Homogeneous p-adic vector bundles on abelian varieties that are analytic tori

Homogeneous p-adic vector bundles on abelian varieties that are analytic tori Thomas Ludsteck June, 2008 Abstract We investigate homogeneous p-adic vector bundles on abelian varieties that are analytic tori. We show that for each homogeneous vector bundle on such a variety there exists an integer N > 0 such that the pullback of this vector bundle via the N-multiplication is attached to an integral representation of the topological fundamental group. 1 Introduction Denote by C p := Q p the completion of an algebraic closure of Q p, denote its ring of integers by o and let v be the p-adic valuation on C p normalized by v(p) = 1. Denote by the Berkovich analytification of an abelian variety that is defined over C p. Assume that is an analytic torus, i.e. is isomorphic to the quotient T/Λ of T := (G g m) an by a p-adic lattice Λ. On such a torus one can define an analogue of the classical Riemann-Hilbert correspondence (see e.g. Y. André [And03] III). In this correspondence a discrete representation ρ : π top 1 (, 1) GL r (C p ) gives rise to a vector bundle with connection M ρ on and vice versa. It was shown by G. Faltings [Fal83] and M. van der Put - M. Reversat [PuRe88], that there is an bijective correspondence between homogeneous vector bundles on and so-called Φ-bounded representation of π top 1 (, 1) if one omits connections. A vector bundle M on is said to be homogeneous or translation invariant if it satisfies t sm M for the translation by s map t s and all s. Besides this topological Riemann-Hilbert correspondence on analytic spaces, there is also a theory of étale parallel transport on algebraic curves and abelian varieties due to C. Deninger - A. Werner [DeWe05b], [DeWe05a] and the p-adic impson correspondence [Fal05] of G. Faltings. It was shown by G. Herz [Her07] and the author [Lud08] that the topological and the algebraic parallel transport are compatible. This comparison has been worked out only for algebraic curves, and we assume that the higher dimensional case (e.g. abelian varieties) should be analogous. But this is not the aim of our paper. An important characteristic of this comparison is that it works only for vector bundles M ρ that are attached to an integral representation, i.e. ρ maps into GL r (o). The aim of this paper is to show the following theorem: Theorem 1.1. Let M ρ be a homogeneous vector bundle on, attached to a representation ρ : π top 1 (, 1) GL r(c p ). Then there exists an integer N > 0 such that N M ρ, the pullback of M ρ via the N-multiplication on, is isomorphic to 1

a vector bundle attached to an integral representation ρ : π top 1 (, 1) GL r (o). The representation ρ can be computed explicitly from the representation ρ. As an application one could compare the topological and the algebraic parallel transport for the vector bundle N M ρ. It is known that there are many homogeneous vector bundles on that are not attached to integral representations ([Fal83],[PuRe88]). For the proof of Theorem 1.1 we will use the temperate fundamental group of. The topological fundamental group is not sufficient, because the N- multiplication on is not a topological cover of unless N = 1. We can restrict us to indecomposable representations, and those can be written as a tensor product of a one-dimensional representation and an unipotent representation. We will then show that the p n -th power of an unipotent matrix will have integer coefficients for n big enough, what takes care of the unipotent representations. Using an explicit criterion to decide if two vector bundles attached to two different representations are isomorphic solves the one-dimensional case. 2 Homogeneous p-adic vector bundles on abelian varieties 2.1 Fundamental groups of analytic tori An analytic torus (see e.g. [Ber90] ection 6.3 or [PuRe88]) is the quotient = T/Λ of the g-th power of the multiplicative group T := (G g m) an by a lattice Λ T (C p ) = (C p) g. A subgroup Λ T (C p ) is called a lattice if the map l : T (C p ) Q g ; (z 1,..., z g ) (v(z 1 ),..., v(z zg )) (1) sending coordinates to its p-adic valuations identifies Λ with a lattice in Q g. We will further assume that is the analytification of an abelian variety defined over C p. These tori are C p -manifolds in the sense of Y. André ([And03] III ection 1.1.1 and Remark 1.1.2 e)). For these C p -manifolds Y. André defined several fundamental groups. We recall briefly the definition of these groups, and refer to [And03] III ection 1,2 for details. For a torus denote by Cov et, Covtemp and Cov top the categories of étale, temperate and topological coverings ([And03] III Definition 1.2.3. and Definition 2.1.1). Using Grothendieck s theory of the fundamental group one can define fundamental groups π 1(, 1) corresponding to the categories Cov, where = et, temp or top and 1 is the neutral element. There are canonical maps π et 1 (, 1) π temp 1 (, 1), π et 1 (, 1) π top 1 (, 1) and πtemp 1 (, 1) π top 1 (, 1) between these fundamental groups corresponding to the fully faithful embeddings Cov top Cov et, Cov temp Cov et and Cov top Cov temp. We will now discuss some properties of these fundamental groups for analytic tori. 2

Lemma 2.1. The canonical morphism is injective. π temp 1 (, 1) π alg 1 (, 1) Proof. This is known in the one-dimensional case ([And03] III Proposition 2.1.6). As in the proof of this proposition it suffices to show that the topological fundamental group of a finite étale Galois covering of is free and finitely generated. By Riemann s existence theorem [Lüt93] is also the analytification of an abelian variety. The topological fundamental group of is a free abelian group of finite rank [Ber90] Corollary 6.5.2 Proposition 2.2. There exists a canonical exact sequence 1 π temp 1 (T, 1) π temp 1 (, 1) π top 1 (, 1) 1. There are non-canonical isomorphisms and π top 1 (, 1) Λ Z g, (2) π temp 1 (T, 1) π alg 1 (T, 1) Ẑg (3) π temp 1 (, 1) Ẑg Z g (4) Proof. In the case that dim = 1 this was shown in [And03] III ection 2.3. The sequence is exact because of [And03] III Corollary 1.4.12. The isomorphism 2 follows from the definition of. If G g m denotes the algebraic torus of dimension g defined over C p, then there is an isomorphism π alg 1 (Gg m, 1) = π alg 1 (G m, 1)... π alg 1 (G m, 1) ([Gro67] X Corollaire 1.7). From this we can deduce that the finite étale covers of G g m are induced from endomorphisms (z 1,..., z g ) (z n1 1,..., zng g ). By Riemann s existence theorem [Lüt93] we know that finite étale covers of T = (G g m) an are induced from algebraic covers of (G g m), hence the underlying space of a finite étale cover of T is isomorphic to T. Because T has trivial topological fundamental group we conclude that all temperate coverings of T are algebraic. This implies the isomorphism 3. To show the isomorphism 4 it suffices to show that the exact sequence splits. By Lemma 2.1 the group π temp 1 (, 1) can be injected into the abelian group π alg 1 (, 1), so it is also abelian and a Z-module. The group π top 1 (, 1) is a free Z-module and so the sequence splits by the universal property of projective modules Lemma 2.3. Let N > 0 be an integer. The N-multiplication on induces an endomorphism N of π temp 1 (, 1) that is given by the N-multiplication on π temp 1 (, 1). Proof. This is known for the algebraic fundamental group π alg 1 (, 1) or for finite étale Galois-covers of. But finite temperate covers are finite étale. o for all subgroups H π temp 1 (, 1) of finite index the N multiplication on induces the N multiplication on the finite group π temp 1 (, 1)/H. The lemma follows now from the explicit description π temp 1 (, 1) Ẑg Z g of Proposition 2.2 3

2.2 A p-adic Riemann Hilbert correspondence on analytic tori In analogy to the complex Riemann-Hilbert correspondence Y. André defined in [And03] III ection 3.4 a p-adic Riemann-Hilbert correspondence. This correspondence is defined on C p -manifolds, but we will need it only for tori. For = et, temp or top let ρ : π 1(, 1) GL r (C p ) denote a representation that is continuous with respect to the discrete topology of GL r (C p ). The kernel H π 1(, 1) of the representation ρ corresponds to a -Galois covering u : with group G = π 1 (, 1)/H. The vector bundle attached to ρ is defined by (O Cp C r p) G, where ρ acts via the diagonal action. The vector bundle attached to ρ will be denoted by M ρ. We will need a criterion to decide if two vector bundles attached to two different representations are isomorphic: Lemma 2.4. Let / be an (Berkovich-) étale Galois covering with group G (e.g temperate or topological) and let ρ 1, ρ 2 : G GL r (C p ) be two continuous (discrete topology) representations. Then the following two conditions are equivalent: a) M ρ1 = Mρ2 ; b) There is an (Berkovich-) analytic function f : GL r (C p ) such that f(γz) = ρ 2 (γ)f(z)ρ 1 (γ) 1 γ G, z. Proof. We adapt a proof from the complex situation ([Flo01] Lemma 2) for our purposes: An isomorphism between M ρ1 and M ρ2 is an analytic global section of Mρ 1 M ρ2 = M t ρ 1 1 ρ2, consisting of invertible matrices. o it corresponds to an analytic f : GLn (C p ) such that f(γ z) = ( t ρ 1 1 ρ 2 )(γ)f(z) = ρ 2 (γ)f(z)ρ 1 (γ) 1 This lemma has the following application for line bundles. Proposition 2.5. Let ρ : Λ C p be an one-dimensional representation. Let M ρ be the line bundle on attached to ρ. Then there exists an integer N > 0, such that N M ρ is isomorphic to a line bundle attached to an integral representation ρ : Λ o. Proof. Choose some generators λ 1,..., λ g of Λ and denote their images (via the map l of 1) in Q g by λ 1,..., λ g. These elements generate a lattice in Q g, hence the linear system of equations 0. 0 = v(ρ(λ 1 )) λ 11 λ1g. +.. v(ρ(λ g )) λ g1 λgg r 1. r g (5) 4

has an unique solution (r 1,..., r g ) Q g. Choose an integer N > 0 such that the numbers n 1 := N r 1,..., n g := N r g are elements of Z. We want to make use of Lemma 2.4 so let us define an analytic function f : (C p) g C p by setting z = (z 1,..., z g ) z n1 1... z ng g. Define a representation ρ : Λ C p by the rule ρ (λ) := f(λz) f(z) ρn (λ). Because of this relation and Lemma 2.4 we see that the line bundle N M ρ is isomorphic to the line bundle attached to the representation ρ. To prove the proposition it suffices to show that the representation ρ is integral, i.e. v(ρ (λ)) = 0 for all λ Λ. It suffices to show this for the generators λ 1,..., λ g. We fix an i and calculate the valuation of ρ (λ i ) v(ρ (λ i )) = v( f(λ iz) f(z) ρ N (λ i )) = v( λn1 i1... λng ig f(z) f(z) = n 1 v(λ i1 ) +... + n g v(λ ig ) + N v(ρ(λ i )) = n 1 λ i1 +... + n g λ ig + N v(ρ(λ i )) = 0. ) + v(ρ N (λ i )) The last equation holds true because we have chosen the function f according to the solution vector (r 1,..., r g ) of equation 5. 2.3 ome representation theory We will show that indecomposable representations of the lattice Λ are the tensor product of a character and an unipotent representation. We will further show that unipotent representations of Λ will become integral after pullback by some integer N > 0. Lemma 2.6. Let ρ : Λ GL r (C p ) be an indecomposable representation, i.e. every direct decomposition of ρ is trivial. Then for every λ Λ the matrix ρ(λ) has exactly one eigenvalue. Proof. Fix a λ Λ. The matrix B := ρ(λ) defines an endomorphism of the representation ρ, i.e. B ρ(λ ) v = ρ(λ ) B v for all λ Λ and all v C r p. If B had two distinct eigenvalues µ τ, then the endomorphism B µe r End(ρ) would be neither an automorphism nor nilpotent. But this is not possible because ρ is indecomposable and this would contradict [Ati56] Lemma 6 Corollary 2.7. The representation ρ : Λ GL r (C p ) can be written as ρ = δ β for a one-dimensional representation δ and an unipotent representation β. Proof. Let λ 1,..., λ g be generators of Λ. For a fixed i the matrix ρ(λ i ) has Jordan normal form µ i U i for the unique eigenvalue µ i of ρ(λ i ) (Lemma 2.6) and an upper triangle matrix U i with ones on the diagonal. This matrix U i is then clearly unipotent. We can then define δ by setting δ(λ i ) := µ i and β by setting β(λ i ) := 1 µ i ρ(λ i ). The matrix 1 µ i ρ(λ i ) is conjugated to the matrix U i and so it is unipotent. This shows the lemma 5

Lemma 2.8. Let U GL r (C p ) be an unipotent matrix. Then there exists an integer K > 0 such that for all n K the p n -th power of U and its inverse have integer coefficients i.e. U pn, (U pn ) 1 GL r (o). Proof. Let H > 0 be an integer satisfying P H = 0 for the nilpotent matrix P = U E r. For some integer n 1 satisfying p n > H we can take the p n -th power of U and obtain H ( ) p n U pn = (P + E r ) pn = E r + P k. (6) k k=1 }{{} T U,n := If v denotes the standard p-adic valuation as in the introduction and if 1 k H < p n then we have the following estimate: ( p n v( )) = v( (pn 1)!p n k (p n k)!k! ) v(pn ) v(k!) n v(h!). As n becomes big the term T U,n from equation (6) becomes small in the p-adic norm. We can choose an integer K > 0 such that for all n K we can write T U,n = p ( 1 p T U,n) and the term 1 p T U,n has integer coefficients. This implies immediately that the matrix U pn has integer coefficients, i.e. U pn GL r (o). Its inverse (U pn ) 1 has also integer coefficients, because one can apply the geometric series that converges for arguments divisible by p Corollary 2.9. Let ρ : Λ GL r (C p ) be an unipotent representation. Then there exists an integer N > 0 such that the representation Λ N Λ ρ GL r (C p ) is integral, i.e. it maps into GL r (o). Proof. If Λ Z then we can choose an integer N = p n > 0 as in Lemma 2.8 for a generator of Λ and the corollary follows. In the general case we choose integers N 1,..., N g for generators of Λ. Then N := N 1... N g will do the job 3 Proof of the main theorem and final remarks We have now all ingredients to prove Theorem 1.1. Proof. We can restrict us to indecomposable vector bundles and representations. o we assume that M ρ and ρ are indecomposable. The representation ρ can be written as ρ = δ β for an one dimensional representation δ and an unipotent representation β (Corollary 2.7). The Riemann-Hilbert correspondence between representation and vector bundles is compatible with pullbacks and tensor products. The vector bundle M ρ can be thus written as M ρ M δ β M δ M β. If the assertion holds for the line bundle M δ with integer N δ and for M β with integer N β, then the assertion holds for M δ M β with integer N ρ := N δ N β, because pullbacks of integral representation remain integral. We can restrict us to the special representations δ and β. The one-dimensional case δ was treated in Proposition 2.5. The unipotent case β follows from Corollary 2.9. The representation ρ can be computed explicitly from ρ because all calculation involved were explicit. 6

Remark 3.1. - a) If dim = 1, then is a Tate curve, and we can apply the comparison theorem of G. Herz [Her07] and T. Ludsteck [Lud08]. If M ρ is a vector bundle attached to the representation ρ then the DeWe-representation attached to N M ρ is the pro-finite completion of the integral representation ρ. b) An interesting question one might ask is wether an analogue of Theorem 1.1 holds for Mumford curves. A Mumford curve X is a smooth projective algebraic curve, such that its analytification X an admits a p-adic uniformization (see e.g. [Ber90] Theorem 4.4.1). If M denotes a semistable algebraic vector bundle of degree 0 on X does there exist a finite étale covering π : Y an X an such that the pullback of its analytification π (M ) an is attached to an integral representation of the topological fundamental group of Y an? A positive answer would answer a question of G. Faltings [Fal05], C. Deninger and A. Werner [DeWe05b]. To be more precise, can the algebraic vector bundle M be trivialized modulo p after a finite étale cover of X? We refer the reader to the introduction of [DeWe05b] for a more detailed discussion of this question. Acknowledgement I would like to thank my supervisor Annette Werner for introducing me to this topic and her advice and encouragement. I want to thank G. Herz and C. Florentino for useful discussions on vector bundles. This paper is part of the authors Ph.D.-thesis, that was funded and supported by Universität tuttgart. Contact Thomas Ludsteck Universität tuttgart IAZ Pfaffenwaldring 57 70569 tuttgart Germany Firstname.Lastname@googlemail.com (replace by real name) References [And03] Y. André, Period mappings and differential equations. From C to C p, Tôhoku-Hokkaidô lectures in arithmetic Geometry (2003), 246 p. [Ati56] M. Atiyah, On the Krull-chmidt theorem with application to sheaves, Bulletin de la.m.f. 84 (1956), 307 317. [Ber90] V. Berkovich, pectral theory and analytic geometry over non- Archimedean fields, Mathematical urveys and Monographs 33, A.M.., 1990. [DeWe05a] C. Deninger, A. Werner, Line bundles and p-adic characters, G. van der Geer, B. Moonen, R. choof (Eds.), Number Fields and Function Fields Two parallel Worlds, Birkhäuser, Basel (2005). [DeWe05b] C. Deninger, A. Werner, Vector bundles on p-adic curves and parallel transport, Ann. cient. Éc. Norm. up., 4e série, 38 (2005), 553 597. 7

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