On the Mumford-tate Conjecture of Abelian Fourfolds

We prove the Mumford-Tate conjecture for absolutely simple abelian fourfolds with trivial endomorphism algebras. The main goal of this paper is to prove the Mumford-Tate conjecture for certain abelian fourfolds. Let A/F be an abelian variety defined over a number field F of dimension n. Fix an algebraic closure F̄ of F and a complex embedding F̄ → C. Let V = H1(A/C,Q) be the first singular homology group of A/C with coefficients in Q. Then we denote by MT (A)/Q (resp. Hg(A)/Q) the Mumford-Tate group (resp. Hodge group) associated to the natural Hodge structure of V . On the other hand, for any rational prime l, let TlA(F̄ ) be the l-adic Tate module of A and set Vl = TlA(F̄ )⊗Zl Ql. Then we have a Galois representation ρl : Gal(F̄ /F ) → AutQl(Vl). We define an algebraic group Gl/Ql as the Zariski closure of the image of ρl inside the algebraic group AutQl(Vl), and let G ◦ l/Ql be its identity component. By comparison theorem, we have an isomorphism V ⊗Q Ql ∼= Vl. Under this isomorphism, the Mumford-Tate conjecture states that: Conjecture 0.1. For any prime l, we have the equality Gl/Ql = MT(A)×Q Ql. In this paper, we are interested in the case that A/F is an absolutely simple abelian fourfolds (so in particular n = 4). Let g/Q (resp. gl/Ql) be the Lie algebra of the algebraic group MT (A)/Q (resp. Gl/Ql). Then let h/Q (resp. hl/Ql) be the subalgebra of g/Q (resp. gl/Ql) consisting of elements of trace 0. So we have g = h ⊕ Q · Id (resp. gl = hl ⊕ Ql · Id). In [9], Moonen and Zarhin computed the Lie algebras h/Q and hl/Ql . From their result, the endomorphism End (A/F̄ ) together with its action on the Lie algebra Lie(A/F̄ ) determines the Lie algebras h/Q and hl/Ql uniquely except in the case that End (A/F̄ ) = Q. When End(A/F̄ ) = Q, we have two possibilities for h: either h = sp4 over Q̄ or h = sl2× sl2× sl2 over Q̄. And similarly we have two possibilities for hl: either hl = sp4 over Q̄l or hl = sl2 × sl2 × sl2 over Q̄l. The first case happens when A/F comes from a ’generic’ element in the Siegel moduli space while the second happens when A/F comes from an analytic family of abelian varieties constructed by Mumford in [11]. On the other hand, Deligne proved the inclusion hl ⊆ h⊗QQl. To prove the Mumford-Tate conjecture for the abelian variety A/F with End (A/F̄ ) = Q, it is sufficient to prove the following: Theorem 1. If hl = sl2×sl2×sl2 over Q̄l, then A/F comes from a Shimura curve constructed by Mumford in [11]. This theorem is the main result in this paper. The argument is based on two results: one is to use local Galois representation to determine the Serre-Tate coordinates of an ordinary abelian variety; the other is formal linearity of Shimura variety of Hodge type. Here we give a sketch of the proof. Suppose that the abelian variety A/F satisfies hl = sl2 × sl2 × sl2 over Q̄l. By a theorem of Pink, there exists a set V of finite places of F of density 1 at which the abelian variety A/F has good ordinary reduction. For v ∈ V , let kv be the residue field of F at v with characteristic p = pv and we use Av/kv to denote the reduction of A/F at v. If the abelian variety A/F comes from a Shimura curve Z ↪→ A4,1,n The author is partially supported by Hida’s NSF grant DMS 0753991 and DMS 0854949 through UCLA graduate