Ching-Li Chai 1. Let p be a prime number, fixed throughout this note. Our central question is:

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1 1. Introduction Methods or p-adic monodromy Ching-Li Chai 1 Let p be a prime number, ixed throughout this note. Our central question is: How to show that the p-adic monodromy o a modular amily o abelian varieties is large? Some clariication is called or: A modular amily o abelian varieties will be interpreted as a subvariety Z o a modular variety M o PEL type over an algebraically closed ield k F p deined by ixing some invariant or geometric ibers o the Barsotti-Tate group A[p ] M with prescribed symmetries attached to the universal abelian scheme A M. Example o such invariants include the p-rank, Newton polygon, or the isomorphism type o the Barsotti-Tate group with prescribed symmetries. Typically such a subvariety Z is stable under all prime-to-p Hecke correspondences on M; moreover any two geometric ibers o A[p ] Z are isogenous via a quasi-isogeny which preserves the prescribed endomorphisms and polarizations. The étale shea o such quasi-isogenies gives rise to a homomorphism ρ p = ρ p,z rom the undamental group o Z to the group o Q p -points o a linear algebraic group G over Q p, deined up to conjugation. Oten the target o the p-adic monodromy homomorphism ρ p is an open subgroup o G(Q p ). We abuse notation and denote this target group by G(Z p ). The expectation is that the p-adic monodromy is large, or maximal. In other words, the image o ρ is expected to be equal to the target group G(Z p ), or equal to an open subgroup o G(Z p ) i we are less ambitious. The irst example o maximality o p-adic monodromy is a theorem o Igusa in [18], when Z is the open dense subset o the modular curve corresponding to ordinary elliptic curves. See 2.4 (i) or the statement o Igusa s theorem, and 2.3 (ii), or comments on generalizations o Igusa s method based on local p-adic monodromy. For generalizations to higher-dimensional modular varieties, the best-known examples in chronological order are the ordinary locus o a Hilbert modular variety and the ordinary locus o a Siegel modular variety. 1 Partially supported by grant DMS rom the National Science Foundation 1

2 In the second example, Z is the open dense subset o a Hilbert modular variety M F attached to a totally real number ield F, which classiies ordinary abelian varieties with endomorphisms by O F. The target o the p-adic monodromy homomorphism ρ p,f is the group (O F Z Z p ) o local units. Ribet showed that ρ p,f is surjective; see [24] and [9]. Ribet s method in [24] and [9] is global and arithmetic in nature; it uses Frobenii attached to points over inite ields o the moduli space M F. In the third example, Z is the ordinary locus o a Siegel modular variety A g,n, and the p-adic monodromy is equal to GL g (Z p ). See 2.2 or the precise statement, and [11] and [12] or proos. The proos o Thm. 2.2 in [11] and [12] are based on considerations o local p adic monodromy; see Rem In this article we explain three methods or proving the maximality o p-adic monodromy. Instead o pushing or the most general case with each method, we choose to illustrate the methods or the third example above, when Z is the ordinary locus in the Siegel modular variety A g,n. In other word, we oer three proos 2 o Thm Each method can be applied to more general situations, such as a lea or a Newton polygon stratum in a modular variety o PEL-type. See 6 or the case o a lea in a Siegel modular or the ordinary locus o a modular variety o quasi-split PEL-type U(n, n). See also [7, 5] or more inormation about the irreducibility o non-supersingular leaves and the maximality o their p-adic monodromy in the case o Siegel modular varieties. The irst o the three proos o 2.2 generalizes o an argument o Ribet in [24] and [9] to the situation when the target o the p-monodromy representation is noncommutative. The other two proos ollow a common thread o ideas, in that they both use Hecke correspondences with a given hypersymmetric point (in the sense o [5]) as a ixed point; these Hecke correspondences orm the local stabilizer subgroup H x0 o the given hypersymmetric point. In the second proo one applies the local stabilizer subgroup H x0 o a hypersymmetric point x 0 to a modular subvariety B x 0 with known p-adic monodromy to produce many subvarieties o Z with known p-adic monodromy. In the third proo one examines the action o the local stabilizer subgroup o a hypersymmetric point x 0 on a tower o inite étale covers o Z which deines the p-adic monodromy representation ρ Z ; the result is that the image o the local stabilizer subgroup H x0 is contained in the image o the p-adic monodromy. The second proo was sketched in [6]. The third proo was inspired by Hida s work on p-adic monodromy in [14], 3 and drew on an argument in [4]. The three proos are explained in 3, 4, 5 respectively. Sketches o the ideas o the proos can be ound at the beginning o these sections. In 6 we indicate how the methods in this article can be applied to show the maximality o p-adic monodromy groups in various situations, including the ordinary locus o the modular variety o quasi-split PEL-type U(n, n) and leaves in Siegel modular varieties. I we compare the above three methods or proving maximality o p-adic monodromy, the second and the third methods using Hecke correspondences have the aura o a pure- 2 In the second proo we assume that p > 2. 3 See also [16]. 2

3 thought proo ; this is especially true or the third method. Since the last two methods depend on the existence o hypersymmetric points in Z, there are situations when they do not apply, while Ribet s method is not burdened by such restrictions; see Rem Together these three methods support the contention that p-adic monodromy o a modular amily o abelian varieties is easy. 4 It is a pleasure to thank Haruzo Hida or the communications explaining his ideas on p-adic monodromy, one o which led to the third proo. Indeed the third proo, which happens to be a sibling o the second proo, is a characteristic-p rendering o his vision. Thanks are due to the reeree, or a very careul reading, many suggestions and reerences. 5 The ideas o the irst and the second proo occurred while the author was working on the Hecke orbit problem, and I would like to express my gratitude to Frans Oort and Chia-Fu Yu or their support, inspiration, and encouragement. I would like to thank M. Harris and R. Langlands or prompting me to understand Hida s argument, and to D. Lee and E. Urban or listening to the second proo. The irst and the second proo were presented during the Spring School on Abelian Varieties held in Utrecht and Amsterdam in May, 2006; the second and the third proo were presented in the Workshop on Automorphic Galois Representations, L-unctions and Arithmetic held at Columbia University in June, I thank the organizers o these conerences, as well as Utrecht University, University o Amsterdam, and Columbia University or their hospitality. 2. Notation (2.1) Let p be a prime number. Let A (p) = l p Q l be the ring o inite prime-to-p adeles. Denote by Z (p) the subring o Q consisting o all rational numbers whose denominator is prime to p. Let n 3 be a positive integer, (n, p) = 1. Let A ord g,n be the moduli space over F p o g-dimensional ordinary principally polarized abelian varieties in characteristic p with symplectic level-n structure. Let A[p ] et A ord g,n be the maximal étale quotient o the Barsotti-Tate group A[p ] A ord g,n attached to the universal abelian scheme A A ord g,n; it is an étale Barsotti-Tate group o height g over A ord g,n. Let x 0 = [(A 0, λ 0, η 0 )] be an F p -point o A ord g,n, where λ 0 is a principal polarization o an ordinary abelian variety A 0 over F p, and η 0 is a level-n structure on A 0. Let T 0 = T p (A 0 [p ] et ) be the p-adic Tate module attached to the maximal étale quotient A 0 [p ] et o the Barsotti-Tate group A 0 [p ] attached to A 0. Notice that T 0 = Z g p non-canonically. 4 In contrast, the important question on the semisimplicity o p-adic monodromy o an arbitrary amily o abelian varieties in characteristic p seems inaccessible at present. 5 Including the reerence to [25] and that the irreducibility o the Igusa tower over a Shimura variety o type U(n, 1) was proved by P. Boyer. 3

4 Let ρ p = ρ p,a ord g,n : π 1(A ord g,n, x 0 ) GL(T 0 ) = GL g (Z p ) be the p-adic monodromy representation deined by the base point x 0 o A ord g,n. I we view π 1 (A ord g,n, x 0 ) as a quotient o the Galois group G Ag,n o the unction ield o A ord g,n, then ρ p corresponds to the limit o the natural actions o G Ag,n on the generic iber o the inite étale group schemes A[p m ] et A ord g,n, m N. See 2.7 or another deinition o p-adic monodromy. (2.2) Theorem The image o the p-adic monodromy homomorphism ρ p,a ord g,n GL(T 0 ) = GL g (Z p ). is equal to (2.3) Remark The special case o 2.2 when g = 1 is a classical theorem o Igusa in [18]. See [19], Thm. 4.3 on page 149 or an exposition o Igusa s theorem. (2.4) Remark (i) Igusa s proo uses the local monodromy at a point o A 1,n which corresponds to a supersingular elliptic curve. In this case the image o the local p-adic monodromy is already equal to the target group Z p o the p-adic monodromy representation. (ii) Ekedahl s proo in [11] shows that the p-adic local monodromy at a superspecial point o A g,n (i.e. a point which corresponds to the product o g copies o a supersingular elliptic curve) is already equal to GL g (Z p ). This is an exact generalization o Igusa s proo. In [11] Ekedahl used curves o genus two instead o abelian varieties, but it is clear that one can also use deormation theory o abelian varieties. See [21] or the case o Picard modular varieties, and [1] or deormations o p-divisible groups with large local p-adic monodromy. There is one disadvantage o the Igusa-Ekedahl method: substantial eort is required when one uses this method to compute the p-adic monodromy o a subvariety Z deined by p-adic properties o the universal Barsotti-Tate groups. The work is in constructing explicit local coordinates o this subvariety Z at a basic point z and the computation o the Galois group o suitable inite extensions o the unction ield o the ormal completion o Z at z. (iii) The proo in [12] is also based on local monodromy. It uses the arithmetic compactiication theory to show that the p-adic local monodromy at a zero-dimensional cusp o the minimal compactiication o A g,n is equal to SL g (Z p ). This method applies to ewer situations or two reasons. First, the modular variety M may be proper, i.e. the boundary o M may be empty. Even when the modular variety has a boundary, the Zariski closure o the modularly deined subvariety Z may not intersect the boundary o M. Secondly, when the boundary o M is not empty, the local monodromy at the boundary may be still be too small. For instance in the case o a Hilbert modular variety M F,n over F p with (n, p) = 1, the local monodromy at a cusp in the minimal compactiication o M F,n is the p-adic completion o the subgroup o units in O F which are 1 (mod n).6 Notice that the image o the local monodromy at a cusp is contained in the subgroup Ker ( ) N F/Q : (F Q p ) Q p o the target group (F Q p ) o ρ p ; this ([F : Q] 1)-dimensional subgroup is the target 6 This is what the argument in [2] shows. The statement in [2] about the p-adic monodromy o a Hilbert modular variety is wrong. 4

5 group o the local p-adic monodromy. A priori, the dimension o the local p-adic monodromy group at the cusps o a Hilbert modular variety M F,n may be smaller than [F : Q] 1, the dimension o its target; it is equal to [F : Q] 1 i and only i the Leopoldt conjecture holds or F! (2.5) In the rest o this article, we will take the base point x 0 = [(A 0, λ 0, η 0 )] o A ord g,n to be o the orm A 0 = E 1 Spec(Fp) Spec(F p) E 1, where E 1 is an ordinary elliptic curve over F p, and λ 0 is the g-old product o the principal polarization o E 1. Let E 1 [p ] mult (resp. E 1 [p ] et ) be the multiplicative part o E 1 [p ] (resp. the maximal étale quotient o E 1 [p ]). Let T := T p (E 1 [p ] et ), which is a ree Z p -module o rank one. Then T 0 := T p (A 0 [p ] et ) is the direct sum o g copies o T, and GL Zp (T 0 ) is canonically isomorphic to GL g (End(T)) = GL g (Z p ). Let O = End(E 1 ) and K = End(E 1 ) Z Q. So End(A 0 ) (resp. End(A 0 ) Z Q) is canonically isomorphic to M g (O) (resp. M g (K)). It is well-known that K is an imaginary quadratic ield, and O is an order o K such that O Z Z p = Zp Z p. The two actors o O Z Z p correspond to the action o O on E 1 [p ] mult and E 1 [p ] et respectively. Via the above isomorphisms, the Rosati involution attached to the principal polarization o λ 0 corresponds to the involution C t C on Mg (O) and M g (K). On M g (O Z Z p ) = M g (Z p ) M g (Z p ), the Rosati involution is (C 1, C 2 ) ( t C 2, t C 1 ), (C 1, C 2 ) M g (Z p ) M g (Z p ). Denote by H the unitary group attached to the semisimple Q-algebra End(A 0 ) Z Q = M g (K) and the involution. The reductive linear algebraic group H over Q is characterized by the property that H(R) = { x (End(A 0 ) Z R) x (x) = (x) x = 1 }. Under the natural isomorphism End(A 0 ) Z Q = M g (K) and the isomorphism End(A 0 ) Z Q p = Mg (Q p ) M g (Q p ) induced by K Q Q p Q p Q p, H(Q) is identiied with the set o all matrices C M g (K) such that C (C) = (C) C = Id g, and H(Q p ) is identiied with the subset o all pairs (C 1, C 2 ) M g (Q p ) M g (Q p ) such that C 1 tc 2 = Id g = C 2 tc 1. The second projection rom M g (O Z Z p ) to M g (Z p ) induces an isomorphism pr : H(Q p ) GL g (Q p ). We abuse notation and write H(Z p ) or the compact open subgroup pr 1 (GL g (Z p )); it is the set o all elements β H(Q p ) such that β induces an automorphism o A 0 [p ]. Denote by H(Z (p) ) the subgroup H(Q) H(Z p ) o H(Q). The image o H(Z (p) ) in H(Z p ) is a dense subgroup o H(Z p ) or the p-adic topology. (2.6) Two eatures o the base point x 0 deserve attention. (1) The abelian variety A 0 is hypersymmetric in the sense that End(A 0 ) Z Q p End(A 0 [p ]) Zp Q p. See [5] or more discussion on the notion o hypersymmetric abelian varieties. 5

6 (2) The order End(A 0 ) o the semisimple Q-algebra End(A 0 ) Z Q is maximal at p. In view o (1) above, this means that End(A 0 [p ]) is a maximal order o the semisimple Q p -algebra End(A 0 [p ]) Zp Q p. (2.7) Remarks on the deinition o p-adic monodromy. Let (B, µ) S be a principally polarized abelian scheme over an irreducible normal F p - scheme S such that all ibers o B are ordinary abelian varieties. Let s 0 be a geometric point o S. When B is ordinary, one can deine the p-adic monodromy o B S to be the homomorphism π 1 (S, s 0 ) GL Zp (T p (B s0 [p ] et )) attached to the smooth Z p -shea T p (B[p ] et ) over S, where B s0 is the iber o B S at s 0. There is an equivalent deinition which works or more general situations, such as when (B, µ)[p ] S is iberwise geometrically constant. Let G := Aut ((B s0, µ s0 )[p ]). Consider the shea I := Isom S per ((B s0, µ s0 )[p ], (B, µ)[p ]), which is a right G-torsor over the perection S per o S. Notice that B[p ] splits uniquely over S per as the direct product o its maximal toric part B[p toric] and its maximal etale quotient B[p et ]. Then the p-adic monodromy ρ B/S : π 1 (S, s 0 ) G is the homomorphism rom π 1 (S per, s 0 ) = π 1 (S, s 0 ) to the proinite p-adic Lie group G attached to the G-torsor I. We indicate why the two deinitions are equivalent. Let I be the subshea o the étale shea Isom S (B s0 [p ] toric, B[p ] toric ) S Isom S (B s0 [p ] et, B[p ] et ) which are compatible with the polarizations µ s0 [p ] and µ[p ]. Let I et := Isom S (B s0 [p ] et, B[p ] et ). The subgroup G consisting o all elements o Aut(B s0 [p ] toric ) Aut(B s0 [p ] et ) which are compatible with the polarizations is naturally identiied with G. The projection G GL(T p (B s0 [p ] et )) is an isomorphism because the principal polarization µ s0 identiies B s0 [p ] toric as the dual o B s0 [p ] et. Similarly, the natural projection map I I et is an isomorphism. Moreover the homomorphisms ρ I : π 1 (S, s 0 ) G = G and ρ I et : π 1 (S, s 0 ) Aut(B s0 [p ] et ) = G attached to I and I et are equal. Notice that the homomorphism ρ I : π 1 (S, s 0 ) G GL(T p (B s0 [p ] et )) is nothing but the p-adic monodromy representation coming rom the action o the Galois group o the unction ield κ(s) o S on the p-power torsion points B[p ](κ(s) alg ). On the other hand, because the étale topology is insensitive to nilpotent extensions, the homomorphisms ρ I : π 1 (S, s 0 ) G and ρ I : π 1 (S, s 0 ) G are equal. So the two deinitions are equivalent. 3. Ribet s method revisited (3.1) Sketch o idea. The goal o this section is to prove Thm. 2.2 using Ribet s method in [24], [9]. Since the target o the p-adic monodromy homomorphism or the ordinary locus o Hilbert modular varieties is commutative, it was oten thought that Ribet s method would have diiculty producing inormation beyond the abelianized p-adic monodromy. This is not the case at 6

7 all. Indeed Ribet s method can be used to compute the p-adic monodromy o leaves in a Hilbert modular variety M F,n or instance, where the target o the p-adic monodromy homomorphism may be non-commutative; see [7, Thm. 4.5]. The proo o 2.2 in this section consists o a ew lemmas in group theory, ollowed by the body o the proo in 3.7. For instance Lemma 3.4 says that a suitable congruence condition modulo p 2 on the characteristic polynomial o a semisimple conjugacy class in GL 2 (Z p ) ensures that the reduction modulo p o such a semisimple conjugacy class is a non-trivial unipotent conjugacy class in GL 2 (F p ). According to these lemmas, we only need to show that the image o ρ contains certain congruent classes modulo p 2, in the case g = 2. The basic idea o the proo is as ollows. Suppose we have two g-dimensional ordinary principally polarized abelian varieties A 1, A 2 over a common inite ield F q, then the dierence Fr A1 Fr 1 A 2 gives an element o the geometric undamental group π 1 (A ord g,n, x 0 ), and its image under the p-adic monodromy homomorphism ρ p gives a conjugacy class o GL g (Z p ). See Rem. 3.8 or urther discussion about Frobenius at closed points. O course there is ambiguity when one tries to orm the dierence between two conjugacy classes, but i we restrict ourselves to the case when the action o Fr A2 modulo p N lies in the center o GL g (Z/p N Z), then the dierence gives a well-deined conjugacy class in GL g (Z p ) modulo p N. There is a technical dierence with [24] and [9]: we use Honda-Tate ([17], [26], [27]; see also [28]) to produce abelian varieties over inite ields with real multiplications, instead o using [8], which depends on Honda-Tate. One reason or this choice is to emphasize that Ribet s method applies to more general situations where the abelian varieties involved may not be ordinary. (3.2) Reduction to the case g = 2. Notation as in 2.1. Recall that the target o the p-adic monodromy ρ is GL(T p (A 0 [p ])) = GL g (End(T)) = GL g (Z p ). The group GL g (Z p ) contains many copies o standardly embedded GL 2 (Z p ) in block orm. For any 1 i 0 < j 0 g, the associated standardly embedded GL 2 (Z p ) consists o all elements (a ij ) GL g (Z p ) such a ii = 1 i i i 0, j 0, and a ij = 0 i i j and (i, j) (i 0, j 0 ), (j 0, i 0 ); denote by H J this standardly embedded subgroup o GL g (Z p ). In Lie theory these standardly embedded SL 2 are called root subgroups. It is easy to see that the g 1 standardly embedded subgroups H {1,2},..., H {g 1,g} o GL g (Z p ) generate GL g (Z p ) or any g 2. A standardly embedded GL 2 (Z p ) inside GL g (Z p ) can be realized in terms o geometry o moduli spaces as ollows. I we ix a level-n structure or the elliptic curve E 1 over F p, then or each subset J {1, 2,..., g} with two elements o the orm J = {j, j + 1} where j is an integer with 1 j g 1, we have an embedding i J : A 2,n A g,n such that the image o π 1 (A ord 2,n, s 2 ) under ρ p : π 1 (A ord g,n, x 0 ) GL g (Z p ) is contained in the subgroup o J-blocks o GL g (Z p ) which is isomorphic to GL 2 (Z p ). Here s 2 denotes the F p -point o A ord 2,n which corresponds to the abelian surace E 1 E Spec(Fp) 1 with the product principal polarization µ 1 µ 1 and product level-n structure η 1 η 1. This embedding i J is deined by the amily (E 1, µ 1, η 1 ) j 1 (A, λ, η) (E 1, µ 1, η 1 ) g j 1 over A 2,n, where (A, λ, η) A 2,n denotes the universal principally polarized abelian scheme with level-n 7

8 structure over A 2,n. It is clear that the image o the composition π 1 (A ord 2,n, s 2 ) π 1 (A g,n, x 0 ) ρ GL g (Z p ) is contained in the J-block subgroup GL g (Z p ), and this image subgroup is naturally isomorphic to the image o the p-adic monodromy representation π 1 (A ord 2,n, s 2 ) GL 2 (Z p ). Thereore it suices to prove Thm. 2.2 in the case when g = 2. This reduction step to the case g = 2 using the standardly embedded copies o A ord 2,n in A ord g,n already appeared in [11]. Remark Although not needed in this article, we note that it is possible to deine the standard embedding i J : A 2,n A g,n or any subset J {1,..., g} with two elements, as ollows. Suppose that J = {i 0, j 0 }, 1 i 0 < j 0 g. Consider the principally polarized abelian scheme (B, λ B, η B ) := (A, λ, η) (E 1, µ 1, η 1 ) g 2 over A 2,n. Change the level-n structure η B : B[n] (Z/nZ) 2n = (Z/nZ) n (Z/nZ) n to τ {1,2},{i0,j 0 }, where τ {1,2},{i,j} is the symplectic automorphism o (Z/nZ) n (Z/nZ) n induced by the permutation σ {1,2},{i0,j 0 } o {1,..., n} which sends 1 to i 0, 2 to j 0, and σ {1,2} (i) < σ {1,2} (j) i 3 i < j g. The resulting triple (B, λ B, τ {1,2},{i0,j 0 } η B ) deines the standard embedding i J. (3.3) Lemma Denote by the subgroup o GL 2 (Z p ), isomorphic to Z p Z p, consisting o all diagonal matrices in GL 2 (Z p ). Let π 0 : GL 2 (Z p ) GL 2 (F p ) be the natural surjection given by reduction modulo p. Let H be a closed subgroup o GL 2 (Z p ) which contains the subgroup such that π 0 (H) = GL 2 (F p ). Then H = GL 2 (Z p ). Proo. Denote by D the subgroup o M 2 (F p ) consisting o all diagonal 2 2 matrices with entries in F p. Denote by U m the subgroup 1+p m M 2 (Z p ) o GL 2 (Z p ) or every positive integer m. A simple calculation show that M 2 (F p ) is generated by Ad(GL 2 (F p )) D, the set o all GL 2 (F p )-conjugates o D. The last statement implies that the composition H (1 + p m M 2 (Z p )) U m U m /U m+1 is surjective or every m > 0. Hence H surjects to GL 2 (Z/p m Z) or every m > 0, and Lemma 3.3 ollows. (3.4) Lemma Let A be an element o GL 2 (Z p ) such that tr(a) 2 (mod p 2 ), det(a) 1 (mod ( p) ) and det(a) 1 (mod p 2 ). Then the image A o A in GL 2 (F p ) is conjugate to 1 1. In other words, A is a non-trivial unipotent element in GL (F p ). Proo. Since ( the ) characteristic polynomial ( o A ) is equal to T 2 2T + 1 F p [T ], A is either equal to or is conjugate to. In particular, ater conjugating A by a suitable element o GL 2 (Z p ), we may and do assume that ( ) 1 + pa b A = c 1 + pd 8

9 or suitable elements a, b, c, d Z p. The assumptions on A imply that a+d 0 (mod p) and bc pz p. The act that bc pz p implies that A is either an upper-triangular non-trivial unipotent matrix or a lower-triangular non-trivial unipotent matrix. We choose and ix a generator b o the cyclic group F p 2. Denote by (T ) the characteristic polynomial o b over F p. (3.5) Lemma Let be the subgroup o all diagonal matrices in GL 2 (F p ). Let H be a subgroup o GL 2 (F p ) containing. Assume that H contains a non-trivial unipotent element ū o GL 2 (F p ), and also an element v GL 2 (F p ) such that ( v) = 0 in M2 (F p ). Then H = GL 2 (F p ). Proo. The assumptions on H implies that it intersects non-trivially with every conjugacy class o GL 2 (F p ). It is a standard exercise in group theory that i a subgroup S o a inite group G intersects non-trivially with every conjugacy class in G, then S = G. (3.6) Remark (i) The point o 3.4 is that, by imposing congruence conditions on the characteristic polynomial o a semisimple element o GL 2 (Q p ) which belongs to GL 2 (Z p ), one can make sure that the reduction o this element is a nontrivial unipotent element o GL 2 (F p ). O course this is a general phenomenon in the context o reductive group over local ields and not restricted to GL 2. (ii) The author claims no novelty whatsoever about Lemmas 3.3, 3.4 and 3.5. As pointed out by the reeree, similar statements already appeared in Lemma 5 and Lemma 1 o [25]. (3.7) First Proo o Thm As explained in 3.2, it suices to prove the case when g = 2. Choose an identiication o GL(T 0 ) with GL 2 (Z p ). Moreover, Igusa s theorem implies that the image o the p-adic monodromy homomorphism ρ p contains a conjugate o the subgroup o diagonal matrices in GL 2 (Z p ): Consider the standard embedding i : A ord 1,n Spec(Fp) Aord 1,n A ord 2,n. The image o the restriction o ρ p to π 1 (A ord 1,n Spec(Fp) Aord 1,n, x 0 ) is (a conjugate o) by Igusa and the unctoriality o π 1. Let H be the image o ρ p, and let H be the image o H in GL 2 (F p ). By Lemmas 3.3 and 3.5, it suices to show that H contains a non-trivial unipotent element ū GL 2 (F p ) and also an element v such that ( v) = 0 in M 2 (F p ). Choose a quadratic polynomial (T ) Z[T ] such that (T ) splits over R and (T ) (T ) (mod p). Choose a quadratic polynomial g(t ) Z[T ] such that g(t ) splits over R and that g(t ) T 2 2T p (mod p 2 ). Let F = Q[T ]/((T )) and let F g = Q[T ]/(g(t )); they are the real quadratic ields deined by (T ) and g(t ) respectively. Let a be the image o T in F and let a g be the image o T in F g. Choose a positive integer n 1 such that (n 1, np) = 1 and every element o the strict ideal class group o F is represented by an ideal o O F which divides n 1 O F. Similarly, choose a positive integer n 2 such that 9

10 (n 2, np) = 1 and every element o the strict ideal class group o F g is represented by an ideal o O Fg which divides n 2 O Fg. Choose a suitable power q = p r o p such that A ord 2,n, M ord F,n, M ord F are all deined over F g,n q. Here M ord F,n (resp. Mord F g,n ) denotes the ordinary locus o the Hilbert modular surace with level-n structure attached to the real quadratic ield F (resp. F g ). The p-adic monodromy representation ρ p attached to the modular variety A ord 2,n over F p extends to a homomorphism ρ arith p rom the arithmetic undamental group π 1 (A ord 2,n /Fq, x) to GL(T 0 ) = GL 2 (Z p ). Analogous statements hold or the Hilbert modular suraces M ord F,n and M ord F g,n. Replacing r by a suitable multiple i necessary, we may and do assume that q 1 (mod n 2 n 2 3), where n 3 is the least common multiple o n 1 and n 2. Apply the argument in [24], [9]: Choose an element b 1 O F such that b 1 a (mod p) and b 1 2 (mod n 2 n 2 1). Also, choose an element b 2 O F such that b 2 1 (mod p) and b 2 2 (mod n 2 n 2 1). For s suiciently large, the quadratic polynomials T 2 b 1 T + q s F [T ] and T 2 b 2 T + q s F [T ] deine q s -Weil numbers π 1, π 2 in totally imaginary quadratic extensions K 1, K 2 o F such that K i Q Q p = (F Q Q p ) (F Q Q p ) or i = 1, 2. Moreover, the image o π i in exactly one o the two actors is b i (mod p). The congruence condition modulo n 2 n 2 1 guarantees that π i 1 nn 1 is integral, i.e. it is an element o O Ki, i = 1, 2. Thereore we obtain abelian varieties A 1, A 2 over F q s with endomorphism by O F. The inite étale group schemes A i [n] over F q s is constant because π i 1 is integral. Changing A n i by a suitable F q s-rational O F -linear isogeny, we may assume that the polarization shea o A i is trivial or i = 1, 2. So we obtain F q s-points o A ord 2,n /Fq. The dierence Fr A1 Fr 1 A 2 o their q s -Frobenius gives an element o the geometric undamental group o A ord 2,n whose image in GL 2 (F p ) is a root o the irreducible polynomial (T ) over F p. Hence H contains an element v such that ( v) = 0 in M 2 (F p ). A similar argument shows that the image o ρ p contains an element o GL 2 (Z p ) whose characteristic polynomial is congruent to g(t ) modulo p 2. So H contains a non-trivial unipotent element by Lemma 3.4. We conclude that H = GL 2 (Z p ) by Lemma 3.5. (3.8) Remark At the end o the second-to-last paragraph o the proo in 3.7, the statement that the dierence o two Frobenius elements at closed points gives an element o the geometric undamental group is a consequence o the ollowing general act: Let G 1, G 2 be Barsotti-Tate ( groups over F q, and consider the natural action o Gal(F p /F q ) on the set I := Isom Fp G1 Spec(Fq) Spec(F p ), G 2 Spec(Fq) Spec(F p ) ). Then the action o the arithmetic Frobenius element φ = φ q on I is given by α Fr G2 /F q α Fr 1 G 1 /F q or all α I. Notice that in the above ormula the partial compositions α Fr G2 /F q α and α Fr 1 G 1 /F q are quasi-isogenies but not elements o I. 10

11 The above assertion can be seen rom the ollowing diagram G 1 Spec(Fq) Spec(F p ) β G 2 Spec(Fq) Spec(F p ), id G1 σ G 1 Spec(Fq) Spec(F p ) α id G2 σ G 2 Spec(Fq) Spec(F p ) Fr G1 id Fp G 1 Spec(Fq) Spec(F p ) Fr G1 id Fp β G 2 Spec(Fq) Spec(F p ) where β and β are deined by the requirement that the diagram commutes. By deinition o the Galois action on I, we have φ α = β. On the other hand, the composition Fr Gi id Fp id Gi σ is equal to the absolute q-frobenius morphism or G i Spec(Fq) Spec(F p ). The commutativity o the diagram implies that β = β. We have proved the assertion. In the context o 3.7, consider the shea I := Isom A ord,per ((A s0, λ s0 )[p ], (A, λ)[p ]) 2,n over the perection A ord,per 2,n o A ord 2,n as in 2.7. Let s i be the closed point corresponding to the principally polarized abelian varieties (A i, λ i ), i = 1, 2. Then we obtain two conjugacy classes Fr 1 A 0 Fr Ai, i = 1, 2, in the image o the arithmetic undamental group π 1 (A ord 2,n), both lying above the arithmetic Frobenius element φ q. So the image o the geometric undamental group π 1 (A ord 2,n Spec(Fp)Spec(F p )) contains the conjugacy class Fr A1 Fr 1 A 2. That this dierence is well-deined modulo p N has been explained in Hecke translation o Shimura subvarieties (4.1) Notation and sketch o idea. We ollow the notation in 2.5. In addition, we assume that p > 2. See Remark 4.5 or the case when p = 2. The idea o our second proo o Thm. 2.2 is as ollows. The unitary group H attached to the semisimple algebra End(A 0 ) Z Q = M g (K) with involution gives rise to Hecke correspondences on A ord g,n with x 0 as a ixed point. The product o g copies o the modular curve A ord 1,n is diagonally embedded in A ord g,n as a subvariety B. The image ρ p (π 1 (B, x 0 )) o the undamental group o B under ρ p is the subgroup D o diagonal matrices in GL g (Z p ), D = (Z p ) g. Take an element γ H(Z (p) ). Such an element γ gives rise to a prime-to-p Hecke correspondence on A ord g,n which has x 0 as a ixed point; the image o B under this Hecke correspondence is a subvariety γ B in A ord g,n such that ρ p (π 1 (γ B, x 0 )) is equal to Ad(γ), the conjugation o D by the image o γ in GL g (Z p ). An exercise in group theory tells us that subgroups o the orm Ad(γ) D generate GL g (Z p ). This proo was sketched in [6]. (4.2) Second proo o Thm. 2.2 when p > 2. Let B be the product o g copies o A 1,n, diagonally embedded in A g,n. Recall that E 1 is an ordinary elliptic curve over F p, A 0 is the product o g copies o E 1, and λ 0 is the product 11

12 principal polarization on A 0. We have O Z Z p = Zp Z p, corresponding to the natural splitting o E 1 [p ] into the product o its multiplicative part E 1 [p ] mult and its maximal étale quotient E 1 [p ] et. So we have an isomorphism End(A 0 ) = M g (O), and a splitting End(A 0 ) Z Z p = Mg (Z p ) M g (Z p ) corresponding to the splitting o A 0 [p ] into the product its multiplicative and étale parts. Denote by pr : (End(A 0 ) Z Z p ) GL(T 0 ) = GL g (Z p ) the projection corresponding to the action o End(A 0 ) Z Z p on the étale actor A 0 [p ] et o A 0 [p ]. The Rosati involution on End(A 0 ) interchanges the two actors o End(A 0 ) Z Z p. Recall that H denotes the unitary group attached to (End(A 0 ) Z Q, ); in particular H(Z p ) is a compact open subgroup o H(Q p ) isomorphic to GL(T 0 ) under the projection map pr. Moreover the image o H(Z (p) ) in H(Z p ) = GL(T 0 ) is dense in H(Z p ). By Igusa s theorem in [18], the p-adic monodromy group o the restriction to B, i.e. ρ (Im (π 1 (B, x 0 ) π 1 (A g,n, x 0 ))), is naturally identiied with the product o g copies o Z p diagonally embedded in GL(T 0 ) = GL g (Z p ). Denote by D this subgroup o GL(T 0 ). (4.3) Lemma The image o the p-adic monodromy homomorphism ρ p : π 1 (A ord g,n, x 0 ) GL(T 0 ) is a closed normal subgroup o GL(T 0 ) which contains the subgroup D = (Z p ) g. Proo o Lemma 4.3. Every element u H(Z (p) ) deines a prime-to-p isogeny rom A 0 to itsel respecting the polarization λ 0. Such an element u H(Z (p) ) gives rise to a prime-to-p Hecke correspondence h on A g,n having x 0 as a ixed point, and an irreducible component B o the image o B under h such that B x 0. By the unctoriality o the undamental group, the image o the undamental group π 1 (B, x 0 ) o B in π 1 (A ord g,n, x 0 ) is mapped under the p-adic monodromy representation ρ to the conjugation o D by the element pr(h) GL(T 0 ). In particular, ρ(π 1 (A ord g,n, x 0 )) is a closed subgroup o GL(T 0 ) which contains all conjugates o D by elements in the image o pr : H(Z (p) ) H(Z p ) = GL(T 0 ). Recall that the image o H(Z (p) ) in H(Z p ) is a dense subgroup. So ρ(π 1 (A ord g,n, x 0 )) is a closed normal subgroup o GL(T 0 ) = GL g (Z p ) which contains the subgroup D o all diagonal elements. (4.4) End o the second Proo. An exercise in group theory shows that the only closed normal subgroup which contains the subgroup o all diagonal matrices in GL g (Z p ) is GL g (Z p ) itsel: Let N be such a normal subgroup. Then N contains all matrices o the orm h u h 1 u 1 = (Ad(h) u) u 1, where h D is a diagonal matrix in GL g (Z p ) and u is an upper triangular unipotent matrix in GL g (Z p ). Since p > 2, not every element o Z p is congruent to 1 modulo p, thereore N contains all upper triangular unipotent matrices in GL g (Z p ). Similarly N contains all lower triangular unipotent matrices in GL g (Z p ). These unipotent matrices and D generate GL g (Z p ). 12

13 (4.5) Remark When p = 2, the smallest closed normal subgroup o GL g (Z 2 ) which contains the group D o all diagonal matrices in GL g (Z 2 ) is the principal congruence subgroup U 1 o GL g (Z 2 ) o level 1, i.e. the subgroup consisting o all matrices in M g (Z 2 ) which are congruent to Id g modulo 2. In other words, the 2-adic monodromy generated by the undamental group o the Hilbert modular variety attached to E = Q Q and its Hecke translates by the stabilizer at x 0 is equal to the principal congruence subgroup U 1 o GL g (Z 2 ). One way to get the ull target group GL g (Z 2 ) is to use Hecke translates o the Hilbert modular variety attached to a totally real number ield F with [F : Q] = g which is not totally split above p. (4.6) Remark In the proo o the key Lemma 4.3 it is important to know the image o the undamental group π 1 (B, x 0 ) under the p-adic monodromy representation ρ p on the nose ; knowing it only up to conjugation is useless. One unction o the chosen base point x 0 is to help identiying the subgroup ρ p (π 1 (B, x 0 )) o GL g (Z p ); this is possible because B passes through x 0. The crucial property o A 0 that End(A 0 ) is large allows us to construct many subvarieties o the orm B. The act that End(A 0 ) is so large that the Z p -points o the unitary group attached to (End(A 0 ), ) is already isomorphic to the target GL g (Z p ) o the homomorphism ρ has the ollowing consequence. Subgroups o the orm ρ p (π 1 (B, x 0 )) generate a normal subgroup o GL g (Z p ). 5. Hecke correspondence and p-adic monodromy (5.1) Notation and sketch o idea. We keep the notation in 2.5. Fix a positive integer m > 0. Consider the inite étale cover π n;m : A g,n;m A ord g,n over F p with Galois group GL g (Z/p m Z), where A g,n;m is the moduli space which classiies ordinary g-dimensional principally polarized ordinary abelian varieties (A, λ) with a level-n structure, plus an isomorphism ψ : (Z/p m Z) g A[p m ] et, and the map π m sends (A, λ, η, ψ) to (A, λ, η). We have the prime-to-p towers Ãg,n(p);m := (A g,nb;m ) b I and Ãord g,n(p) := ( ) A ord g,nb, where the indexing set I consists o all positive integers such b I that (b, p) = 1. Moreover we have a natural morphism π m rom the tower Ãg,n(p);m to the tower à ord g,n(p) which is compatible with π n;m. The group Sp 2g (A (p) ) operates on the two towers, and π m is Sp 2g (A (p) )-equivariant. The Sp 2g(A (p) )-action induces prime-to-p Hecke correspondences on A g,n;m and A ord g,n, and the inite étale morphism π m;n is Hecke equivariant. The statement o Thm. 2.2 is equivalent to the assertion that A g,n;m is irreducible or every m > 0, since A ord g,n is known to be irreducible. It suices to show that any two points y 1, y 2 A g,n;m above the base point x 0 belong to the same irreducible component o A g,n;m. Choose an element h H(Z (p) ) = H(Q) H(Z p ) such that h p,et ψ 1 = ψ 2. Here ψ 1 (resp. ψ 2 ) is the isomorphism (Z/p m Z) g A 0 [p m ] et attached to y 1 (resp. y 2 ), and h p,et is the automorphism o A 0 [p m ] et induced by the element h H(Z (p) ). Such an element h exists because H(Z (p) ) is dense in H(Z p ) or the p-adic topology. Let h (p) l p H(Q l) be the inite prime-to-p component o h. Then y 1 belongs to the 13

14 image o y 2 under the prime-to-p Hecke correspondence given by h (p). Now one can apply (the argument o) the main result o [4] to conclude that y 1 and y 2 lie on the same irreducible component o A g,n;m. This inishes the sketch o the third proo o Thm The actual proo consists o Lemma 5.4 and Prop. 5.5: It is clear that together they imply Thm (5.2) We recall the notion o abelian varieties up to prime-to-p isogenies. Denote by AV k the category o abelian varieties over k such that morphisms are homomorphisms o abelian varieties. Recall that an isogeny α : A B between abelian varieties is said to be primeto-p i Ker(α) is killed by an integer N not divisible by p. Denote by AV (p) k the category o abelian varieties over k up to prime-to-p isogenies, obtained by AV k by ormally inverting all prime-to-p isogenies. The latter category has the same objects but more morphisms: Let A, B be abelian varieties over k. Then Hom (p) AV ([A], [B]) := Hom k (A, B) Z Z (p). Composition o k is deined in the obvious way. In the above [A] (resp. [B]) denotes the image o A (resp. B) under the obvious unctor π : AV k AV (p) k. We will use this notation when we want to consider an abelian variety as an object in AV (p) morphisms in AV (p) k An alternative deinition o morphisms in AV (p) k in AV (p) k k. is as ollows. A morphism rom A to B is a diagram o the orm β 1 α1 1 : A α 1 β 1 A 1 B in AVk, where α 1 is a prime-to-p isogeny and β 1 is a homomorphism rom A 1 to B. Two arrows β 1 α1 1 : A α 1 β 1 A 1 B and β 2 α2 1 : A α 2 β 2 γ 1 A 2 B are equal i and only i there exists prime-to-p isogenies A3 A1, A 3 γ 2 A2 such that β 1 γ 1 = β 2 γ 2. Composition o arrows in AV (p) k is deined, because β 1 γ 1 or every diagram A 1 B B1 in AV k where γ 1 is a prime-to-p isogeny, there exists a β 2 γ 2 diagram A 1 A2 B1 such that β 1 β 2 = γ 1 γ 2. (5.3) Hecke correspondence on A g,n;m We explain the action o the group Sp 2g (A (p) ) on the tower à g,n(p);m ; the action o Sp 2g (A (p) ) on Ãord g,n(p) is similar but simpler. See 5 6 o [20] or urther discussion o prime-to-p Hecke correspondences on the prime-to-p tower o modular varieties o PELtype. It is clear that the projective system Ãg,n(p);m is isomorphic to the projective system à g,(p);m := (A g,n;m ) N I, where the indexing set I consists o all integers N such that N 3 and (N, p) = 1. We will describe geometric points o the tower Ãg,(p);m and the action o Sp 2g (A (p) ) on geometric points o à g,(p);m. Let k F p be an algebraically closed ield. We ix an isomorphism χ : Ẑ(p) Ẑ(p) (1) over F p. Also we ix a standard symplectic pairing, : (Ẑ(p) ) 2g (Ẑ(p) ) 2g Ẑ(p) given by the ormula (v 1, v 2 ), (w 1, w 2 ) = t v 1 w 2 t v 2 w 1, or v 1, v 2, w 1, w 2 (Ẑ(p) ) g. The set Ãg,(p);m(k) o k-points o the pro-f p -scheme Ãg,(p);m is the set o isomorphism classes o quadruples (A, λ, η (p), ψ), where (A, λ) is a principally polarized g-dimensional ordinary 14

15 abelian variety, η (p) : (A (p) /Z p) 2g = (Q l /Z l ) 2g A[l ] =: A[non-p] l p l p is a symplectic isomorphism, and ψ : (Z/p m Z) g A[p m ] et is an isomorphism. In the above, the principal polarization λ induces a compatible system o pairings A[N] A[N] (Z/NZ)(1) χ 1 Z/NZ, (N, p) = 1, while the standard pairing, on (Ẑ(p) ) 2g induces a compatible system o pairings in the sense that, N : (N 1 Z/Z) 2g (N 1 Z/Z) 2g Z/NZ, N 2 v, N 2 w N1 v, w N1 N 2 (mod N 1 ) or all v, w ((N 2 N 1 ) 1 Z/Z) 2g. Identiying A[N] with H 1 (A, Z/NN) or (N, p) = 1, a prime-to-p level structure η (p) as above gives rise to symplectic isomorphisms and ˆη : (Ẑ(p) ) 2g H 1 (A, Z (p) ) η : (A (p) )2g H 1 (A, A (p) ). We will use a slightly dierent description o the set o geometric points o à g,(p);m. Let k = k alg F p be as above. Then the set o k-points o à g,(p),m is in natural bijection with the set o all isomorphism classes o quadruples ([A], λ, η, ψ), where [A] is a g-dimensional ordinary abelian variety regarded as an object o AV (p) k (as an object AV (p) k H 1 (A, A (p), λ is a principal polarization on [A], i.e. a separable polarization on A as an object in AV k ), η : (A (p) )2g ) is a symplectic isomorphism, and ψ : (Z/pm Z) A[p m ] et is an isomorphism. In the second description o à g,(p),m (k) above, more symplectic isomorphisms η are allowed, because η is not required to come rom an isomorphism ˆη : (Ẑ(p) ) 2g H 1 (A, Ẑ(p) ); this is compensated by the act that there are more isomorphisms in the category AV (p) k than in the category AV k. An isomorphism rom ([A 1 ], λ 1, η 1, ψ 1 ) to ([A 2 ], λ 2, η 2, ψ 2 ) is a prime-to-p isogeny α : A 1 A 2 such that α η 1 = η 2, α ψ 1 = ψ 2 and α t λ 2 α = λ 1. We indicate how to go rom the second description o à g,(p);m (k) to the irst description o à g,(p);m (k). Let ([A], λ, η, ψ) be a quadruple as in the previous paragraph. Then there exists an abelian variety B over k and a prime-to-p-isogeny α : B A, both deined up to unique isomorphisms, such that α 1 η : (A (p) ) induces an isomorphism ˆη : (Ẑ(p) ) 2g )2g H 1 (B, A (p) H 1 (B, Ẑ(p) ) and an isomorphism η (p) : (A (p) /Ẑ(p) ) 2g B[non-p]. Moreover λ B := α t λ α is a principal polarization o B as an abelian variety. Let ψ B = α 1 ψ. Then the quadruple (B, λ B, η (p), ψ B ) is well-deined up to unique isomorphism and gives a point o à g,(p);m (k) according to our irst description. 15

16 The right action o Sp 2g (A (p) ) on Ãg,(p);m(k) is quite easy to describe in terms o the second description: An element γ Sp 2g (A (p) ) sends a point [([A], λ, η, ψ)] Ãg,(p);m(k) to the point [([A], λ, η γ, ψ)]. Things are a bit more complicated with the irst description. Let [(A, λ, η (p), ψ)] be a point o à g,(p);m (k) according to the irst description. Let ˆη : (Ẑ(p) ) 2g H 1 (A, Ẑ(p) ) and η : (A (p) )2g H 1 (A, A (p) ) be the symplectic isomorphisms attached to η. There exists an abelian variety B over k and a prime-to-p isogeny α : B A, unique up to unique isomorphisms, such that α 1 η γ : (A (p) )2g H 1 (B, A (p) ) induces symplectic isomorphisms ˆη B : (Ẑ(p) ) 2g H 1 (B, Ẑ(p) ) and η (p) B : (A(p) /Ẑ(p) ) 2g B[non-p]. Then λ B := α t λ α is a principal polarization o B. Let ψ B := α 1 ψ. Then [(B, λ B, η (p) B, ψ B)] is well-deined, and is the image o [(A, λ, η (p), ψ)] under γ. (5.4) Lemma Notation as above. Recall that x 0 = [(A 0, λ 0, η 0 )] A ord g,n(f p ), A 0 = E 1 E 1, λ 0 is the product principal polarization on A 0, and η 0 is a level-n structure on A 0. Let y 1 = [(A 0, λ 0, η 0, ψ 1 )], y 2 = [(A 0, λ 0, η 0, ψ 2 )] be two F p -points o π 1 m (x 0 ). Let h be an element o H(Z (p) ) such that h p,et ψ 1 = ψ 2, where h p,et denotes the automorphism o A 0 [p m ] et induced by h. Then y 2 belongs to the Sp 2g (A (p) )-Hecke orbit o y 1 on the GL g (Z/p m Z)-cover A g,n;m o A ord g,n. Proo. The element h H(Z (p) ) is a prime-to-p isogeny rom A 0 to itsel which respects the polarization λ 0. Let η (p) : (A (p) /Ẑ(p) ) 2g A 0 [non-p] be a symplectic isomorphism which extends η 0. Let η : (A (p) )2g H 1 (A 0, A (p) ) be the symplectic isomorphism attached to η(p). The F p -point [([A 0 ], λ 0, η, ψ i )] o the tower à g,n(p);m lies above the F p -point y i o A g,n;m, i = 1, 2. Here we have ollowed the notation in 5.3, and [A 0 ] is the object in AV (p) k attached to A 0. By deinition, the prime-to-p isogeny h induces an isomorphism rom ([A 0 ], λ 0, η, ψ 1 ) to ([A 0 ], λ 0, h (p) η, ψ 2 ). Since h (p) η = η ( η 1 h (p) η), we see rom the deinition o prime-to-p Hecke correspondences on A g,n;m that y 1 belongs to the image o y 2 under the prime-to-p Hecke correspondence induced by the element η 1 h (p) η Sp 2g (A (p) ). Remark A prominent eature o the above argument is the similarity to the product ormula: I one changes the prime-to-p level structure η by the prime-to-p component h (p) o a rational element h H(Z (p) ) and the p-power level structure ψ by the p-adic component h p o h, one gets back to the same point o A g,(p);m. (5.5) Proposition Notation as in 5.1. Let z 1, z 2 be two F p -points o A g,n;m which belong to the same Sp 2g (A (p) )-Hecke orbit on A g,n;m. Then z 1 and z 2 belong to the same irreducible component o the smooth F p -scheme A g,n;m. 16

17 Notation as in 5.4. We oer two proos: one by quoting [4], the other by explaining the relevant part o the argument in [4]. Proo A. By Prop o [4], z 1 and z 2 belong to the same irreducible component o the smooth F p -scheme A g,n;m. We need to explain why quoting [4] is legitimate. In [4], the subvariety W is assumed to be a subscheme o A g,n, while in the present situation A g,n;m is a inite étale cover o A ord g,n. However one can examine the argument in [4] and convince onesel that the same proo works in the present situation. Proo B. Let l 1,, l r be distinct prime numbers, all dierent rom p, such that z 1 and z 2 belong to the same G L -Hecke orbit on A g,n;m, where G L denotes the product group G L := Sp 2g (Q l1 ) Sp 2g (A lr ). Let L = r i=1 l i. Consider the L-adic subtower Ãg,nL ;m := ( Ag,nL ;m) j o the tower (A j N g,nb;m) b N pn, and let π 0 (Ã) := lim π 0(A j g,nl j ;m) be the inverse limit o the set o irreducible components o A g,nl j ;m. The group G L operates on the tower à g,nl ;m, inducing the G L -Hecke correspondences on A g,n;m. Since A g,n;m is a inite étale cover o A ord g,n, the image o the L-adic monodromy representation attached to the universal abelian scheme over A g,n;m is an open subgroup o G L, hence π 0 (Ã) is a inite set. The action o the group G L on à deines a natural action o G L on π 0 (Ã). By [4, Lemma 3.1], every subgroup o inite index in G L is equal to G L itsel. Thereore G L operates trivially on the inite set π 0 (Ã). So z 1 and z 2 belong to the same irreducible component o A g,n;m. Remark The above proo o Thm. 2.2 was inspired by Hida s work on p-adic monodromy in [14]. 6. Remarks and Comments (6.1) Remark Let Γ be a inite dimensional semisimple Q-algebra, and let O Γ be an order o Γ. Recall that an O Γ -linear abelian variety (A, ι) over F p is Γ-hypersymmetric i End OΓ (A) Z Q p End OΓ Z Z p (A[p ]) Zp Q p ; see [5, De. 6.4]. Let M be a modular variety o PEL-type over F p with O Γ as the ring o prescribed endomorphisms. Then there may exist a Newton stratum Z (resp. a lea) on M with no Γ-hypersymmetric point. This happens when M is a Hilbert modular variety M F associated to a totally real number ield F, Γ = F, and Z is a Newton stratum in M F (or a lea in M F ) such that every point o Z corresponds to an O F -linear abelian variety with some but not all slopes equal to 1. Then the methods in 4 and 5 do not help in proving 2 maximality o p-adic monodromy or Z M F, while the method in 3 does; see [7, 5] and also

18 (6.2) We indicate how the methods in 4 and 5 can be used to prove the maximality o p-adic monodromy, or equivalently the irreducibility o the Igusa tower, or the ordinary locus o a modular variety o quasisplit U(n, n) type. This irreducibility statement is useul or constructing p-adic L-unctions or GL(n); see [13]. It is a special case o [14], Cor. 8.17; see also [15] 10. We reer to [13], [14] and [15] or more inormation on the U(n, n) type modular variety and related algebraic groups. (6.2.1) Notation. Let K be a totally imaginary extension o a totally real number ield F such that p is unramiied in K and every prime ideal in O F splits in K. Let m 3 be an integer relatively prime to p. The modular variety M = M K,U(n,n),m over F p classiies quadruples (A S, ι, λ, η), where S is a scheme over F p, A S is an abelian scheme o relative dimension 2n[F : Q], ι : O K End S (A) is a ring homomorphism, λ : A A t is a principal polarization such that the Rosati involution induces complex conjugation on K, and η is a level m-structure. Moreover one requires that the Kottwitz condition in [20] is satisied or the quasisplit U(n, n) PEL-type or K/F. Under the present assumptions on K and p, the last condition or ordinary abelian varieties can be made explicit as ollows. Suppose that A S is ordinary, i.e. the Barsotti-Tate group A[p ] S is an extension o étale Barsotti-Tate group by a multiplicative Barsotti-Tate group. Then the Kottwitz condition means that the Barsotti-Tate group A[ w ] S has relative dimension n [K w : Q p ] and height 2n [K w : Q p ] or every place w o O K above p. Let M ord be the locus o M over which the universal abelian scheme is ordinary; it is an open dense subscheme o M. Let x 0 = [(A 0, ι 0, λ 0, η 0 )] M ord (F p ) be an F p -point o the ordinary locus M ord. Let 0 be the Rosati involution on End OK Z p (A 0 [p ]) attached to the polarization λ 0. The central (O K Z Z p )-algebra End (OK Z p)(a 0 [p ] mult ) is isomorphic to v p M n(o K OF O Fv ), where v runs through all places o F above p. Similarly End (OK Z p)(a 0 [p ] et ) is isomorphic to v p M n(o K OF O Fv ). (6.2.2) Denote by ν the map End (OK Z p)(a 0 [p ]) = End (OK Z p)(a 0 [p ] mult ) End (OK Z p)(a 0 [p ] et ) (O K Z Z p ) which corresponds to the map M n (O K OF O Fv ) M n (O K OF O Fv ) v p v p v p (O K OF O Fv ) deined by ( ) ( (Bv ) v p, (C v ) v p detok O (B Fv v) det OK O (C Fv v) ) v p or all elements ( ) ( ) ( (B v ) v p, (C v ) v p v p M n(o K OF O Fv ) v p M n(o K O OF Fv)). Denote by U the group consisting o all elements u ( End OK Z p (A 0 [p ]) ) 18 such that