Q which split in the extension L/Q
are of density zero. To prove the theorem, it is enough to show that given any ε > 0 the
upper density of the set Sρ of ramified primes for ρ is less than ε. Consider the Nε that
the proposition provides. Further, note that in ρNε only finitely many primes ramify.
From this it readily follows that the upper density of Sρ is less than ε. Hence we have
Theorem 1. Thus it only remains to prove the proposition.
2.2 Tame inertia
Proposition 1 relies on the structure of the Galois group Gq of the maximal tamely
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ramified extension ofQq. This is used to calculate the densities cρ,n for large enough n.
The concept of largeness of n for our purposes is independent of the representation ρ and
the prime q and depends only on K and the dimension of the representation. Roughly, the
idea of the proof of Proposition 1 is that for these large n only semistable (i.e., the image
of inertia is unipotent) lifts intervene in the calculation of the cρ,n, and the conjugacy
classes in the image of ρn of the Frobenius classes associated to the primes in Sρ,n lie in
the O/mn-valued points of an analytically defined subset of im(ρ) of smaller dimension.
We flesh out this idea below.We implicitly use the fact that although one cannot speak of
eigenvalues of an element of GLm(A), for a general ring A, its characteristic polynomial
makes good sense.
The group Gq is topologically generated by two elements σq and τq that satisfy
the relation
σqτqσ
−1
q = τ
q
q (1)
and such that σq induces the Frobenius on residue fields and τq (topologically) generates
the tame inertia subgroup.
2.3 Reduction to the semistable case
Lemma 1. Let θ : Gq → GLm(K) be any continuous representation. Then the roots of the
characteristic polynomial of θ(τq) are roots of unity. Further, the order of these roots of
unity is bounded by a constant depending only on K. 
Proof. Using Krasner’s lemma, we know that there are only finitely many degree m ex-
tensions of K. Let K ′ be the finite extension of K which is the compositum of all the
degreem extensions of K. By extending scalars to K ′, we can assume that θ(τq) is upper
triangular. Let θ1 , . . . , θm be the diagonal entries. Using equation (1), we deduce that
{θ1 , . . . , θm} = {θ
q
1 , . . . , θ
q
m}.
From this it follows that the θi are roots of unity (of order dividing qm! − 1). Hence the
last statement of the lemma follows from the fact that there are only finitely many roots
of unity in K ′. 
Corollary 1. Let θ : Gq → GLm(K) be any continuous representation. Assume that the
characteristic polynomial of θ(τq) is not equal to (x − 1)m. Then there exists an integer
N(m,K) depending only on m and K such that the reduction modulomN(m,K) of any
conjugate of θ into GLm(O) is ramified. 
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Proof. Choose N(m,K) such that if ζ ∈ K ′∗ is a root of unity satisfying (ζ − 1)m ≡
0 (modmN(m,K) ), then ζ = 1 for K ′ as in the proof of Lemma 1.Then the corollary follows
by considering reductions of characteristic polynomials. 
Corollary 2. In a continuous, semisimple representation of ρ : GL → GLm(K), the set of
primes q for which ρ(τq) is not unipotent is finite. 
Corollary 3. Any continuous, semisimple abelian representation of GL → GLm(K) is
finitely ramified. 
2.4 The GL2 case
At this point, for the sake of exposition, we briefly indicate the proof of Theorem 1 when
m = 2 and ρ(GL) is open in GL2(K) with determinant ε the p-adic cyclotomic character.
(Note that in the case when the Lie algebra of ρ(GL) is abelian the ramification set is
finite by Corollary 3.)
Consider Sρ,n forn > N(2, K), and let q ∈ Sρ,n. Let ρ˜q be any lift of ρn|Dq to GL2(K)
which is ramified at q. By the above considerations, it follows that ρ˜q(τq) is unipotent,
which we can assume to be upper triangular. Since ρ˜q(σq) normalises ρ˜q(τq), we can
assume that
• ρ˜q(τq) is of the form ( 1 ∗0 1 ) and ρ˜q(τq) is nontrivial,
• ρ˜q(σq) is of the form (α ∗0 β ).
Observe that α = β because of relation (1). Thus we can further assume by
conjugating by an element of the form ( 1 10 y ) ∈ GL2(K) that ρ˜q(σq) is of the form (α 00 β ).
Then we see from equation (1) that αβ−1 = q.
Consider the invariant functions tr and det defined on the space of conjugacy
classes of GL2(O) or GL2(O/mn) given by the trace and determinant functions. We see
from our work that primes q ∈ Sρ,n for n > N(2, K) are such that the conjugacy classes
ρn(Frobq) satisfy the relation
tr2 = (1+ det)2 .
From this we conclude, using the fact that the image of ρ is open in GL2(K), Cebotarev
density theorem, and the second paragraph of [S1, p. 586], that cρ,n → 0 as n → ∞.
Proposition 1 follows in this case, and the proof of Theorem 1 is complete in the special
case of open image in GL2(K) with determinant ε. 
2.5 The general case
We reduce the general situation to the case when im(ρ) is a semisimple p-adic Lie group
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contained in GLM(Qp) for someM. First, byWeil restriction of scalars, we may assume
that K = Qp (with possibly a differentm). Let G be the Zariski closure of the image of ρ.
Since ρ is semisimple, G is reductive; let Z be the centre of the connected component of
G. Let ρs : GL → (G/Z)(Qp) be the corresponding representation. Because of Corollary 2,
we see that the ramification set of ρ and ρs differs by a finite set, and thus we can work
with ρs. Now embed G/Z into GLM /Qp for some M. Thus we have reduced to the case
when im(ρ) is a semisimple p-adic Lie group contained in GLM(Qp) for someM.
We look at Sρ,n for n > N(M,Qp), and we let q ∈ Sρ,n. Let ρ˜q be any lift of ρn|Dq
to GLM(K)which is ramified at q. By Corollary 1,we can assume that ρ˜q(τq) is unipotent
(and nontrivial), which we can further assume to be upper triangular.
Consider the canonical filtration of ρ˜q(τq) acting on the vector space Q
M
p , with
the dimension of the corresponding graded components m1 , . . . ,mi. By conjugating by
an element in the Levi, over a finite extension of Qp, of the corresponding parabolic
subgroup defined by ρ˜q(τq) (of the form GLm1× · · · ×GLmi), we can assume that ρ˜q(Gq)
is upper triangular.
Lemma 2. If fq(x) is the characteristic polynomial of ρ˜q(σq), then fq(x) and fq(qx) have
a common root. 
Proof. Let U be the subgroup of unipotent upper triangular matrices of SLM(Qp), and
let
U = U0 ⊃ U1 ⊃ · · · ⊃ 1
be the descending central filtration. Let i be the smallest integer such that ρ˜q(τq) /∈ Ui+1 .
By looking at the conjugation action of ρ˜q(σq) onUi/Ui+1 andusing relation (1), it follows
that there are two eigenvalues αq, βq of ρ˜q(σq) such that αqβq
−1 = q. Hence we have
the lemma. 
Consider ρ ′ = ρ ⊕ ε : GL → GLM(Qp) × GL1(Qp). Let G ′ = G × GL1 . Choose
an integral model for ρ ′, that is, ρ ′(GL) ⊂ GLM(Zp) × GL1(Zp), induced by the chosen
integral model of ρ, and denote by ρ ′n its reduction modm
n. We normalise the isomor-
phism of class field theory so that a uniformiser is sent to the arithmetic Frobenius. (So
ε(Frobq) = q.)
Let
(A,b) ∈ GLM(Qp)× GL1(Qp),
and let f(x) be the characteristic polynomial ofA. Let F be the invariant polynomial func-
tion on GLM×GL1 with Zp-coefficients defined by the resultant of the two polynomials
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606 Khare and Rajan
f(x) and f(bx).
By choosing b different from the ratios of eigenvalues of an element of G, we
deduce that no connected component of G ′ is contained inside the variety F = 0. Thus
we see that {F = 0} ∩G ′ is a subvariety of smaller dimension than the dimension of G ′.
Lemma 3. The image ρ ′(GL) is an open subgroup of G ′(Qp) = G(Qp)× GL1(Qp). 
Proof. Since im(ρ) is a semisimple p-adic group, we deduce from Chevalley’s theorem
(see [Bo , Corollary 7.9]) that im(ρ) is open in G(Qp). From this we further deduce that
the commutator subgroup of im(ρ) is of finite index in im(ρ). Thus the intersection of
the fixed fields of the kernel of ρ and ε is a finite extension ofQ. Certainly im(ε) is open
in Q∗p, and hence the lemma follows. 
From the openness of im(ρ ′), we see that
lim
n→∞
|im(ρ ′n)|
pnd
is a nonzero positive constant, where d is the dimension of G ′.
On the other hand, using the notation and results of [S1 , Section 3], if we denote
by Y˜n the elements x ∈ im(ρ ′n) that satisfy F(x) ≡ 0 (modpn), then from the second
paragraph of [S1, p. 586] it follows that |Y˜n| = O(pn(d−δ) ), where δ is a positive constant.
By Lemma 2, we see that ρ ′n(Frobq) ∈ Y˜n for q ∈ Sρ,n. Then applying the Cebotarev
density theorem we conclude that
cρ,n ≤ |Y˜n|
|im(ρ ′n)|
,
and hence cρ,n → 0 as n → ∞. This finishes the proof of Proposition 1 and hence of
Theorem 1. 
Remarks. To prove Theorem 1, instead of defining Sρ,n the way we did, we could have
have worked with the smaller subset consisting of primes that are unramified in ρn,
but ramified in the ρ of Theorem 1. In the notation of [S1, p. 586], we would then be
working with Yn rather than Y˜n. By [S1, Section 3,Theorem 8],we obtain a better estimate
cρ,n = O(p
−n).Thismay be useful to getmore precise quantitative versions ofTheorem 1.
We have defined Sρ,n the way we have for its use in [K].
An analog of Theorem 1 is valid for function fields of curves over finite fields of
characteristic  = p, and the same proof works. On the other hand, for function fields of
characteristic p,Theorem 1 is false, and in this case there are examples of semisimple p-
adic Galois representations ramified at all places. It is easy to construct such examples
using the fact that the Galois group in this case has p-cohomological dimension less
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than or equal to 1 (see [S3, Chapter II.2, Proposition 3]).
References
[Bo] A. Borel, Linear Algebraic Groups, 2d ed., Grad. Texts in Math. 126, Springer, NewYork, 1991.
[K] C. Khare, Limits of residually irreducible p-adic Galois representations, preprint, 2001,
http://www.math.tifr.res.in/˜shekhar
[KR] C. Khare and R. Ramakrishna, Finiteness of Selmer groups and deformation rings, preprint,
2001, http://www.math.tifr.res.in/˜shekhar
[Ra] C. S. Rajan, On strong multiplicity one for -adic representations, Internat. Math. Res. Notices
1998, 161–172.
[R] R. Ramakrishna, Infinitely ramified Galois representations, Ann. of Math. (2) 151 (2000), 793–
815.
[S1] J.-P. Serre, “Quelques applications du théorème de densité de Chebotarev” in Oeuvres: Collected
Works,Vol. 3, Springer, Berlin, 1986, 563–641.
[S2] , Abelian -adic Representations and Elliptic Curves, 2d ed., Adv. Book Class., Addison-
Wesley, Redwood City, N. Y., 1989.
[S3] , Galois Cohomology, Springer, Berlin, 1997.
Khare: School of Mathematics,Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai
400 005, India; shekhar@math.tifr.res.in
Rajan: School of Mathematics,Tata Institute of Fundamental Research, Homi Bhabha Road,Mumbai
400 005, India; rajan@math.tifr.res.in
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The density of ramified primes in semisimple p-adic Galois representations

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