Irreducibility of the Igusa Tower over a Shimura Variety by Haruzo Hida, Study notes of Cryptography and System Security

Download Irreducibility of the Igusa Tower over a Shimura Variety by Haruzo Hida and more Study notes Cryptography and System Security in PDF only on Docsity! IRREDUCIBILITY OF THE SIEGEL–IGUSA TOWER HARUZO HIDA We prove the irreducibility of the Igusa tower over a Shimura variety by studying the decomposition group of the prime p inside the automorphism group of the arithmetic automorphic function field. This is the method employed in [PAF] Sections 6.4.3 and 8.4, which uses characteristic 0 results to prove the characteristic p assertion. As explained below (and in more details in [H06]), one can also give a purely characteristic p proof following the same line. There are some other arguments (of purely in characteristic p) to prove the same result (covering different families of reductive groups giving the Shimura variety) as sketched in [C] for the Siegel modular variety. Here is a general axiomatic approach to prove the relative irreducibility of an étale covering π : I → S of an irreducible variety S over an algebraically closed field F. Suppose the following two axioms: (A1) A group G = M × G1 acts on I and S compatibly so that M ⊂ Aut(I/S), G1 ⊂ Aut(S) and G1 acts trivially on π0(I). (A2) M acts on each fiber of I/S transitively; so, M acts transitively on π0(I). Let I◦/F be an irreducible component of I/F. We want to prove I ◦ = π−1(S) = I (relative irreducibility). Then Gal(I◦/S) ⊂M , and if M = Gal(I◦/S), we get I◦ = π−1(S). Let D be the stabilizer of I◦ ∈ π0(I) in G. Pick a point x ∈ I (which can be a generic point), and look at the stabilizer Dx ⊂ G of x. Since gx(x) ∈ I◦ (gx ∈ M) by the transitivity of the action, we have gxDxg −1 x ⊂ D. Then we show that M = G/G1 is generated by {gxDxg−1x |x ∈ I}, which implies M = Gal(I◦/S). In the study of the Igusa tower I/S, F is an algebraic closure of the finite field Fp, and S is the ordinary locus of the reduction modulo p the Shimura variety (of level away from p) which is defined over a valuation ring of mixed characteristic with residue field F. We can take G to be a closed subgroup of the automorphism group of the p-integral Shimura variety (studied by Shimura, Deligne and Milne-Shih in depth). In the case of the Siegel–Shimura variety, we have G = M(Zp)×G1 with G1 = Sp2n(A(p∞)) for the standard Levi-subgroup M of the Siegel parabolic subgroup P of Sp(2n). Then we can take x to be the generic point of I◦/F containing the infinity cusp (so, x = I ◦). The group M(Zp) is isomorphic to GLn(Zp) via GLn(Zp) 3 a 7→ ( a 0 0 ta−1 ) ∈ M(Zp). Ir- reducible components of I are in bijection with p-adic valuations v of the function field of the characteristic 0 Shimura variety (of level Γ1(p ∞)) unramified over v0 extending Date: November 5,2007. A one hour talk given on 7/18/2006 at the conference on “p-Adic Modular Forms and Applications” at CIRM, Luminy (France), July 17–21, 2006. The author is partially supported by the NSF grants: DMS 0244401 and DMS 0456252. 1 IRREDUCIBILITY OF THE SIEGEL–IGUSA TOWER 2 the valuation v0 given by the q-expansion coefficients at the infinity cusp of automorphic functions of level away from p (Lemma 2.2). This observation of the one-to-one corre- spondence between such valuations and irreducible components was first made by Deuring when he factored the modular equation of X0(p m) over P1(J) into irreducibles (and he could have proven the irreducibility of the elliptic Igusa tower by his argument before Igusa if he had the tower; of course, the special case for X0(p) of his result is the congru- ence relation of Kronecker). We show that the stabilizer D of a valuation v∞ unramified over v0 contains M(Z(p)) for Z(p) = Zp ∩Q (Lemma 1.1) and that G1 = Sp2n(A(p∞)) fixes π0(I) (Lemma 2.3), and therefore, Dx ⊃M(Z(p))Sp2n(A(p∞)), which implies Dx = G and hence M(Zp) = Gal(I◦/S). This choice of the generic point works well for Shimura varieties of ResF/QSp(2n) and ResF/QSU(n, n) for totally real fields F (as was done in [H05] Section 10). If the Shimura variety of PEL type comes from a simply-connected inner form of a symplectic or unitary G1 over a totally real field, we can have at least two choices of the points x ∈ I: (cm) all CM points x ∈ I(F), whose stabilizer covers all maximal tori Tx of G anisotropic at ∞ (this choice is taken in [H06]); (gn) Taking the Serre–Tate coordinates T = (T1, . . . , Td) around x and take the valu- ation vx( ∑ α c(α, f)T α) = Infα ordp(c(α, f)). Then the decomposition group D of vx contains Tx (for all x ∈ I◦), and D is the stabilizer of the generic point of I◦ containing x (this choice is taken in [PAF] 8.4.4). Again we can prove D ⊃ G1(A(p∞)). If we make the choice (cm), by p-adic approximation, {Tx(Z(p))}x∈I◦(F) and G1(A(p∞)) topologically generate M × G1(A(p∞)) and we get M = Aut(I◦/S). If we make choice (gn), again D contains {Tx(Z(p))}x∈I◦(F), and the desired result follows by the same argument. Our axiomatic argument covers all type A and type C classical groups (as detailed in [PAF] 8.4, [H05] Section 10 and [H06]). 1. Siegel modular function fields Fix a prime p and an algebraic closure F of Fp. We fix a strict henselizationW ⊂ Q of Z(p) with quotient field K and residue field F. For any integer N prime to p, we regard Z[µN ] as sitting inside W. We have a continuous embedding ip : W ↪→ Qp. We fix a complex embedding i∞ :W ↪→ C. Put G = GLn(Zp)×Sp2n(A(p∞)) (with GLn identified with the standard Levi subgroup M of the Siegel parabolic subgroup of Sp(2n)). We consider the Mumford moduli M(N)/Z[ 1 N ] for an integer N prime to p which classifies triples (X,λ, φN )/A, where X is an abelian scheme over A of relative dimension n, λ : X → tX is a principal polarization, φN : (Z/NZ)2n ∼= X[N ] with 〈φN (x), φN(y)〉λ = ζ txJy N for J = ( 0 −1n 1n 0 ) and a fixed primitive root of unity ζN ∈ µN (by the pairing 〈·, ·〉λ : X[N ] ×X[N ] → µN induced by the polarization). If we consider level pm-structure φp of type Γ = Γ?(p m) (? = 0, 1) given as follows: φp is a subgroup isomorphic to µ n pm étale locally if Γ = Γ0(p m) and φp : µ n pm ↪→ X[pm] if Γ = Γ1(pm), we can think of the moduli space M(N,Γ)/B for the base ring B = F and B = Q which classifies quadruples (X,λ, φN , φp)/A over B-algebras A. The quasi-projective schemes M(N) (resp. M(N,Γ)) can be regarded as a scheme over Spec(Z[ 1 N , µN ]) (resp. over Spec(Q[µN ])) by the pairing 〈·, ·〉λ. Note that Z[ 1N , µN ] ⊂ W if p - N . IRREDUCIBILITY OF THE SIEGEL–IGUSA TOWER 5 Indeed, writing det : n︷ ︸︸ ︷ XV̂ [p m]et ×V̂ · · · ×V̂ XV̂ [p m]et → ∧n Z/pmZ XV̂ [p m]et for the determinant map, ÎV,m is identified with det −1( ∧n Z/pmZ XV̂ [pm]et − ∧n Z/pm−1Z XV̂ [pm−1]et). Lemma 2.1. The formal scheme ÎV,m/Spf(V̂ ) is an étale finite covering with a natural GLn(Zp)-action, and ÎV,m is a GLn(Z/pmZ)-torsor over Spf(V̂ ). We may regard the moduli scheme M(N,Γ)/F as a scheme over M(N)[ 1 H ] (forgetting the level p-structure). Since ÎV,m is étale faithfully flat over Spf(V̂ ), it is affine, and we may write ÎV,m = Spec(V̂m). Then V̂m is a semi-local normal V̂ -algebra étale faithfully flat over V̂ and hence is a product of finitely many valuation rings unramified over V̂ . We put MΓ/F = lim←−p-N M(N,Γ)/F. Then MΓ1(pm)/F = IsomS(µ n pm, X[pm]◦) =: Im gives rise to the Igusa tower I∞  · · · Im  · · · I1  S over S. By the definition of the action of (a, g) ∈ G: (X,λ, η, φp) 7→ (X,λ, η ◦ g, φp ◦ a), G := M(Zp)× Sp2n(A(p∞)) acts on IV,m (m = 1, 2, . . . ,∞), Spec(V ) (by Lemma 1.1 (2)), Spf(V̂ ), FΓ, Im, MΓ/F and MΓ/K. Thus we can make the étale quotient ÎΓ0(pm) := ÎV,m/GLn(Z/pmZ). Again we have ÎΓ0(pm) = Spf(V̂Γ0(pm)), and by Lemma 2.1, V̂Γ0(pm) is a valuation ring unramified over V̂ sharing the same residue field. Thus V̂Γ0(pm) = V̂ . Indeed, there is a unique connected subgroup of X (isomorphic to µnpm étale locally) if (X,λ, φN )/A gives rise to an A-point of M(N,Γ0(p m))/F. Thus M(N,Γ0(p m))/F = S/F. This shows that the residue field of V̂Γ0(pm) is the function field of S and that the quotient field of VΓ0(pm) = V̂Γ0(pm) ∩ FΓ0(pm) is FΓ0(pm). This shows Lemma 2.2. We have the following one-to-one onto correspondences:{ v : FΓ1(pm) → Z ∣∣v|FΓ0(pm) = vΓ0(pm) } ↔ Max(V̂m)↔ π0(Im), where v is a p-adic valuation of FΓ1(pm) unramified (of degree 1) over v0 and Max(V̂m) is the set of maximal ideals of V̂m. Proof. By the above argument, V̂m⊗W F = ∏ I◦m∈π0(Im) F(I◦m); so, π0(Im) injects into the set of valuations in the lemma. Since GLn(Z/pmZ) = Gal(FΓ1(pm)/FΓ0(pm)) acts transitively on π0(Im) and the set of valuations (by Hilbert’s theory of extension of valuations), the injection is onto.  Lemma 2.3. The action of G1 := Sp2n(A(p∞)) fixes vm = vΓ1(pm) and each element of π0(Im). Proof. Since FΓ1(pm)/FΓ0(pm) is a finite Galois extension, the set of extensions of vΓ0(pm) to FΓ1(pm) is a finite set, and by the above lemma, it is in bijection with π0(Im). Thus the action of Sp2n(A(p∞)) on π0(Im) gives a finite permutation representation of Sp2n(A(p∞)). Since Sp2n(k) of any field k of characteristic 0 does not have nontrivial finite quotient group IRREDUCIBILITY OF THE SIEGEL–IGUSA TOWER 6 (because it is generated by divisible unipotent subgroups), the action of Sp2n(A(p∞)) fixes every irreducible component of π0(Im).  3. Proof of irreducibility of Im Let v∞ = vΓ1(p∞), and define Dv = { x ∈ (M(Zp)× Sp2n(A(p∞)))/{±1} ∣∣v∞ ◦ τ (x) = v∞ } . Since M(Z(p)) and Sp2n(A(p∞)) fixes v∞ (Lemmas 1.1 and 2.3) and M(Z(p))Sp2n(A(p∞)) is dense in (M(Zp)× Sp2n(A(p∞))), we have Theorem 3.1. We have Dv = M(Zp)× Sp2n(A(p∞))/{±1}. Let K(p) be an open compact subgroup of Sp2n(A(p∞)) and K = K(p)×GSp2n(Zp). Put MK = M (p)/K(p) (which is the level K Siegel modular variety). Let IK = I∞/K (p), which is the Igusa tower over MK. Since I∞ is irreducible by Gal(I ◦/S) = GLn(Zp) = M(Zp) (the above theorem), IK is irreducible. Thus we have reproved Corollary 3.2 (Chai-Faltings). The Siegel-Igusa tower IK over MK/F is irreducible for any open compact subgroup K of Sp2n(A(∞)) maximal at p. References [C] C.-L. Chai, Methods for p-adic monodromy, preprint 2006, (downloadable at http://www.math.upenn.edu/~chai) [DAV] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Springer, New York, 1990 [PAF] H. Hida, p-Adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathe- matics, 2004, Springer [H05] H. Hida, p-Adic automorphic forms on reductive groups, Astérisque 298 (2005), 147–254 [H06] H. Hida, Irreducibility of the Igusa tower, preprint, 2006 Department of Mathematics, UCLA, Los Angeles, CA 90095-1555 E-mail address: hida@math.ucla.edu