Problems on Growth of Hecke fields

Transcription

1 Problems on Growth of Hecke fields Haruzo Hida Department of Mathematics, UCLA, Los Angeles, CA , U.S.A. A list of conjectures/problems related to my talk in Simons Conference in January 2014 (Puerto Rico). The author is partially supported by the NSF grant: DMS and DMS

2 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 unramified at p; 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. 1

3 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.

4 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. By unramifiedness of p in F, we know Im( N ) = Γ. 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.

5 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.

6 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). 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 ).

7 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.

8 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 weight k(p) 2.

9 8. Vertical Conjectures on Hecke fields Conjecture 1 (V-Conjecture). The component I is a non CM component if and only if for any infinite set A of arithmetic points in Spec(I) of varying weight with fixed level Np r (r bounded by some B > 0), lim [Q(a P(l)) : K] = for each prime l of F. P A Known facts: If I is a CM component, [Q(a P (l)) : Q] is bounded independent of arithmetic P and prime l. Suppose F = Q and l = (p) and that there exists P A with the corresponding motive M P is rank 2 with coefficients in Q. I has CM if and only if limsup P B A [Q(a P (p)) P B : Q] =, where B runs over all finite subsets of A (see JAMS 24 (2011)).

10 9. CM and Eisenstein ring and bounded degree. Let L /Q be a finite extension inside Q p. Consider the group G = L Γ, and put Z (p) = Q Z p. We may regard the group algebra Z (p) [G] as a subalgebra of Iwasawa algebra Λ = Z p [[T]] by sending α G to t log p(α)/log p (γ) for t = 1 + T. In any CM and Eisenstein component, the subalgebra generated over Z (p) by Hecke operators a(l) is a finite extension of Z (p) [G] for some choice of L. A consequence of the above conjecture is: Conjecture 2. For every prime l of F, a(l) in a non CM cuspidal component I is transcendental over Z (p) [G]. Known fact: As in my talk, for a density one subset Ξ of primes of F including those over p, the conjecture holds (see my preprint posted as

11 10. Mod p transcendence. Let I be a non CM component. As before for G = L Γ, embed F p [G] in F p [[T]] J. Take an irreducible component J of I Zp F p and write al) for the image of T(l) in J. Then J is an algebra over F p [[T]] = Λ Zp F p. Let F p ((T)) (resp. Q) be the quotient field of F p [[T]] (resp. F p [G]). Fix an algebraic closure F p ((T)) of F p ((T)). Conjecture 3. Suppose p > 3 and that N is square free. For any prime l of F, the subring F p [G][a(l)] of F p ((T)) for has transcendental degree 1 over Q ν, where a(l) is the image of T(l) in J. Known fact. Nothing!

12 11. Consequence for the cyclotomic 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 Z (p) [G] for L = 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) Z[G] but a(l) is transcendental over Z[G].

13 13. Derivative of a(p). Question 1. Let P be any height 1 prime of I over pλ. Assume that I is a cuspidal non CM component. Then for a prime p p of F, da(p) dt 0 mod P in I? A similar version could be true for a(l) for l N. Note that the derivative da(p) dt evaluated at an arithmetic point is closely related to the L-invariant of the adjoint Galois representation acting on sl 2 (I) via conjugation of ρ I (σ) by my old result (IMRN 2007, and recent preprint of Harron Jorza). For an Eisenstein component, I with the image a (p) of U(p), we see easily that da (p) dt = 0, and again this implies the vanishing of the µ-invariant of the Deligne Ribet p-adic L function.