∗
Growth of Hecke fields
over a slope 0 family
Haruzo Hida
Department of Mathematics, UCLA,
Los Angeles, CA 90095-1555, U.S.A.
∗A
conference talk on January 27, 2014 at Simons Conference (Puerto Rico).
The author is partially supported by the NSF grant: DMS 0753991 and
DMS 0854949. Posted as http://www.math.ucla.edu/˜hida/HF.pdf
Sug Woo Shin and Templier generalized Serre’s method to measure the growth of Hecke fields in an analytic way for growing
level. We study such a growth on a thin p-adic analytic family
of slope 0 Hilbert modular forms.
Fix an odd prime p > 2. Analyzing prime factorization of Weil
numbers in the union of algebraic extensions with bounded degree of the cyclotomic field K of all p-power roots of unity,
we show that there are only finitely many Weil l-numbers of a
given weight for a prime l (upto roots of unity). Applying this
fact to Hecke eigenvalues of cusp forms in slope 0 p-adic analytic families of cusp forms of p-power level, we show that the
field generated by most of T (l)-eigenvalue over the family has
unbounded degree over K.
This implies that, for a number field L, most of Hecke operators
in a non CM irreducible component of the (spectrum of) the big
Hecke algebra is transcendental over the group algebra Z[G] of
G := L× ∩ Γ inside the Iwasawa algebra Λ = Zp[[Γ]] for Γ =
1 + pZp.
1
§1. Notation
To describe the Hilbert modular cyclotomic ordinary big p-adic
Hecke algebra, we introduce some notation. Fix
•
•
•
•
•
A prime p (we assume p is odd for simplicity);
A totally real field F with integer ring O;
an integral ideal N of F prime to p;
two field embeddings C ←- Q ,→ Qp;
Γ = 1 + pZp ⊂ Z×
p and K = Q(µp∞ ) ⊂ Q.
Let Sk, := Sk (Npr , ; C) denote the space of (parallel) weight k
adelic Hilbert cusp forms of level Npr with Neben character modulo Npr . Thus is the central character of the automorphic
representation generated by each Hecke eigenform in the space.
Regard as a character of the strict ray class group ClF (Np∞) =
m ) module Np∞.
lim
Cl
(
N
p
F
←−m
§2. Hecke algebra
Let the rings
Z[] ⊂ C and Zp[] ⊂ Qp
be generated by (n) for F -ideals n over Z and Zp.
The Hecke algebra over Z is
h = Z[][T (n)|0 6= n ⊂ O] ⊂ End(Sk,).
Put hk, = h ⊗Z[] Zp[].
Sometimes our T ((p)) (resp. T (p)) is written as U (p) (resp. U (p)
for a prime p|p) as the level is divisible by p.
§3. Cyclotomy hypothesis
We have the norm map N : ClF (Np∞) → Z×
p induced by the
association: a 7→ |O/a| for ideal a prime to Np. Further projecting
down to the maximal torsion-free quotient Γ of Z×
p , we have
hN i : ClF (Np∞) → Γ. For simplicity, we assume Im(hN i) = Γ
(i.e., F/Q is not wildly ramified at p). We fix a splitting
ClF (Np∞) = ∆ × Γ.
Fix a finite order character δ : ∆ → Z×
p.
We assume that = δε with ε factoring through hN i (so our
character has a fixed part δ and varying cyclotomic part ε).
Let Λ = Zp [[Γ]] (the Iwasawa algebra). Fix a generator γ = 1 + p
of Γ. Identify Λ with Zp[[T ]] by γ 7→ 1 + T = t.
§4. Big Hecke algebra
The ordinary part hord
k, ⊂ hk, is the maximal ring direct summand on which U (p) is invertible; so,
hord = e · h for e = lim U (p)n! .
n→∞
We have a unique ‘big’ Hecke algebra h = hδ such that
• h is free of finite rank over Zp[[T ]] with T (n) ∈ h (T (p) = U (p))
• Let γ = 1 + p. If k ≥ 2 and ε : Γ → µp∞ is a character,
∼ hord , T (n) 7→ T (n),
h ⊗Λ,t7→ε(γ)γ k Zp[εδk ] =
k,εδk
where δk = δω 2−k for the Teichmüller character ω.
§5. Galois representation
Each irreducible component
Spec(I) ⊂ Spec(h)
has a Galois representation unramified outside Np
ρI : Gal(Q/F ) → GL(2)
with coefficients in I (or its quotient field) such that
Tr(ρI(F robl)) = a(l)
(for the image a(l) in I of T (l)) for all primes l - Np. Usually ρI
has values in GL2(I), and we suppose this for simplicity.
We regard P ∈ Spec(I)(Qp) as an algebra homomorphism P :
I → Qp, and we put ρP = P ◦ ρI : Gal(Q/F ) → GL2(Qp).
§6. CM component and CM family
We call a Galois representation ρ CM if there exists an open
subgroup G ⊂ Gal(Q/F ) such that the semi-simplification (ρ|G)ss
has abelian image over G.
We call I a CM component if ρI is CM.
If I is a CM component, it is known that for a CM quadratic
extension M in which all p|p splits, there exists a Galois character
∼ IndF ϕ.
ϕ : Gal(Q/M ) → I× such that ρI =
M
∼ IndF ϕ for some arithmetic point P , I is a CM compoIf ρP =
M P
nent.
§7. Analytic family
A point P of Spec(I)(Qp) is called arithmetic if P (t) = εδk (γ)γ k
for k ≥ 2 and ε : Γ → µp∞ . If P is arithmetic, we have a Hecke
eigenform fP ∈ Sk,εδk such that
fP |T (n) = aP (n)fP (n = 1, 2, . . . )
for aP (n) := P (a(n)) = (a(n) mod P ) ∈ Qp. We write εP = εδk
and k(P ) = k for such a P .
Thus I gives rise to an analytic families
FI = {fP |arithemtic P ∈ Spec(I)}, ΦI = {ρP |P ∈ Spec(I)}.
Write Q(fP ) for the subfield generated by aP (l) for all primes l
(the Hecke field).
Pick an infinite set A of arithmetic points P ∈ Spec(I)(Qp) with
fixed weight k(P ) = k ≥ 2.
§8. Theorems on Hecke fields
Theorem 1 (H-theorem). Recall K = Q(µp∞ ). Then I is a non
CM component if and only if there exists a set of primes Ξ of
F with Dirichlet density one such that for any infinite set A of
arithmetic points in Spec(I) of a fixed weight k ≥ 2,
lim [K(aP (l)) : K] = ∞ for each l ∈ Ξ.
P ∈A
If I is a CM component, [K(aP (l)) : K] is bounded independent
of arithmetic P and prime l.
The H-theorem essentially follows from:
Main Theorem: Let Σ be a subset of primes of F with positive
density outside pN. If there exists an infinite set Al ⊂ Spec(I)(Qp)
of arithmetic points P with fixed weight kl ≥ 2 for each l ∈ Σ
such that [K(aP (l)) : K] ≤ Bl for K = Q(µp∞ ) with a bound Bl
(possibly dependent on l) for all P ∈ Al, then I has CM.
§9. CM and Eisenstein ring and bounded degree.
Consider the torus T = Reso/ZGm for a number field L with
integer ring o. Take a non-trivial character 1 6= ν ∈ X ∗(T ). We
×
×
assume ν : T (Zp ) = o×
p = (o ⊗Z Zp ) → Zp = Gm (Zp ). Here ν can
be the norm character NL/Q or, if L is a CM field with a p-adic
Q
ϕ
CM type Φ, ν : (L ⊗Q Qp)× → Q×
given
by
ν(ξ)
=
ϕ∈Φ ξ .
p
Define an integral domain R = Rν (resp. Rν ) by the subalgebra
of Λ (resp. Λ/pΛ = Fp[[T ]]) generated over Z(p) = Q ∩ Zp (resp.
Fp) by tlogp(ν(α))/ logp(γ) for all 0 6= α ∈ o×
.
(p)
For CM component, the ring generated over Z(p) by a(l)’s is
essentially of the above form Rν for a suitable ν. The field
K(ν) generated by ν(α) (α ∈ L) is a finite extension of K. If
P (t) = δk (γ)γ k = ζγ k , then
P (tlogp(ν(α))/ logp(γ)) = ζ logp(ν(α))/ logp(γ) ν(α)k ∈ K(ν, δ)
and therefore [K(aP (l)) : K] is bounded if I has CM.
§10. Transcendence.
Let Q be the quotient field of Λ and fix its algebraic closure Q.
Corollary I. Regard I ⊂ Q and Rν ⊂ Λ ⊂ I. Take a set Σ of
prime ideals of F outside pN of positive density. If Qν (a(l)) ⊂ Q
for all l ∈ Σ is a finite extension of Qν for the quotient field Qν
of Rν , then I is a component having complex multiplication by a
CM quadratic extension M/F .
Indeed, [Qν (a(l)) : Qν ] < ∞ ⇒ [K(aP (l)) : K] ≤ [Qν (a(l)) : Qν ] for
all l ∈ Σ and P ∈ A. By Main Theorem, I has CM. An obvious
consequence of the above corollary is
Corollary II. Let the notation be as in the above theorem. If I
is a non-CM component, for a density one subset Ξ of primes of
F , the subring Q(Rν )[a(l)] of Q for all l ∈ Ξ has transcendental
degree 1 over Q(Rν ).
§11. A conjecture.
Recall Rν = Fp[tlogp(α)/ logp(γ) |α ∈ o] ⊂ Fp[[T ]] = Λ/pΛ. Write Qν
for the quotient field of Rν .
Conjecture. Let eI be a reduced irreducible non CM component
of Spec(h ⊗Zp Fp) embedded in an algebraic closure F of Fp((T ))
as a Λ-algebra. Then for a density one subset Ξ of primes of F ,
the subring Qν [a(l)] of Fp((T )) for all l ∈ Ξ has transcendental
degree 1 over Qν , where a(l) is the image of T (l) in eI.
This conjecture implies the vanishing of the Iwasawa µ-invariant
of the Deligne–Ribet p-adic L-function.
§12. Consequence for the µ-invariant.
We can think of the non-cuspidal big Hecke algebra H including
e of
Eisenstein components E. For the Eisenstein component E
Spec(H ⊗Zp Fp) (the special fiber of E), the image b(l) of T (l) is
N
in Rν for ν : F × −→ Q×.
κ
Write ρE = id ⊕κ for a character Gal(Q/F ) −
→ E× . Then we
have the corresponding Deligne-Ribet p adic L-function Lp(κ).
Regarding Lp ∈ Λ, we have Lp(κ)(γ k−1 − 1) + L(1 − k, ψω −k ) for
some fixed ψ. By Wiles, for each prime factor P |Lp(κ) in λ,
we have a cuspidal component I such that P |a(l) − b(l) for all l.
Write a(l) and b(l) for the image of a(l) and b(l) in Fp((T ))
Take P = (p) ⊂ Λ, this is impossible by the conjecture for all l ∈
Ξ; so, p - Lp(κ) (i.e., the Iwasawa µ-invariant of Lp(κ) vanishes).
Indeed, b(l) ∈ Rν but a(l) is transcendental over Qν .
§13. Weil numbers, Start of the proof of Main Theorem.
For simplicity, we assume that F = Q. For a prime l, a Weil
l-number α ∈ C of integer weight k − 1 ≥ 0 satisfies
(1) α is an algebraic integer; (2) |ασ | = l(k−1)/2 for all conjugates.
We say that α is equivalent to β if α/β ∈ µp∞ (Q).
Theorem 2 (Finiteness Theorem). Let d be a positive integer.
Let Kd be the set of all finite extensions of K = Q[µp∞ ] of degree
d inside Q. If l 6= p, there are only finitely many Weil l-numbers
S
×
of a given weight in the set-theoretic union L∈Kd L× (in Q ) up
to equivalence.
If we insist L/K tame l-ramification, there is only finitely many
isomorphism class of L⊗Z Zl as K-algebra; so, possibilites of prime
factorization of Weil l-numbers of weight k − 1 are finite. If not,
the prime factorization of αd! is a product of tamely ramified
primes; so, there are finitely many possibilties of αd! up to roots
of unity.
§14. Some more notation
We introduce one more notation:
(A) If p is a prime factor of p, let Ap be the image of U (p) in I, and
for a prime l - Np, fix a root Al in Q of det(T − ρI(F robl)) = 0.
n
Take and fix pn-th root t1/p of t = 1 + T in Q and consider
n
1/p
Λn := Λ[µpn ][[T ]][t
]⊂Q
n
which is independent of the choice of t1/p .
§15. Frobenius Eigenvalue formula
Replacing I by its finite extension, we assume that Al ∈ I for a
prime l.
Proposition 1 (Frobenius eigenvalue formula). Let Ll,P = K(Al,P )
(Al,P = P (Al)) for each arithmetic point P with k(P ) = k ≥ 2.
Fix a prime ideal l as in (A). Suppose the there exists an infinite
set A of arithmetic points of weight k ≥ 2 of Spec(I) such that
(Bl) if l ∈ Σ, Ll,P /K is a finite extension of bounded degree independently of P ∈ A.
Then we have Al ∈ Λn ∩ I in Q for 0 ≤ n ∈ Z, and there exist s ∈ Qp
and 0 6= c0 ∈ Q such that Al(X) = c0(1 + T )s for 0 6= s ∈ Qp.
§16. A rigidity lemma
We prepare a lemma to prove the proposition. Let W be a p-adic
valuation ring finite flat over Zp and Φ(T ) ∈ W [[T ]]. Regard Φ
as a function of t = 1 + T ; so, Φ(1) = Φ|T =0. We start with a
lemma whose characteristic p version was studied by Chai:
Lemma 1 (Rigidity). Suppose that there is an infinite subset
Ω ⊂ µp∞ (K) such that Φ(Ω) ⊂ µp∞ . Then there
exist ζ0 ∈ µp∞
P
s
n.
and s ∈ Zp such that ζ0−1Φ(t) = ts = ∞
T
n=0 n
Note here that if
b ×G
b = Spf(W [t, t\
−1, t0 , t0−1])
Z⊂G
m
m
is a formal subtorus, it is defined by the equation t = t0s for
s ∈ Qp. Thus we need to prove that the graph of the function
b ×G
b
t 7→ ζ0−1Φ(t) in G
m
m is a formal subtorus. For simplicity,
assume Φ(1) = 1 (so ζ0 = 1) in the following proof.
§17. Proof of the rigidity lemma.
b →G
b .
Step 1: Regard Φ as a morphism of formal schemes G
m
m
Step 2: For any σ in an open subgroup 1+pmZp ⊂ Gal(W [µp∞ ]/W ) ⊂
z
σ
z
Z×
p , we have Φ(ζ ) = Φ(ζ ) = σ(Φ(ζ)) = Φ(ζ) (ζ ∈ Ω); so,
Φ(tz ) = Φ(t)z
if ζ σ = ζ z for z ∈ 1 + pmZp.
b ×G
b
Step 3: The graph Z of t 7→ Φ(t) in G
m
m is therefore stable
under (t, t0) 7→ (tz , t0z ) for z = 1 + pmZp.
Step 4: Pick a point (t0, t00 = Φ(t0)) of infinite order in Z, then
mz
1+pm z 0 1+pm z
0
0
p
(t0
, t0
) = (t0, t0)(t0, t0)
∈Z
for all z ∈ Zp. Thus Z has to be a coset of a formal subgroup
m
generated by (t0, t00)p . Since (1, 1) ∈ Z, Z is a formal torus, and
we find s ∈ Zp with Φ(t) = ts.
§18. Proof of Frobenius eigenvalue formula
We give a sketch of a proof assuming I = W [[T ]]. Suppose
[Ll,P : K] < Bl < ∞ for P ∈ Al. By Finiteness theorem, we have
S
only a finite number of Weil l-numbers of weight k in P ∈Al Ll,P
up to multiplication by p-power roots of unity, and hence
Al(P ) = Al,P for P ∈ Al hits one of such Weil l-number α of
weight k infinitely many times, up to roots of unity.
After a suitable variable change T 7→ Y = γ −k (1 + T ) − 1 and
division by a Weil number, A(Y ) satisfies the assumption of the
rigidity lemma. We have for s1 ∈ Zp
A(Y ) = c0(1 + Y )s1 ,
and A(T ) = c0 (1+T )s. From this, it is not difficult to determine
s explicitly.
§19. Recall of the assumptions of Main theorem
Suppose that there exist a set Σ of primes of positive density as
in Main theorem. By the assumption of the theorem, for each
l ∈ Σ, we have an infinite set Al of arithmetic points of a fixed
weight k = kl ≥ 2 of Spec(I) such that
(B) if l ∈ Σ, [Ll,P : K] ≤ Bl < ∞ for all P ∈ Al.
By absurdity, we assume that I is a non CM component. Pick
distinct P |(t − γ k ) and Q|(t − ζγ k ) in Al (ζ ∈ µp∞ ). The strategy
is as follows. By the eigenvalue formula, for l ∈ Σ for a positive
density set Σ, we have Al(t) = cl tsl ; so,
αl = Al(γ k ) = ζ −sl Al(ζγ k ) = ζ −sl βl
for any ζ ∈ µp∞ . Here αl (resp. βl) is a chosen eigenvalue of
sym⊗m ∼
ρP (resp. ρQ). From this, if ζ m = 1, we conclude ρP
=
sym⊗m
ρQ
⊗ χ with a finite order character χ for P = (t − γ k ) and
Q = (t − ζγ k ) (by a result of Rajan). This never happens if I is
non CM.
§20. Proof, Step 1: Use of Eigenvalue formula
By (B) and Frobenius eigenvalue formula applied to l ∈ Σ, we
have A := Al(t) = cltsl for sl ∈ Qp and 0 6= cl ∈ Q.
Note that we have A ∈ W [µpn ][[T ]][tp
−n
− 1]]. Since rankΛ I ≥
−n
rankΛ W [µpn ][[T ]][tp − 1]], the integer n is also bounded inden
pendent of l. Thus by the variable change t 7→ tp , we may
assume that I = W [[T ]] and A ∈ W [[T ]].
We now vary l ∈ Σ. We may assume that k = kl is independent
of l as we already know the shape of A = Al.
§21. Proof, Step 2: Compatible systems
Pick a p-power root of unity ζ 6= 1 of order 1 < m = pe, and
write αf,l = αl = Al(γ k ) and αg,l = βl = Al(ζγ k ). They are Weil
l-numbers of weight fl. Write f = fP for P = (t − γ k ) and g for
the cusp form fQ for Q = (t − ζγ k ). Since βl = ζlαl (l ∈ Σ) for a
root of unity ζl = ζ s we have βlm = αm
l .
Consider the compatible system of Galois representation associated to f and g. Pick a prime q (whose residual characteristic q 6=
p sufficiently large) completely split over Q in Q(f, g) = Q(f )(g).
Write ρf,q (resp. ρg,q ) for the q-adic member of the system associated to f (resp. g). Thus ρ?,q has values in GL2(Zq ).
§22. Proof, Step 3: Taking m-th power trace.
m
m
By βlm = αm
l for all l ∈ Σ, we have Tr(ρf,q(F robl)) = Tr(ρg,q (F robl))
for all prime l ∈ Σ prime to pN, where Tr(ρm
?,q)(g) is just the trace
of m-th matrix power ρm
?,q(g).
m ) match on
Since the continuous functions Tr(ρm
)
and
Tr(ρ
g,q
f,q
f := {F rob |l ∈ Σ}, we find that Tr(ρm ) = Tr(ρm ) on the
Σ
l
g,q
f,q
f
closure Σ of Σ.
§23.Proof, Step 4: Taking symmtric tensor.
Since Tr(ρm) = Tr(ρsym⊗m) − Tr(ρsym⊗(m−2) ⊗ det(ρ)), we get
f
over Σ,
sym⊗m
Tr(ρf,q
sym⊗(m−2)
) − Tr(ρf,q
⊗ det(ρf,q))
sym⊗(m−2)
⊗ det(ρg,q )).
sym⊗(m−2)
⊗ det(ρf,q)))
sym⊗m
= Tr(ρg,
) − Tr(ρg,q
q
which implies
sym⊗m
Tr(ρf,q
sym⊗(m−2)
⊕ (ρg,q
⊗ det(ρg,q )))
sym⊗m
= Tr(ρg,q
over Σ.
⊕ (ρf,q
§24. Identity of Symmetric tensors.
Non-CM property of f and g tells us that Im(ρ?,q) contains an
open subgroup of GL2(Zq ); so, the Zariski closure of Im(ρ?,q) is
the connected group GL(2) for ? = f, g.
Since Σ has positive Dirichlet density, by Rajan (IMRN, 1998),
there exists an open subgroup Gal(Q/K) of Gal(Q/F ) such that
as representations of Gal(Q/K)
sym⊗m
ρf,q
sym⊗(m−2)
⊕(ρg,q
∼ ρsym⊗m⊕(ρsym⊗(m−2) ⊗det(ρ )).
⊗det(ρg,q )) =
g,q
f,q
f,q
In particular, we get the identity of their p-adic members
sym⊗m ∼ sym⊗m
⊗ χ for a finite order character χ.
= ρg,p
ρf,p
§25. Contradiction, conclusion.
α ∗ ?
0 β? with unramified β? by Wiles.
∼ ρsym⊗m ⊗ χ
=
g,p
∼
We know ρ?,p|Gal(Q /Q ) =
p
p
sym⊗m
Then we have from ρf,p
j m−j
{αf βf
|j = 0, . . . , m} = {αjg βgm−j χ|j = 0, . . . , m}.
Note that we have β?([γ, Qp]) = 1,
αf ([u, Qp]) = uk−1 and αg ([u, Qp]) = ζuk−1
for u ∈ Z×
p and the root of unity ζ. Therefore we conclude
j m−j
αf βf
= αjg βgm−j χ ⇔ (αf /αg )j = (βg /βf )m−j χ
for all j. Thus (αf /αg )j = χ on the inertia group at p as β? is
unramified. By taking j = 0, χ is unramified at p; so, taking
j = 1, we conclude αf /αg = 1 on the inertia group at p. This is
a contradiction, as ζg = ζ 6= 1.