Image of Λ-adic Galois representations modulo p

Let p ≥ 5 be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its mod p Galois representation contains an open subgroup of SL2(Fp[[T ]]) for the canonical “weight” variable T . This fact appears to be deep, as it is almost equivalent to the vanishing of the μ-invariant of the Kubota–Leopoldt p-adic L-function and the anticyclotomic Katz p-adic L-function. Another key ingredient of the proof is the anticylotomic main conjecture proven by Rubin/Mazur–Tilouine. Fix a prime p ≥ 5, field embeddings C i∞ ←↩ Q ip ↪→ Qp ⊂ Cp and a positive integer N prime to p. We sometimes identify Cp and C by a fixed field isomorphism compatible with the above embeddings. Consider the space of modular forms Mk+1(Γ0(Np), χ) with (p N, r ≥ 0) and cusp forms Sk+1(Γ0(Np), χ). Let the ring Z[χ] ⊂ C and Zp[χ] ⊂ Qp be generated by the values χ over Z and Zp, respectively. The Hecke algebra H = Hk+1(Γ0(Np ), χ; Z[χ]) over Z[χ] is H = Z[χ][T (n)|n= 1, 2, · · · ] ⊂ End(Mk+1(Γ0(Np), χ)). For any Z[χ]-algebra A ⊂ C, Hk+1(Γ0(Np), χ;A) = H ⊗Z[χ] A is actually the A-subalgebra of End(Mk+1(Γ0(Np), χ)) generated over A by the T (n)’s. Simply we put Hk+1,χ = Hk+1,χ/A = Hk+1(Γ0(Np ), χ;A) := H ⊗Z[χ] A for a p-adically closed subring A ⊂ Cp containing Zp[χ] (with uniformizer $). Fix a complete discrete valuation ring W finite over Zp inside Qp (though nothing changes even if W is just a complete discrete valuation ring over Zp inside Cp). Our T (p) is often written as U(p) as the level is divisible by p. The ordinary part Hk+1,χ/W ⊂ Hk+1,χ/W is then the maximal ring direct summand on which U(p) is invertible. We write e for the idempotent of Hk+1,χ/W ; so, e is the p-adic limit in Hk+1,χ/W of U(p) n! as n → ∞. Via the fixed isomorphism Cp ∼= C, the idempotent e not only acts on the space of modular forms with coefficients in W but also on the classical space Mk+1(Γ0(Np), χ). We write the image of the idempotent asMord k+1 for modular forms and Sord k+1 for cusp forms. Let χ1 = χN × the tame p-part of χ. Then, by [H86a] and [H86b], we have a unique ‘big’ Hecke algebra H = Hχ1/W such that (H1) H is free of finite rank over Λ := W [[T ]] equipped with T (n), U(m) ∈ H with (n,Np) = 1, m|Np, (H2) if k ≥ 1, regarding a character ε : Zp → μp∞(Qp) as a W -algebra homomorphism εk of Λ onto W [ε] sending (1 + T ) to γε(γ) for γ = 1 + p ∈ Zp, H⊗Λ,εk W [ε] ∼= Hk+1,εχk/W [ε] for χk := χ1ω, sending T (n) ⊗ 1 to T (n) and U(m)⊗ 1 to U(m), where ω is the Teichmüller character. Here W [ε] ⊂ Qp is the W -subalgebra generated by the values of ε. The corresponding objects for cusp forms are denoted by the corresponding lower case characters; so, h = Z[χ][T (n)|0< n ∈ Z] ⊂ End(Sk+1(Γ0(Np), χ)), hk+1,χ/A = hk+1(Γ0(Np ), χ;A) := h⊗Z[χ]A, the ordinary part hk+1,χ/W ⊂ hk+1,χ/W and the big cuspidal Hecke algebra hχ1/W . Replacing modular forms by cusp forms (and upper case symbols by Date: November 18, 2012. 2010 Mathematics Subject Classification. 11F11, 11F25, 11F33, 11F80. The author is partially supported by the NSF grant: DMS 0753991 and DMS 0854949, and part of this work was done during the author’s stay in January to March 2010 at the Institut Henri Poincaré Centre Emile Borel. The author thanks this institution for hospitality and support. 1 IMAGE OF Λ-ADIC GALOIS REPRESENTATIONS MODULO p 2 lower case symbols), we can construct the “big” cuspidal Hecke algebra h = hχ1 . Then, similarly to the case of modular forms, we have the following characterization of the cuspidal ‘big’ Hecke algebra h = hχ1/W : (1) h is free of finite rank over Λ equipped with T (n), U(m) ∈ h with (n,Np) = 1, m|Np, (2) h⊗Λ,εk W [ε] ∼= hk+1,εχk/W [ε] for k ≥ 1 (and ε as above). We often identify Λ with the completed group algebra W [[Γ]] of Γ = 1 + pZp = γ Zp by (1 + T )↔ γ. We have a surjective Λ-algebra homomorphism H h sending T (n) to T (n) (and U(m) to U(m)). Each (reduced) irreducible component Spec(I) ⊂ Spec(H) has a continuous 2-dimensional semi-simple representation ρI of Gal(Q/Q) with coefficients in the quotient field Q(I) of I (see [H86b]). This representation preserves an I-lattice L ⊂ Q(I) (i.e., an I-submodule of Q(I) of finite type which spans Q(I) over Q(I)), and as a map of Gal(Q/Q) into the profinite group AutI(L), it is continuous. Write a(l) for the image of T (l) (l Np) in I and a(p) for the image of U(p). The representation ρI restricted to the p-decomposition group Dp is reducible. We write ρ ss I for its semi-simplification over Dp. As is well known now (e.g., [GME] §4.3), ρI satisfies (Gal) Tr(ρI(Frobl)) = a(l) (l Np), ρ ss I ([γ ,Qp]) ∼ ( (1+T ) 0 0 1 ) and ρ I ([p,Qp]) ∼ ( ∗ 0 0 a(p) ) , where γ = (1 + p) ∈ Zp for s ∈ Zp and [x,Qp] is the local Artin symbol. Since σ 7→ Tr(ρI(σ)) has values in I, for each prime ideal P of I, the reduction modulo P of Tr(ρI) gives rise to a pseudo-representation (of Wiles) with values in I/P , which produces a semi-simple representation ρP : Gal(Q/Q) → GL2(κ(P )) with Tr(ρP ) = (Tr(ρI) mod P ), where κ(P ) is the quotient field of I/P (the residue field of P ). When P is the maximal ideal of I, we simply write ρ for ρP . We call a prime ideal P ⊂ I a prime divisor if Spec(I/P ) has codimension 1 in Spec(I). If a prime divisor P of Spec(I) contains (1 + T − εχk(γ)γ ) with k ≥ 1, by (2) we have a Hecke eigenform fP ∈Mk+1(Γ0(Np), εχk) such that its eigenvalue for T (q) is given by aP (q) := P (a(q)) ∈ Qp for all primes q. Here we regard P as a W -algebra homomorphism P : I → Qp. A prime divisor P ∈ Spec(I)(Qp) with P (1 + T − εχk(γ)γ ) = 0 for k ≥ 1 is called an arithmetic point (or prime). We write εP = ε and k(P ) = k ≥ 1 for an arithmetic P . Thus I gives rise to an analytic family FI = {fP |arithmetic P ∈ Spec(I)(Qp)} of slope 0 Hecke eigenforms. A component I (or the associated family) is called cuspidal if Spec(I) ⊂ Spec(h). A cuspidal component I is called a CM component if there exists a nontrivial character ξ : Gal(Q/Q)→ I such that ρI ∼= ρI ⊗ ξ. If a cuspidal I is not a CM component, we call it a non CM component. Pick and fix a non CM component I of prime-to-p level N . Then we have (R) ρ|Dp ∼= ( ∗ 0 δ ) with δ unramified. Consider the following conditions: (s) There exists σ ∈ Dp such that ρ(σ) has two eigenvalues α, β ∈ Fp with α 6= β; (v) ρI(Dp) contains a unipotent element g ∈ GL2(I) with g 6≡ 1 mod mI. We consider an involution ∗ of Fp[[T ]] over Fp such that φ(T ) = φ((1+T )−1−1). Write Fp[[T ]] for the subring fixed by ∗; so, Fp[[T ]] is a quadratic extension of Fp[[T ]]. Similarly we write F((T )) for the subfield of Fp((T )) fixed by ∗. Put Φ(N) = N ∏ l|N (1 − 1 l2 ) for an integer N > 1 and its prime factors l. Theorem. Take a non CM cuspidal component I of cube-free prime-to-p level N , and let P ∈ Spec(I) be a prime divisor above (p) ⊂ Zp[[T ]]. Suppose p Φ(N). If either (s) or (v) is satisfied, then the image of ρP contains an open subgroup of SL2(Fp[[T ]]). If neither (s) nor (v) holds, there exists an inner form S/Fp[[T ]] of SL(2)/Fp [[T ]] such that ρP contains an open subgroup of S(Fp((T )) ). We will prove a slightly stronger result as Theorem 6.2. Here is an outline of the proof. For a prime divisor P above (p) ⊂ Zp[[T ]], let κ(P ) be an algebraic closure of κ(P ). The Zariski closure of the image Im(ρP ) ∩ SL(2) in SL(2)/κ(P) is an algebraic subgroup GP of SL(2)/κ(P) defined over κ(P ). Let GP be the connected component of GP . Then G ◦ P is either a Borel subgroup, a torus or a unipotent group. If GP is in a Borel subgroup or is a torus, we prove that P has to be either an Eisenstein ideal or the family has congruence modulo P with a CM component I having CM by an imaginary quadratic field M . In the Eisenstein case, by a result of Mazur–Wiles [MW] and Ohta [O3] combined with the mod p admissible representation theory of Vigneras [V] IMAGE OF Λ-ADIC GALOIS REPRESENTATIONS MODULO p 3 (see Corollary 2.2), P divides the Iwasawa power series of a Kubota–Leopoldt p-adic L-function (under p φ(N) = N ∏ l|N (1− l)). This is impossible as the Kubota–Leopoldt p-adic L-function has trivial μ-invariant [FeW]. In the CM case, P divides Lp(Ad(ρI′ )) = h · Lp(Ψ−I′ ) (congruence criterion) for the class number h of M (see Proposition 5.6), where Lp(Ψ − I ) is the anticyclotomic p-adic Hecke L-function constructed by de Shalit, Yager and Katz (see [K], [HT] and [H07]). By [Fi] and [H10], the anticyclotomic p-adic Hecke L-function has trivial μ-invariant (under p Φ(N)); so, if p h, this proves the theorem. If p|h, by computation of the congruence power series, we prove that the congruence between CM components exhausts the p-part of the congruence power series (Theorem 5.1), and thereby, we conclude that GP is SL(2), and (Gal) implies, by a result of Pink [P], that ρP to have the open image property. This type of results, asserting that Im(ρP ) contains an open subgroup of SL2(Zp) for arithmetic points P of a non CM family was proven in an Inventiones paper by Ribet [Ri] and [Ri1] long ago. Thus fullness of the Galois image of special prime divisors (arithmetic or residual characteristic p) characterizes non-CM components (the importance of such characterization was brought up in the author’s first lecture in the Galois trimester 2010 at Poincaré Institute in Paris, which is summarized in the last section of [H11b]). In a forthcoming paper [H13], for non CM components I, we come close to the determination of Im(ρI) ∩ SL2(Λ). Conjecture. Let FI be a non CM cuspidal parallel weight Hilbert modular family (in [H88b]) of prime-to-p level N for a totally real field F . Suppose p ≥ 5, and let P be a prime divisor of I over (p) ⊂ Zp[[T ]]. Then we have (1) The mod P Galois representation ρP is irreducible over Gal(Q/F ). (2) Suppose p ΦF (N) = N(N) 2 ∏ l|N(1 − 1 N(l)2 ) and that N is prime to p and cube free. If either dimF F [μp] > 2 or the strict class number of F is odd, the image of ρP contains a subgr