Growth of Hecke fields over a slope 0 family

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 = Z p [[Γ]] for Γ = 1 + pz p. 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 Q p ; Γ = 1 + pz p Z p and K = Q(µ p ) Q. Let S k,ɛ := S k (Np r, ɛ; C) denote the space of (parallel) weight k adelic Hilbert cusp forms of level Np r with Neben character ɛ modulo Np r. Thus ɛ is the central character of the automorphic representation generated by each Hecke eigenform in the space. Regard ɛ as a character of the strict ray class group Cl F (Np ) = lim m Cl F(Np m ) module Np.

2. Hecke algebra Let the rings Z[ɛ] C and Z p [ɛ] Q p be generated by ɛ(n) for F-ideals n over Z and Z p. The Hecke algebra over Z is Put h k,ɛ = h Z[ɛ] Z p [ɛ]. h = Z[ɛ][T(n) 0 n O] End(S k,ɛ ). 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 : Cl F (Np ) Z p induced by the association: a O/a for ideal a prime to Np. Further projecting down to the maximal torsion-free quotient Γ of Z p, we have N : Cl F (Np ) Γ. For simplicity, we assume Im( N ) = Γ (i.e., F /Q is not wildly ramified at p). We fix a splitting Cl F (Np ) = Γ. Fix a finite order character δ : Z p. We assume that ɛ = δε with ε factoring through N (so our character has a fixed part δ and varying cyclotomic part ε). Let Λ = Z p [[Γ]] (the Iwasawa algebra). Fix a generator γ = 1+p of Γ. Identify Λ with Z p [[T]] by γ 1 + T = t.

4. Big Hecke algebra The ordinary part h ord k,ɛ h k,ɛ is the maximal ring direct summand on which U(p) is invertible; so, h ord = e h for e = lim n U(p) n!. We have a unique big Hecke algebra h = h δ such that h is free of finite rank over Z p [[T]] with T(n) h (T(p) = U(p)) Let γ = 1 + p. If k 2 and ε : Γ µ p is a character, h Λ,t ε(γ)γ k Z p [εδ k ] = h ord k,εδ k, T(n) T(n), where δ k = 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 ρ I : Gal(Q/F) GL(2) with coefficients in I (or its quotient field) such that Tr(ρ I (Frob l )) = a(l) (for the image a(l) in I of T(l)) for all primes l Np. Usually ρ I has values in GL 2 (I), and we suppose this for simplicity. We regard P Spec(I)(Q p ) as an algebra homomorphism P : I Q p, and we put ρ P = P ρ I : Gal(Q/F) GL 2 (Q p ).

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 ρ I 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 ϕ : Gal(Q/M) I such that ρ I = Ind F M ϕ. If ρ P = Ind F M ϕ P for some arithmetic point P, I is a CM component.

7. Analytic family A point P of Spec(I)(Q p ) is calledarithmetic if P(t) = εδ k (γ)γ k for k 2 and ε : Γ µ p. If P is arithmetic, we have a Hecke eigenform f P S k,εδk such that f P T(n) = a P (n)f P (n = 1,2,...) for a P (n) := P(a(n)) = (a(n) mod P) Q p. We write ε P = εδ k and k(p) = k for such a P. Thus I gives rise to an analytic families F I = {f P arithemtic P Spec(I)}, Φ I = {ρ P P Spec(I)}. Write Q(f P ) for the subfield generated by a P (l) for all primes l (the Hecke field). Pick an infinite set A of arithmetic points P Spec(I)(Q p ) with fixed weight k(p) = k 2.

8. Theorems on Hecke fields Theorem 1 (H-theorem). Recall K = Q(µ p ). 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(a P(l)) : K] = for each l Ξ. P A If I is a CM component, [K(a P (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 A l Spec(I)(Q p ) of arithmetic points P with fixed weight k l 2 for each l Σ such that [K(a P (l)) : K] B l for K = Q(µ p ) with a bound B l (possibly dependent on l) for all P A l, then I has CM.

9. CM and Eisenstein ring and bounded degree. Consider the torus T = Res o/z G m for a number field L with integer ring o. Take a non-trivial character 1 ν X (T ). We assume ν : T (Z p ) = o p = (o Z Z p ) Z p = G m(z p ). Here ν can be the norm character N L/Q or, if L is a CM field with a p-adic CM type Φ, ν : (L Q Q p ) Q p given by ν(ξ) = ϕ Φ ξ ϕ. Define an integral domain R = R ν (resp. R ν ) by the subalgebra of Λ (resp. Λ/pΛ = F p [[T]]) generated over Z (p) = Q Z p (resp. F p ) by t log p(ν(α))/log p (γ) for all 0 α o (p). For CM component, the ring generated over Z (p) by a(l) s is essentially of the above form R ν for a suitable ν. The field K(ν) generated by ν(α) (α L) is a finite extension of K. If P(t) = ɛδ k (γ)γ k = ζγ k, then P(t log p(ν(α))/ log p (γ) ) = ζ log p (ν(α))/log p (γ) ν(α) k K(ν, δ) and therefore [K(a P (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 R ν Λ I. Take a set Σ of prime ideals of F outside pn of positive density. If Q ν (a(l)) Q for all l Σ is a finite extension of Q ν for the quotient field Q ν of R ν, then I is a component having complex multiplication by a CM quadratic extension M /F. Indeed, [Q ν (a(l)) : Q ν ] < [K(a P (l)) : K] [Q ν (a(l)) : Q ν ] 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(R ν )[a(l)] of Q for all l Ξ has transcendental degree 1 over Q(R ν ).

11. A conjecture. Recall R ν = F p [t log p(α)/log p (γ) α o] F p [[T]] = Λ/pΛ. Write Q ν for the quotient field of R ν. Conjecture. Let Ĩ be a reduced irreducible non CM component of Spec(h Zp F p ) embedded in an algebraic closure F of F p ((T)) as a Λ-algebra. Then for a density one subset Ξ of primes of F, the subring Q ν [a(l)] of F p ((T)) for all l Ξ has transcendental degree 1 over Q ν, where a(l) is the image of T(l) in Ĩ. 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 Eisenstein components E. For the Eisenstein component Ẽ of Spec(H Zp F p ) (the special fiber of E), the image b(l) of T(l) is in R ν for ν : F N Q. Write ρ E = id κ for a character Gal(Q/F) κ E. Then we have the corresponding Deligne-Ribet p adic L-function L p (κ). Regarding L p Λ, we have L p (κ)(γ k 1 1) L(1 k, ψω k ) for some fixed ψ. By Wiles, for each prime factor P L p (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 F p ((T)) Take P = (p) Λ, this is impossible by the conjecture for all l Ξ; so, p L p (κ) (i.e., the Iwasawa µ-invariant of L p (κ) vanishes). Indeed, b(l) R ν but a(l) is transcendental over Q ν.

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) α σ = l (k 1)/2 for all conjugates. We say that α is equivalent to β if α/β µ p (Q). Theorem 2 (Finiteness Theorem). Let d be a positive integer. Let K d be the set of all finite extensions of K = Q[µ p ] of degree d inside Q. If l p, there are only finitely many Weil l-numbers of a given weight in the set-theoretic union L K d L (in Q ) up to equivalence. If we insist L/K tame l-ramification, there is only finitely many isomorphism class of L Z Z l as K-algebra; so, possibilites of prime factorization of Weil l-numbers of weight k 1 are finite. If not, the prime factorization of α d! is a product of tamely ramified primes; so, there are finitely many possibilties of α d! up to roots of unity.

14. Some more notation We introduce one more notation: (A) If p is a prime factor of p, let A p be the image of U(p) in I, and for a prime l Np, fix a root A l in Q of det(t ρ I (Frob l )) = 0. Take and fix p n -th root t 1/pn of t = 1 + T in Q and consider Λ n := Λ[µ p n][[t]][t 1/pn ] Q which is independent of the choice of t 1/pn.

15. Frobenius Eigenvalue formula Replacing I by its finite extension, we assume that A l I for a prime l. Proposition 1 (Frobenius eigenvalue formula).let L l,p = K(A l,p ) (A l,p = P(A l )) 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 (B l ) if l Σ, L l,p /K is a finite extension of bounded degree independently of P A. Then we have A l Λ n I in Q for 0 n Z, and there exist s Q p and 0 c 0 Q such that A l (X) = c 0 (1 + T) s for 0 s Q p.

16. A rigidity lemma We prepare a lemma to prove the proposition. Let W be a p-adic valuation ring finite flat over Z p 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 µ p (K) such that Φ(Ω) µ p. Then there exist ζ 0 µ p and s Z p such that ζ0 1 Φ(t) = ts = ( ) s n=0 n T n. Note here that if Z Ĝm Ĝm = Spf( W[t, t 1, t, t 1 ]) is a formal subtorus, it is defined by the equation t = t s for s Q p. Thus we need to prove that the graph of the function t ζ0 1 Φ(t) in Ĝm Ĝm is a formal subtorus. For simplicity, assume Φ(1) = 1 (so ζ 0 = 1) in the following proof.

17. Proof of the rigidity lemma. Step 1: Regard Φ as a morphism of formal schemes Ĝm Ĝm. Step 2: For any σ in an open subgroup 1+p m Z p Gal(W[µ p ]/W) Z p, we have Φ(ζz ) = Φ(ζ σ ) = σ(φ(ζ)) = Φ(ζ) z (ζ Ω); so, Φ(t z ) = Φ(t) z if ζ σ = ζ z for z 1 + p m Z p. Step 3: The graph Z of t Φ(t) in Ĝm Ĝm is therefore stable under (t, t ) (t z, t z ) for z = 1 + p m Z p. Step 4: Pick a point (t 0, t 0 = Φ(t 0)) of infinite order in Z, then (t 1+pm z 0, t 0 1+pm z ) = (t0, t 0 )(t 0, t 0 )pmz Z for all z Z p. Thus Z has to be a coset of a formal subgroup generated by (t 0, t 0 )pm. Since (1,1) Z, Z is a formal torus, and we find s Z p with Φ(t) = t s.

18. Proof of Frobenius eigenvalue formula We give a sketch of a proof assuming I = W[[T]]. Suppose [L l,p : K] < B l < for P A l. By Finiteness theorem, we have only a finite number of Weil l-numbers of weight k in P A l L l,p up to multiplication by p-power roots of unity, and hence A l (P) = A l,p for P A l 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 Y = γ k (1 + T) 1 and division by a Weil number, A(Y ) satisfies the assumption of the rigidity lemma. We have for s 1 Z p A(Y ) = c (1 + Y ) s 1, and A(T) = c 0 (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 A l of arithmetic points of a fixed weight k = k l 2 of Spec(I) such that (B) if l Σ, [L l,p : K] B l < for all P A l. By absurdity, we assume that I is a non CM component. Pick distinct P (t γ k ) and Q (t ζγ k ) in A l (ζ µ p ). The strategy is as follows. By the eigenvalue formula, for l Σ for a positive density set Σ, we have A l (t) = c l t s l; so, α l = A l (γ k ) = ζ s la l (ζγ k ) = ζ s lβ l for any ζ µ p. Here α l (resp. β l ) is a chosen eigenvalue of ρ P (resp. ρ Q ). From this, if ζ m = 1, we conclude ρ sym m P = χ with a finite order character χ for P = (t γ k ) and ρ sym m Q 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 := A l (t) = c l t s l for s l Q p and 0 c l Q. Note that we have A W[µ p n][[t]][t p n 1]]. Since rank Λ I rank Λ W[µ p n][[t]][t p n 1]], the integer n is also bounded independent of l. Thus by the variable change t t pn, we may assume that I = W[[T]] and A W[[T]]. We now vary l Σ. We may assume that k = k l is independent of l as we already know the shape of A = A l.

21. Proof, Step 2: Compatible systems Pick a p-power root of unity ζ 1 of order 1 < m = p e, and write α f,l = α l = A l (γ k ) and α g,l = β l = A l (ζγ k ). They are Weil l-numbers of weight f l. Write f = f P for P = (t γ k ) and g for the cusp form f Q for Q = (t ζγ k ). Since β l = ζ l α l (l Σ) for a root of unity ζ l = ζ s we have β m l = α m l. Consider the compatible system of Galois representation associated to f and g. Pick a prime q (whose residual characteristic q p sufficiently large) completely split over Q in Q(f, g) = Q(f)(g). Write ρ f,q (resp. ρ g,q ) for the q-adic member of the system associated to f (resp. g). Thus ρ?,q has values in GL 2 (Z q ).

22. Proof, Step 3: Taking m-th power trace. By βl m = α m l for all l Σ, we have Tr(ρ m f,q (Frob l )) = Tr(ρm g,q(frob l )) for all prime l Σ prime to pn, where Tr(ρ m?,q)(g) is just the trace of m-th matrix power ρ m?,q (g). Since the continuous functions Tr(ρ m f,q ) and Tr(ρm g,q) match on Σ := {Frob l l Σ}, we find that Tr(ρ m f,q ) = Tr(ρm g,q ) on the closure Σ of Σ.

23.Proof, Step 4: Taking symmtric tensor. Since Tr(ρ m ) = Tr(ρ sym m ) Tr(ρ sym (m 2) det(ρ)), we get over Σ, Tr(ρ sym m f,q ) Tr(ρ sym (m 2) f,q det(ρ f,q )) which implies = Tr(ρg,q sym m ) Tr(ρg,q sym (m 2) det(ρ g,q )). Tr(ρ sym m f,q over Σ. (ρ sym (m 2) g,q det(ρ g,q ))) = Tr(ρ sym m g,q (ρ sym (m 2) f,q det(ρ f,q )))

24. Identity of Symmetric tensors. Non-CM property of f and g tells us that Im(ρ?,q ) contains an open subgroup of GL 2 (Z q ); 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) ρ sym m f,q (ρ sym (m 2) g,q det(ρ g,q )) = ρg,q sym m (ρ sym (m 2) f,q det(ρ f,q )). In particular, we get the identity of their p-adic members ρ sym m f,p = ρ sym m g,p χ for a finite order character χ.

25. Contradiction, conclusion. We know ρ?,p Gal(Qp /Q p ) = Then we have from ρ sym m f,p {α j f βm j f ( α? ) 0 β? with unramified β? by Wiles. = ρg,p sym m χ j = 0,..., m} = {α j g βm j g χ j = 0,...,m}. Note that we have β? ([γ, Q p ]) = 1, α f ([u, Q p ]) = u k 1 and α g ([u, Q p ]) = ζu k 1 for u Z p and the root of unity ζ. Therefore we conclude α j f βm j f = α j g βm j g χ (α f /α g ) j = (β g /β f ) m j χ for all j. Thus (α f /α g ) 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 α f /α g = 1 on the inertia group at p. This is a contradiction, as ζ g = ζ 1.