Local indecomposability of Tate modules of abelian varieties of GL(2)-type. Haruzo Hida

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1 Loca indecomposabiity of Tate modues of abeian varieties of GL(2)-type Haruzo Hida Department of Mathematics, UCLA, Los Angees, CA , U.S.A. June 19, 2013 Abstract: We prove indecomposabiity of p-adic Tate modues over the p-inertia group for non CM (partiay p-ordinary) abeian varieties with rea mutipication. I wi aso discuss its appication (given by Bin Zhao) to oca indecomposabiity of Hibert moduar Gaois representations. A tak on in a conference at Lyon Écoe Normae. The author is partiay supported by the NSF grants: DMS and DMS

2 1. Greenberg s Question. Pick a totay rea fied F Q with integer ring O. Take a non CM AVRM A over a number fied k Q with integer ring O; so, O End(A /k ), dim A = [F : Q] and the centraizer of O in End(A /Q ) is O. Pick a prime p p of O and consider p-adic Tate modue T p A. Suppose that A has good reduction à moduo a prime P p of k, and assume that Ã[p](F p ) = O/p (p-ordinary at P). Greenberg s Question: Is T p A indecomposabe over the decomposition group D P? 1

3 2. Soution. Theorem 1. Yes it is indecomposabe. I try to expain my far-fetched proof and its consequences, assuming that p is unramified in F k (this assumption has been removed by Bin Zhao; so, the theorem is unconditiona). Fix a prime p and a finite set of rationa primes p Ξ unramified in F k; fied embeddings C i Q i C for a primes. Write (resp. L) be prime of O (resp. O) induced by Q C.

4 3. Hibert moduar Shimura variety. Let Z (Ξ) = Q Ξ Z, and A (Ξ) be the adee ring away from Ξ { }. Put V = O 2 and V (R) = V Z R for Z (Ξ) -agebras R. Hibert moduar Shimura variety Sh (Ξ) /Z (Ξ) cassifies (A, η (Ξ), λ) made of an AVRM A, eve structure for TA = im N A[N] η (Ξ) : V (A (Ξ) ) = TA Ẑ A(Ξ) and prime-to-ξ poarization cass. Thus Sh (Ξ) (R) = {(A, η (Ξ), λ) /R }/, where is by prime-to-ξ F-inear isogenies. We remove (Ξ) from our notation if no confusion is ikey.

5 4. Aut(Sh). Let G = Res F/Q GL(2). We et G(A (Ξ) ) act on Sh by η η g. If x Sh corresponds (A x, η x, λ x ), et M x = End 0 (A x ) = End(A x ) Q. Then we can embed M x ρ x G(A (Ξ) ) by α η x = η x ρ(α). Then ρ(m x ) gives rise to the stabiizer of x. If M x = F, the action of M x is trivia; but, if M x /F is a CM extension, the action factors through M x /F.

6 5. Serre Tate deformation theory Assume that A = (A, η, λ) has ordinary good reduction at L. Consider its reduction A L = (A L, η L, λ L ) = (A, η, λ) F L for an agebraic cosure F L of F L := O/L, which gives rise to a point x L Sh(F L ). Let W = W(F L ). Then the forma competion Ŝ of Sh aong x L is isomorphic to the Serre Tate deformation space. For any compete oca ring R with residue fied F L, Ŝ (R) = {A := (A, η A, λ A ) /R A /R R R/m R = A L }/ =. As is we known, Ŝ = Ĝm O.

7 6. Serre Tate coordinates Identify Ĝm = Spf( W [t, t 1 ]). Then consider the rigid anaytic space Ŝan associated to Ŝ in the sense of Bertheot. Taking the σ-component of og(t) given by τ,σ : Ĝm O(W ) = (1+m W ) Z O og p σ W σ W, we may identify Ŝan = Sp(C {{τ,σ }} σ ). We have a decomposition ΩŜan /C = σ Ω an σ/c such that Ω an σ is generated by dτ,σ.

8 7. CM action on Ŝ Let M L = End 0 F (A L/F L ) which is a CM quadratic extension of F generated by the N(L)-power Frobenius map φ L. We can embed α M L into G(A(Ξ) ) by α η L = η L ρ(α). Then we have, writing t for t 1 and t a = t a, τ,σ ρ(α) = α σ(1 c) τ,σ ( t ρ(α) = t α1 c ). So we ca τ,σ σ-eigen-coordinate. Any σ-eigencoordinate on Ŝan is proportiona to τ,σ. The origin τ = 0 gives rise to the canonica CM ift A cm of A L. Hereafter we take W = Ξ i 1 (W ) inside Q.

9 8. Point x Ŝp Sh of (A, η, λ). Suppose T p A is decomposabe over D P. If p remains prime over p, by non-cm property, we have t(a) 1 and hence T p A cannot be decomposabe. We may assume that there are more than two prime factors of (p) in F. For simpicity, we assume that [F : Q] = 2; so, (p) = pp. Write σ : F Q p corresponding to p and σ : F Q p corresponding to p. Then we have τ p,σ (x) = 0 and τ p,σ (x) 0.

10 9. Kodaira-Spencer map. Let π : A Sh (resp. π : A Ŝ) be the universa abeian varieties. By the O-action on A, O acts on Ω A/Sh and Ω A/Ŝ. Writing ω = π Ω A/Sh and ω = π ΩÂ/Ŝ, we have the foowing decomposition into σ-eigenspaces: ω = σ ω σ and ω = σ ω σ. The Kodaira-Spencer map induces a canonica isomorphism Ω σ,sh/w = ω 2σ,Ω σ,ŝ /W = ω 2σ.

11 10. Stabiity under CM action. Since τ p,σ ρ(α) = α σ(1 c) τ p,σ, the fiber ωp 2σ (x) at x of the invertibe sheaf ω 2σ is stabe under p/ŝ p the action of ρ(m P ). Since ω 2σ p (x) = ω 2σ (x) W W p, the fiber at x of the goba sheaf ω 2σ (x) is stabe under ρ(m P ). Pick another prime Ξ so that A has ordinary good reduction at L. Then we have ω 2σ (x) = ω 2σ (x) W W. Thus ω 2σ (x) is stabe under ρ(m P ); so, τ,σ (x) = 0 and τ,σ ρ(α) = α σ(1 c) τ,σ.

12 11. CM contradiction. By τ,σ (x) = 0 and τ,σ ρ(α) = α σ(1 c) τ,σ, A L has CM by the same M P = M L. The choice of L is arbitrary, by Chebotarev density appied to the Gaois representation on T p A, we can find with M L M P, a contradiction. Thus T p A must be indecomposabe over D P. The argument works we for any p-ordinary A and genera F.

13 12. Kodaira-Spencer map again. We restart with a CM abeian variety A cm with CM by O. Reca ω = π Ω A/Sh and ω = π ΩÂ/Ŝ, we have the foowing decomposition into σ-eigenspaces: ω = σ ω σ and ω = σ ω σ. The Kodaira-Spencer map induces a canonica isomorphism Ω σ,sh/w = ω 2σ and Ω σ,ŝ /W = ω 2σ. Writing the forma group  of A as /Ŝ  = Ĝm O with Ĝm = Spf( W[s, s 1 ]), the Kodaira-Spencer map is given by dτ,σ ( dsσ s σ ) 2.

14 13. Katz period and proportionaity. Choose an agebraic differentia ω cm with H 0 (A cm,ω A cm /W ) = (W O)ωcm. Assuming A P is ordinary, identifying Âcm = Ĝm O with Ĝm = Spf( W[s, s 1 ]), Katz defined his p-adic period Ω p,σ W by ( ) ωσ cm ds = Ω p,σ s σ comparing its σ-eigen components. Comparing the fibers of the Kodair-Spencer map at τ = 0, we get dτ p,σ /dτ,σ = Ω 2 p,σ /Ω2,σ. Since τ p,σ and τ,σ are proportiona, Theorem 2. τ p,σ /τ,σ = Ω 2 p,σ /Ω2,σ

15 14. Hibert moduar Gaois representation. Let f be a neary p-ordinary weight 2 non CM Hibert moduar Hecke eigenform for a totay rea fied K. Assume that its p-adic Gaois representation ρ f comes from an abeian variety A f of GL(2)-type (e.g., f is the image of Jacquet-Langands correspondence from a Shimura curve). Bin Zhao removed the unramifiedness assumption by the resut of Deigne-Pappas and showed that A f A e for an absoute simpe AVRM over a number fied k /K ; so, ρ f DP is indecompos- Theorem 3 (B. Zhao). abe. Baasubramanyam, Ghate and Vatsa have got a simiar resut under a different set of assumptions.

16 15. Appication to Coeman s probem. Assume p > 3 and et N be a positive number prime to p. Write M k (Γ 1(N)) for the space of eiptic overconvergent p-adic moduar forms. By the theta operator θ = q d dq, we have θ k 1 : M 2 k (Γ 1(N)) M k (Γ 1(N)). Coeman proved that for k 2, every cassica CM cuspida eigenform of weight k and sope k 1 is in the image of θ k 1. Coeman s question is Is there non-cm cassica cusp forms in the image of θ k 1?

17 16. Answer is no for k = 2. By p-adic Hodge theory (a resut of Kisin), if f of p-sope k 1 is in the image of θ k 1, ρ f has to be decomposabe at p (a remark made by Emerton). Thus by the resut of Zhao, we get Theorem 4 (B. Zhao). Suppose k = 2. Then a p-sope 1 cassica Hecke eigen cusp form is in the image of θ if and ony if f has compex mutipication.