Recovering function fields from their decomposition graphs

Recovering function fields from their decomposition graphs Florian Pop In memory of Serge Lang Abstract We develop the global theory of a strategy to tackle a program initiated by Bogomolov in 1990. That program aims at giving a group theoretical recipe by which one can reconstruct function fields K k with td(k k) > 1 and k algebraically closed from the maximal pro-l abelian-by-central Galois group ΠK c of K, where l is any prime number char(k). Key words: anabelian geometry, pro-l groups, Galois theory, function fields, valuations theory, (Riemann) space of prime divisors, Hilbert decomposition theory, Parshin chains, decomposition graphs 2010 AMS subject classification: Primary 12E, 12F, 12G, 12J; Secondary 12E30, 12F10, 12G99 1 Introduction Recall that the birational anabelian conjecture originating in ideas presented in Grothendieck s Esquisse d un Programme [11] and Letter to Faltings [12] asserts roughly the following: First, there should exist a group-theoretical recipe by which one can recognize the absolute Galois groups G K of finitely generated infinite fields K among all the profinite groups. Second, if G = G K is such an absolute Galois group, then the group-theoretical recipe should recover the field K from G K in a functorial way. Third, the recipe should be invariant under open homomorphisms of absolute Galois groups. In particular, the category of finitely generated infinite fields (up to Frobenius twist) should be equivalent to the category of their absolute Galois groups and open outer homomorphisms between these groups. A first instance of this situation is the celebrated Neukirch Uchida theorem, which says that global fields are characterized by their absolute Galois groups. I will not go into further detail about the results concerning Grothendieck s (birational) anabelian geometry, but the interested reader can find more about this in Szamuely s Bourbaki Séminaire talk [34], Faltings Séminaire Bourbaki talk [9], Stix [35], and newer results by Mochizuki [19], Saidi Tamagawa [33], Minhyong Kim [13], and Koenigsmann [15] concerning the (birational) section conjecture. Florian Pop Dept of Mathematics, University of Pennsylvania, 209 S 33rd St, Philadelphia, PA 19104, USA. e-mail: pop@math.upenn.edu 1

2 Florian Pop The idea behind Grothendieck s anabelian geometry is that the arithmetical Galois action on rich geometric fundamental groups (such as the geometric absolute Galois group) makes objects very rigid, so that there is no room left for non-geometric open morphisms between such rich fundamental groups endowed with arithmetical Galois action. On the other hand, Bogomolov [2] advanced at the beginning of the 1990s the idea that one should have anabelian-type results in a total absence of an arithmetical action as follows: Let l be a fixed rational prime number. Consider function fields K k over algebraically closed fields k of characteristic l. For each such function field K k, let ΠK c := Gal(K K) be the Galois group of a maximal pro-l abelian-by-central Galois extension K K. Note that if G (1) = G K and G (i+1) := [G (i),g K ](G (i) ) l for i 1 are the central l terms of the absolute Galois group G K of K, then we have that Π K = G (1) /G (2) is the Galois group of the maximal pro-l abelian subextension K K of K K, and ΠK c = G(1) /G (3) ; and denoting by G ( ) the intersection of all the G (i), it follows that G K (l) := G K /G ( ) is the maximal pro-l quotient of G K. Now the program initiated by Bogomolov [2] has as ultimate goal to recover function fields K k with td(k k) > 1 as above from ΠK c in a functorial way. (Note that Bogomolov denotes ΠK c by PGalc K.) If successful, this program would go far beyond Grothendieck s birational anabelian conjectures, as k being algebraically closed implies that there is no arithmetical action in the game. The program initiated by Bogomolov is not completed yet, and the present manuscript is a contribution towards trying to settle that program; see the historical note below for more about this. Since the present manuscript is quite abstract, let me announce the following concrete result, whose proof relies in an essential way on the main theorem of the present manuscript (see Pop [30] for a complete proof): Target Result. Let K k be a function field with td(k k) > 1 and k an algebraic closure of a finite field. Then the following hold: (1) There exists a group-theoretical recipe which recovers K k from ΠK c. (2) The above group-theoretical recipe is functorial in the following sense: Let L l be a function field with l an algebraically closed field, and let Φ : Π K Π L be the abelianization of some isomorphism Φ c : ΠK c Π L. c Then denoting by Li and K i the perfect closures, there exist an isomorphism of field extensions ı : L i l K i k and an l-adic unit ε Z l such that ε Φ is induced by ı. Moreover, the isomorphism ı is unique up to Frobenius twists, and the l-adic unit ε is unique up to multiplication by p-powers, where p = char(k). (3) For a function field L l as above, let Isom F (L,K) be the set of isomorphisms of field extensions ı : L i l K i k up to Frobenius twists, and let Isom c (Π K,Π L ) be the set of abelianizations of continuous group isomorphisms ΠK c Π L c up to multiplication by l-adic units ε Z l. Then there is a canonical bijection Isom F (L l,k k) Isom c (Π K,Π L ).

Recovering function fields from their decomposition graphs 3 A sketch of a strategy to functorially recover K k from pro-l Galois information, in particular to prove the above target result, can be found essentially already in (the notes of) Pop [25], and has as starting point the following simple idea: Let K be the l-adic completion of the multiplicative group K of K k. 1 Since the cyclotomic character of K is trivial, one can identify the l-adic Tate module T K,l of K with Z l (non-canonically), and let ı K : T K,l Z l be a fixed identification. Via Kummer theory, one has isomorphisms of l-adically complete groups: K = Hom cont (Π K,T K,l ) ı K Hom cont (Π K,Z l ), i.e., K can be recovered from Π K, hence from Π c K via the projection Π c K Π K. On the other hand, since k is divisible, K equals the l-adic completion of the free abelian group K /k. Now the idea of recovering K k is as follows: (a) First, give a recipe to recover the image j K (K ) = K /k of the l-adic completion functor j K : K K /k K inside the known K = Hom cont (Π K,Z l ). (b) Second, interpreting K /k =: P(K) as the projectivization of the infinitedimensional k-vector space (K, +), give a recipe to recover the projective lines l x,y := (kx + ky) /k inside P(K), where x,y K are k-linearly independent. (c) Third, apply the fundamental theorem of projective geometries of Artin [1], and deduce that K k can be recovered from P(K) endowed with all the lines l x,y. (d) Finally, show that the recipes above are functorial, i.e., they are invariant under isomorphisms of profinite groups Π K Π L which are abelianizations of isomorphisms Π c K Π c L. In particular, such isomorphisms Π K Π L originate actually from geometry. The strategy from Pop [25] to tackle the above problems (a), (b), (c), (d), above is in principle similar to the strategies (initiated by Neukirch and Uchida) for tackling Grothendieck s anabelian conjectures. It has two main parts, as follows, the terminology being as introduced later: Local theory: It has as input the Galois/group-theoretical information ΠK c. It should be a recipe which in a first approximation recovers from ΠK c the decomposition/inertia groups of prime divisors of K k in Π K (N.B., not in ΠK c ). The final output of the local theory should be the total decomposition graph G D tot of K k. K This recipe should be invariant under isomorphisms Π K Π L which are induced by some isomorphisms ΠK c Π L c. Global theory: Its input is the total decomposition graph G D tot K recipe which in a first approximation recovers the geometric decomposition graphs of K k. It should be a G DK (together with some of their special properties) for K k from G D tot together K with their sets of rational quotients A K = {Φ κx } κx. In a second approximation, this recipe should recover P(K) and its projective lines from the G DK endowed with their rational quotients A K = {Φ κx } κx. It thus should finally recover the function field K k. Moreover, this recipe should be functorial, i.e., invariant under isomorphisms of total decomposition graphs G D tot G K D tot. L The present manuscript deals mainly with questions of the above global theory, precisely, recovering the geometric decomposition graphs (together with some of

4 Florian Pop their special properties) from the total decomposition graph, and finally proving the main result of the paper, which is to show that morphisms of (total) decomposition graphs which are compatible with rational projections originate in a precise way from geometry. Before announcing the main result of the manuscript, let us briefly introduce the main concepts and objects, which will discussed/studied in detail later. Prime divisor graphs (see Section 3 for more details) Recall that in the context above, a (Zariski) prime divisor of a function field K k is a discrete valuation v of K whose valuation ring is the local ring O X,x1 of the generic point x 1 of some Weil prime divisor of some normal model X k of K k. If so, then the residue field Kv of v is the function field Kv = κ(x 1 ), and therefore, td(kv k) = td(k k) 1. A set of Zariski prime divisors D of K k is called a geometric set if there exists a quasi-projective normal model X k of K k such that D = D X is the set of valuations v x1 defined by the generic points x 1 of all the Weil prime divisors of X. We next generalize the prime divisors of K k as follows: First, for a valuation v of K the following are equivalent: (i) v is trivial on k, and the residue field has td(kv k) = td(k k) r, and there exists a chain of valuations v 1 < < v r := v. (ii) v is the valuation-theoretical composition v = v r v 1, where v 1 is a prime divisor of K, and inductively, v i+1 is a prime divisor of the residue function field κ(v i ) k. A valuation v of K which satisfies the above equivalent conditions is called a prime r-divisor of K k; and a sequence of prime divisors (v r,...,v 1 ) as above will be called a Parshin r-chain of K k. By definition, the trivial valuation will be considered a generalized prime divisor of rank zero, and the corresponding Parshin chain is the trivial Parshin chain. Finally, note that r td(k k), and that in the above notation, one has v i = v i v 1 for all i 1. The total prime divisor graph DK tot of K is the following half-oriented graph: (a) The vertices of DK tot are the residue fields Kv of all the generalized prime divisors v of K k viewed as distinct function fields. (b) For given v = v r v 1 and w = w s w 1, the edges from Kv to Kw are as follows: (i) If v = w, i.e., Kv = Kw, then the trivial valuation v/w = w/v is the only edge from Kv = Kw to itself; and it is by definition a non-oriented edge. (ii) If Kv Kw, then the set of edges from Kv to Kw is non-empty iff s = r + 1 and v i = w i for 1 i r; and if so, then w s = w/v is the only edge from Kv to Kw, and it is by definition an oriented edge. A geometric prime divisor graph for K k is any connected subgraph D K of DK tot which satisfies the following conditions: First, for each vertex Kv of D K, the set D v of all the non-trivial edges of D K originating from Kv is a geometric set of prime divisors of Kv k. Second, all maximal branches of non-trivial edges of D K originate

Recovering function fields from their decomposition graphs 5 at K and have length equal to td(k k). Equivalently, D K is a half-oriented connected graph having K = K 0 as origin and satisfying: (a) The vertices of D K are distinct function fields K i k over k. (b) For every vertex K i, the trivial valuation of K i is the only edge from K i to itself. And the set of non-trivial edges v i originating at K i is a geometric set of prime divisors of K i k, and if v i is a non-trivial edge from K i to K i, then K i = K i v i. (c) The only cycles of the graph are the non-oriented edges, and all the maximal branches consisting of oriented edges only have length equal to td(k k). The functorial behavior of geometric prime divisor graphs is as follows: (1) Embeddings. Let L l K k be an embedding of function fields which maps l onto k. Then the canonical restriction map of valuations Val K Val L, v v L, gives rise to a morphism of the total prime divisor graphs ϕ ı : DK tot D L tot, which moreover is surjective. The relation between geometric prime divisor graphs D K and D L is a little bit more subtle; see Proposition 37: Given geometric prime divisor graphs D K and D L, there exist geometric prime divisor graphs DK 0 and D L 0 containing D K, respectively D L, such that ϕ ı defines a surjective morphism of geometric prime divisor graphs: ϕ ı : D 0 K D 0 L. (2) Restrictions. Given a generalized prime divisor v of K k, let Dv tot generalized prime divisors w of K k with v w. Then the map D tot v D tot Kv, w w/v, be the set of all is an isomorphism of Dv tot onto DKv tot. Moreover, if Kv is a vertex of some geometric prime divisor graph D K for K k, then one has that the maximal subgraph D Kv of D K whose initial vertex is Kv is a geometric graph of prime divisors of Kv. Decomposition graphs (see Section 3 for more details) Let K k be as considered above. Then we have the following, see e.g., Pop [28], Introduction, for a discussion of these facts: For every prime divisor v of K k one has T v = Tl,K, and for every prime r-divisor v one has T v = T r l,k. Further, for generalized prime divisors v and w one has Z v Z w 1 if and only if v,w are not independent as valuations, i.e., O := O v O w K; and if so, then O is the valuation ring of a generalized prime divisor u of K k which turns out to be the unique generalized prime divisor with T u = T v T w, and also the unique generalized prime divisor of K k maximal with the property Z v,z w Z u. In particular, v = w iff T v = T w iff Z v = Z w. Further, v < w iff T v T w strictly iff Z v Z w strictly, and T w /T v = Z s r l if v is a prime r-divisor and w is a prime s-divisor. We conclude that the partial ordering of the set of all the generalized prime divisors v of K k is encoded in the set of their inertia/decomposition groups T v D v. In particular, the existence of the trivial, respectively a non-trivial, edge from Kv to Kw in DK tot is equivalent to T v = T w, respectively to T v T w and T w /T v = Zl.

6 Florian Pop Via the Galois correspondence and the functorial properties of the Hilbert decomposition theory for valuations, we attach to the total prime divisor graph DK tot of K k a graph G D tot whose vertices and edges are in bijection with those of D tot K K as follows: (a) The vertices of G D tot are Π Kv, viewed as distinct pro-l groups (all v). K b) If the edge from Kv to Kw exists, the corresponding edge from Π Kv to Π Kw is endowed with the pair of groups T w/v Z w/v viewed as subgroups of Π Kv ; thus Π Kw = Z w/v /T w/v. The graph G D tot will be called the total decomposition graph of K k, or of Π K K. If D K DK tot is a geometric graph of prime divisors of K k, the corresponding subgraph G DK G D tot will be called a geometric decomposition graph for K k, or for Π K K. Next recall that the isomorphy type of (the maximal abelian pro-l quotient of) the fundamental group Π 1 (X) := π ab,l 1 (X) of complete regular models X k, if such models exist, does depend on K k only, and not on X k. Moreover, one can recover Π 1 (X) as being Π 1 (X) = Π K /T K, where T K is the subgroup of G K generated by all the inertia groups T v with v a prime divisor of K k. This justifies calling the group Π 1,K := Π K /T K the birational fundamental group for K k. As discussed at Fact 57, there always exist quasi-projective normal models X k for K k such that T K = T DX, where T DX is the closed subgroup of Π K generated by all the T v with v D X. We will say that a model X k of K k and/or that D X is complete regular-like if T K = T DX and the rational rank rr ( Cl(X) ) of the divisor class group Cl(X) is positive, and for every normal quasi-projective model X with D X D X one has that rr ( Cl( X) ) = rr ( Cl(X) ) + D X \D X. Note that a complete regular like curve is a complete normal curve and vice-versa. We say that a geometric decomposition graph G DK is complete regular-like if for all vertices v of D K with td(kv k) > 0 one has that the set D v of 1-edges of G DKv k is complete regular-like. As shown in Proposition 22, there exists a group-theoretical recipe by which one can recover the geometric decomposition graphs (and the property of being complete regular-like) from the total decomposition graph G D tot. Further, by Proposition 39, that recipe is invariant under isomorphisms Φ : G D tot G K K DL tot, i.e., every such isomorphism gives rise by restriction to isomorphisms of the (complete regular-like) decomposition graphs for K k onto the (complete regular-like) ones for L l. The functorial properties of the graphs of prime divisors translate to the following functorial properties of the decomposition graphs: (1) Embeddings. Let ı : L l K k be an embedding of function fields which maps l onto k. Then the canonical projection homomorphism Φ ı : Π K Π L is an open homomorphism, and for every generalized prime divisor v of K k and its restriction v L to L one has that Φ ı (Z v ) Z vl is an open subgroup, and Φ ı (T v ) T vl satisfies Φ ı (T v ) = 1 iff v L is the trivial valuation. Therefore, Φ ı gives rise to a morphism of total decomposition graphs, which we denote by the same symbol Φ ı : G D tot K G D tot. L

Recovering function fields from their decomposition graphs 7 In turn, for given geometric decomposition graphs D K and D L, for which ı gives rise to a morphism of geometric decomposition graphs D K D L, the above Φ ı morphism of total decomposition graphs gives rise to a morphisms of geometric decomposition graphs Φ ı : G DK G DL, as defined later in Sections 4 and 5. (2) Restrictions. Given a generalized prime divisor v of K k, let pr v : Z v Π Kv be the canonical projection. Then for every w v we have that T w Z w are mapped onto T w/v Z w/v. Therefore, the total decomposition graph of Kv k can be recovered from that of K k in a canonical way via pr v : Z v Π Kv. Rational quotients (see Section 5 for more details) Let K k be a function field as above satisfying td(k k) > 1. For every non-constant function t K, let κ t be the relative algebraic closure of k(t) in K. Since td(κ t k) = 1, it follows that κ t has a unique complete normal model X t k, which is a projective smooth curve. Therefore, the set of prime divisors of κ t k is actually in bijection with the (local rings at the) closed points of X t, thus with the set of Weil prime divisors of X t. Therefore, the total prime divisor graph Dκ tot t for κ t k is actually the unique maximal geometric prime divisor graph for κ t k. We denote Dκ tot t simply by D κt. Let ı t : κ t K be the canonical embedding, and Φ κt : Π K Π κt the (surjective) canonical projection. Then by the functoriality of embeddings, Φ κt gives rise canonically to a morphism Φ κt : G D tot G κt. Moreover, if G DK is a geometric decompo- K sition graph for K k, then Φ κt restricts to a morphism of geometric decomposition graphs Φ κt : G DK G κt. In the above context, if κ t = k(t), we say that Φ κt is a rational quotient of G D tot K as well as of every geometric decomposition graph G DK for K k. We call such t K general elements of K; and usually denote general elements of K by x, in order to distinguish them from the usual non-constant t K. A birational Bertini-type argument shows that there are many general elements in K; see Lang [18], Ch. VIII, and/or Roquette [32], 4, respectively Fact 43 in Section 5: For any given algebraically independent functions t,t K, not both inseparable, t a,a := t/(a t + a) is a general element of K for almost all a,a k. A set of general elements Σ K is a Bertini set if Σ contains almost all elements t a,a for all t,t as above. We denote by A K = {Φ κx } κx the set of all the rational quotients of K k, and consider subsets A A K containing all the Φ κx A, x Σ, with Σ some Bertini set of general elements, and call them, for short, Bertini-type sets of rational quotients. The relation between rational projections and morphisms of geometric decomposition graphs is as follows: Let ı : L l K k be an embedding of function fields with ı(l) = k, such that K ı(l) a separable field extension, and td(l l) > 1. Then there exists a Bertini-type set B = {Φ κy } κy for L l such that κ x := ı(κ y ) is relatively algebraically closed in K for all κ y. Hence for all Φ κy B and the corresponding Φ κx A K, κ x := ı(κ y ), we get that the isomorphism Φ κx κ y : G κx G κy defined by ı κx κ y := ı κy satisfies the condition Φ κy Φ ı = Φ κx κ y Φ κx. Because of this property, we will say that Φ ı is compatible with rational quotients.

8 Florian Pop Abstract decomposition graphs It is one of our main tasks in the present manuscript to define and study abstract decomposition graphs, which resemble the geometric decomposition graphs G DK (this will be done in Section 2) and to define proper morphisms of such abstract decomposition graphs, in particular their rational quotients (which will be done in Section 4). The abstract decomposition graphs, which endowed with families of rational quotients resemble the complete regular-like geometric decomposition graphs as introduced above, will be called complete regular-like abstract decomposition graphs. The main result of this manuscript is the following; see Theorem 45 for a more general assertion, and Definition 21, Fact/Definition 43 (2), Definition 33 (and Definitions 12 and 9), and Definition/Remark 34 for the definitions of all the terms: Main Theorem. Let K k and L l be function fields with td(k k) > 1. Let G D tot K be their total decomposition graphs, which we endow with Bertini-type sets H D tot L of rational quotients A, respectively B. Then the following hold: (1) There exists a group-theoretical recipe which recovers K k from G D tot endowed K with A. Moreover, this recipe is invariant under isomorphisms in the following sense: Up to multiplication by l-adic units and composition with automorphisms Φ ı of G D tot defined by automorphisms ı : K i k K i k, there exists at most one isomorphism Φ : G D tot H K K D tot of abstract decomposition graphs which is compatible L with the sets of rational quotients A and B. (2) The following more precise assertion holds: Suppose that td(l l) > 1. Let G DK and H DL be geometric complete regular-like decomposition graphs for K k, which endowed with A, respectively B, are viewed as complete regular-like abstract decomposition graphs. Then for every morphism Φ : G DK H DL which is compatible with the sets of rational quotients A and B, there exist an l-adic unit ε Z l and an embedding of field extensions ı : L i l K i k such that Φ = ε Φ ı, where Φ ı : G DK H DL is the canonical morphism defined by ı as above. Further, ı(l) = k, and ı is unique up to Frobenius twists, and ε is unique up to multiplication by powers of p, where p = char(k). We notice that the main theorem above (together with Propositions 22 and 39) reduces the problem of functorially recovering K k from ΠK c, thus completing the proof of the target result above, to recovering the total decomposition graph G D tot K of K k and its rational quotients. In the case that k is an algebraic closure of a finite field, both these problems were solved in Pop [27], but working with the full pro-l Galois group G K (l) instead of ΠK c. Nevertheless, the methods of Pop [27] to recover and its rational quotients used only the set of all the divisorial groups T v Z v G D tot K and

Recovering function fields from their decomposition graphs 9 inside Π K. Using the local theory developed in Pop [28] instead of the local theory of Pop [27], we complete the proof of the target result above in Pop [30]. Historical note The idea to recover K k from ΠK c originates from Bogomolov [2], and a first attempt to do so can be found in his fundamental paper [2]. Although that paper is too sketchy to make clear what the author precisely proposes, a thorough inspection shows that it provides a fundamental tool for recovering inertia elements of valuations v of K (which nevertheless may be non-trivial on k). This is Bogomolov s theory of commuting liftable pairs; see Bogomolov Tschinkel [3] for detailed proofs. On the other hand, it is not at all clear how and whether one could develop a global theory along the lines (vaguely) suggested in [2], and there was virtually no progress on the problem for about a decade. A sketch of a viable global theory at least in the case that k is an algebraic closure of a finite field was proposed in the notes of my MSRI talk in the fall of 1999; see Pop [25]. In the second part of Pop [26], the technical details concerning the global theory hinted at in Pop [25] were worked out. Actually, the present manuscript is an elaboration of parts of Pop [26], and the main theorem here, more precisely Theorem 45, is the Hom-form of the Isom-form of Theorem 5.11 of [26]. However, I should mention that in [26] the mixed arithmetic + geometric situation was considered as well as non-abelian Galois groups, which was/is of interest in the case that k is not algebraically closed. In the case that k is an algebraic closure of a finite field, let me finally mention: In the manuscript Pop [27], a recipe to functorially recover K k from G K (l), in particular a proof of (a slightly stronger form of) the above target result was given. First, the assertion one proves using G K (l) instead of ΠK c is stronger, namely, if Φ : Π K Π L is the abelianization of an isomorphism Φ(l) : G K (l) G L (l), then there exists an isomorphism ı : L i l K i k (unique up to Frobenius twists) which defines Φ; thus one does not need to adjust Φ by multiplying by an l- adic unit ε Z l. I should also observe that the full G K(l) was used in loc.cit. essentially only in order to recover the divisorial subgroups of Π K via the canonical projection G K (l) Π K, whereas all the other steps of the local and global theory are virtually identical with the ones in the case of ΠK c. (The recipe to recover the divisorial subgroups of Π K via ΠK c Π K is given in Pop [28] and uses Bogomolov s theory of commuting liftable pairs as a black box. That recipe is used in Pop [30].) Bogomolov Tschinkel [4], [5], consider the case K = k(x), where X is a projective smooth surface over k. In the initial variant of their manuscript [4], they considered only the case that π 1 (X) is finite, and proved that if ΠK c and Π L c are isomorphic, then K k and L l are isomorphic up to pure inseparable closures, provided k and l are algebraic closures of finite fields with chat 2 (which is less precise than what the above target result gives in this case). Nevertheless, in the published version [5] of their earlier manuscript [4], they announce their main result for surfaces in a form almost identical with the target result above and use a strategy of

10 Florian Pop proof which is in many ways very similar to that announced in Pop [25], and used in Pop [27]. Thanks. First I would like to thank P. Deligne for several useful discussions we had during my visits to IAS Princeton. Actually, the theory of abstract decomposition graphs presented here is inspired by a suggestion of his (made in the letter [8] from September 1995) for an axiomatic approach to the birational anabelian conjecture in the arithmetical case. I also want to thank J.-L. Colliot-Thélène for several discussions we had at MSRI in the fall of 1999, as well as M. Saidi, T. Szamuely, and J. Stix for discussion sessions at Bonn and Penn. Finally, I would like to thank a few others for their interest in this work, including D. Harbater, J. Koenigsmann, P. Lochak, J. Mínáč, H. Nakamura, and A. Tamagawa. It is a great honor for me to to contribute to this volume in memory of Serge Lang. The present manuscript is an expanded version of a manuscript initially submitted at the beginning of 2007. Acknowledgments: The author was supported by NSF grants DMS-0401056 and DMS-0801144. 2 Pro-l abstract decomposition graphs In this section we develop an abelian pro-l prime divisor decomposition theory for abstract function fields which is similar in some sense to the abstract class field theory. Throughout l is a fixed prime number, and δ 0 is a non-negative integer. 2.1 Axioms and definitions Definition 1 A level-δ (pro-l) abstract decomposition graph is a connected halforiented graph G whose vertices are endowed with pro-l abelian groups G i and whose edges v i are endowed with pairs of pro-l abelian groups T vi Z vi satisfying the following: Axiom I): The vertices of G are pro-l abelian free groups G i, and G has an origin, which we denote by G 0 = G. Axiom II): The edges v i and the corresponding T vi Z vi satisfy the following: (i) For every vertex G i there exists a unique non-oriented edge v i0 from G i to itself, and the corresponding pair of pro-l groups is {1} =: T vi0 Z vi0 := G i. For all other vertices G i G i there exists at most one edge v i from G i to G i. If v i exists, we say that v i is the oriented edge from G i to G i, and v i is endowed with a pair T vi Z vi of subgroups of G i such that T vi = Zl and G i = Z vi /T vi. The edges of G are also called valuations of G; in particular, the edges originating from G i are called valuations of G i. The non-oriented edge v i0 from G i to itself is

Recovering function fields from their decomposition graphs 11 called the trivial valuation of G i, whereas the oriented edges v i originating from G i are called non-trivial valuations of G i. The groups T vi Z vi are called the inertia, respectively decomposition, groups of v i ; and G i := Z vi /T vi is called the residue group of v i. (ii) For distinct non-trivial edges v i v i originating from G i, one has Z vi Z vi = 1, hence T vi T vi = 1 holds as well. For every cofinite subset U i of the set of non-trivial edges v i originating from G i, let T Ui be the closed subgroup of G i generated by all the T vi, v i U i. A system (U i,α ) α of such cofinite subsets is called cofinal, if every finite set of valuations v i as above is contained in the complement of U i,α for some α. (iii) There exist cofinal systems (U i,α ) α such that T vi T Ui,α = 1 for all α and all v i U i,α. Axiom III): The non-oriented edges v i0 are the only cycles of the graph G, and all maximal branches of non-trivial edges of G have length equal to δ. Definition/Remark 2 Let G be an abstract decomposition graph of level-δ G on a pro-l group G = G 0. We will say that G is a level-δ G abstract decomposition graph on G. A valuation of G = G 0 will be called a 1-edge of G. If no confusion is possible, we will denote the 1-edges of G simply by v; thus the corresponding pro-l groups involved are denoted by T v Z v and G v := Z v /T v. (1) Consider any δ such that 0 δ δ G. By induction on δ it is easy to see that G has a unique maximal connected abstract decomposition subgraph containing the origin G of G and having all branches of oriented edges of length δ. (2) Let v = (v r,...,v 1 ) be a path of length δ v := r of non-trivial valuations originating at G = G 0. This means by definition that v 1 is a non-trivial valuation of G 0, and if r > 1, then for all i < r one has inductively that G i is the residue group of v i, and v i+1 is a non-trivial valuation of G i. In particular, G r is the residue group of v r. Then there exists a unique maximal connected subgraph G v of G having G v := G r as origin. Clearly, G v is in a natural way an abstract decomposition graph of level δ G δ v on G v. We say that G v is an r-residual abstract decomposition graph of G. In particular, the unique 0-residual abstract decomposition graph of G is G itself. (3) For every path v = (v r,...,v 1 ) of length δ v = r as above, we will say that G v is an r-residual group of G, precisely that G v is the v-residual group of G. One can further elaborate here as follows: For r > 1 we set w = (v r 1,...,v 1 ), and suppose that the inertia/decomposition groups T w Z w G 0 of w have been defined inductively such that the residue group G w := Z w /T w of w is G w = G vr 1. We then define the inertia/decomposition groups T v Z v of v in G 0 as being the preimages of T vr Z vr G vr 1 via Z w Z w /T w = G vr 1. Note that by definition we have Z v /T v =: G v and T v = Z δ v l. We call v = (v r,...,v 1 ) a generalized valuation of G = G 0, or a multi-index of length δ v := r of G. And we will say that δ v is the rank of v or that v is a generalized r-valuation if r = δ v.

12 Florian Pop Given generalized valuations v = (v r,...,v 1 ), w = (w s,...,w 1 ), we will say that w v if s r, and v i = w i for all i s. From the definitions one gets that if w v, then Z v Z w and T w T v. On the other hand, by Axiom II (ii), it immediately follows that the converse of (any of) these assertions is also true. We will say that v and w are dependent if there exists some q > 0 such that v i = w i for i q. For dependent generalized valuations v and w as above, the following are equivalent: (a) q is maximal such that v i = w i for i q. (b) T v T w = Z q l. (c) q is maximal such that Z v,z w are both contained in the decomposition group of some generalized q-valuation of G = G 0. (4) In order to have a uniform notation, we take v = v 0 to be the trivial multiindex, or the trivial path, of G as the unique one having length equal to 0. We further set Z v0 := G 0 and T v0 = {1}. In particular, one has G v0 = Z v0 /T v0 = G 0, which is compatible with the other notations/conventions. Further, v 0 v for all multi-indices v. Definition/Remark 3 Let G be a level-δ G abstract decomposition graph on the abelian pro-l group G = G 0. In notation as above, we consider the following: (1) Define Λ G := Hom ( ) G,Z l. Since G is a pro-l free abelian group, ΛG is a free l-adically complete Z l -module (in l-adic duality with G). From now on suppose that δ G > 0. Recall that T v Z v and G v = Z v /T v denote respectively the inertia, the decomposition, and the residue groups at the 1-edges v of G, i.e., at the valuations v of G. (2) Denote by T G the closed subgroup generated by all the inertia groups T v (all v as above). We set Π 1,G := G/T and call it the abstract fundamental group of G. One has a canonical exact sequence 1 T G Π 1,G 1. Taking continuous Z l -Homs, we get an exact sequence of the form 0 Û G := Hom ( ) Π 1,G,Z can l Λ G := Hom ( ) j G G,Z l Λ T := Hom ( ) T,Z l. We will call Û G := Hom ( Π 1,G,Z l ) the unramified part of ΛG. And if no confusion is possible, we will identify Û G with its image in Λ G. (3) Next we have a closer look at the structure of Λ G. For every 1-edge v as above, the inclusions T v Z v G give rise to restriction homomorphisms as follows: j v : Λ res Zv G ΛZv := Hom ( ) resv Z v,z l ΛTv := Hom ( ) T v,z l. (a) We set Û 1 v = ker(res Zv ) and Û v = ker( j v ) and call them the principal v-units, respectively the v-units, in Λ G. And observe that the unramified part of Λ G is exactly Û G = v ker( j v ). (b) The family ( j v ) v gives rise canonically to a continuous homomorphism v j v of l-adically complete Z l -modules v j v : Λ G Λ T v ΛTv

Recovering function fields from their decomposition graphs 13 Thus identifying Λ T with its image inside v ΛTv, one has j G = v j v on Λ G. We define Div G := v ΛTv and call it the l-adic abstract divisor group of G. (c) Finally, we set Ĉl G = coker( j G ) and call it the l-adic abstract divisor class group of G. And observe that we have a canonical exact sequence 0 Û G Λ G j G Div G can Ĉl G 0. 4) Let Λ G,fin := {x Λ G j v (x) = 0 for almost all v}. We notice that by Axiom II (iii), the Z l -module Λ G,fin is dense in Λ G. Indeed, let (U α ) α be a cofinal system of 1-edges v. Then setting G α = G/T Uα and T α = T /T Uα, we have a canonical exact sequence 1 T α G α Π 1,G 1, and T α is generated by the images T v,α of T v (all v U α ) in G α. Clearly, the image of the inflation homomorphism inf α : Hom ( G α,z l ) Hom ( G,Zl ) is exactly α := {x Λ G j v (x) = 0 for all v U α } = v Uα ker( j v ). Taking inductive limits over the cofinal system (U α ) α, the density assertion follows. We observe that j G ( Λ G,fin ) = Λ G /Û G is a Z l -submodule of the Z l -free module v ΛTv = v Z l v; hence j G ( Λ G,fin ) is a free Z l -module too. Therefore, for every Z l - submodule Λ G, its image j G ( ) under j G is a free Z l -module. The rank of j G ( ) will be called the corank of. We notice that a Z l -submodule Λ G has finite corank iff is contained in ker( j v ) for almost all v. Clearly, the sum of two finite corank submodules of Λ G is again of finite corank. Thus the set of such submodules is inductive, and one has Λ G,fin = (all finite corank ) = α α. (5) We say that G is complete curve-like if the following holds: There exist generators τ v of T v such that v τ v = 1, and this is the only pro-relation satisfied by the system of elements T = (τ v ) v. We call such a system T = (τ v ) v a distinguished system of inertia generators. We notice the following: Let G be complete curve-like, and let T = (τ v ) v and T = (τ v) v be distinguished systems of inertia generators. Then τ v = τ ε v v for some l-adic units ε v Z l, because both τ v and τ v are generators of T v. Hence we have 1 = v τ v = v τ ε v v. By the uniqueness of the relation v τ v = 1, it follows that ε v = ε for some fixed l-adic unit ε Z l. Next consider some δ with 0 < δ δ G. We say that G is level-δ complete curvelike if all the (δ 1)-residual abstract decomposition graphs G v are residually complete curve-like. In particular, level 1 complete curve-like is the same as complete curve-like. (6) For every 1-vertex v consider the exact sequence 1 T v Z v G v 1 given by Axiom II (i). Let inf v : Hom ( G v,z l ) Hom ( Zv,Z l ) be the resulting inflation

14 Florian Pop homomorphism. Since T v = ker(z v G v ), it follows that res Zv (Û v ) is the image of the inflation map infl v. Therefore there exists a canonical exact sequence 0 Ûv 1 j v Û v Hom ( ) G v,z l = ΛGv 0, and we call j v the v-reduction homomorphism. (7) In particular, if δ G > 1, then δ Gv = δ G 1 > 0 for every 1-vertex v, and we have the corresponding exact sequence for the residual abstract decomposition graph G v 0 Û Gv Λ Gv j Gv Div Gv. We will say that G is ample if δ G > 0 and the following conditions are satisfied: (i) j Σ : Λ G v Σ Λ Tv is surjective for every finite set Σ, where j Σ := v Σ j v. (ii) If δ G > 1, then the following hold: (a) j v (Û G ) Û Gv and Û Gv + j v ( Λ G,fin Û v ) = Λ Gv,fin for every v. (b) For every finite-corank submodule Λ G, there exists v such that Û v, and and j v ( ) have equal coranks. Notice that the condition (ii) above is empty in the case δ G = 1. Thus if δ G = 1, then condition (i) is necessary and sufficient for G to be ample. Next consider 0 < δ δ G. We say that G is ample up to level δ if all the residual abstract decomposition graphs G v for v such that 0 δ v < δ are ample. In particular, ample up to level 1 is the same as ample. 2.2 Abstract Z (l) divisor groups Definition 4 (1) Let M be the l-adic completion of a free Z-module. A Z (l) - submodule M (l) M of M is called a Z (l) -lattice in M (for short, a lattice) if M (l) is a free Z (l) -module, it is l-adically dense in M, and it satisfies the following equivalent conditions: (a) M/l = M (l) /l (b) M (l) has a Z (l) -basis B which is l-adically independent in M. (c) Every Z (l) -basis of M (l) is l-adically independent in M. (2) Let N M (l) M be Z (l) -submodules of M such that N and M/N are l- adically complete and torsion-free. We call M (l) an N-lattice in M, if M (l) /N is a lattice in M/N. (3) In the context above, a true lattice in M is a free abelian subgroup M of M such that M (l) := M Z (l) is a lattice in M in the above sense. And we will say that a Z-submodule M M is a true N-lattice in M if N M and M /N is a true lattice in M/N. (4) Let M be an arbitrary Z l -module. We say that subsets M 1,M 2 of M are l- adically equivalent if there exists an l-adic unit ε Z l such that M 2 = ε M 1 inside M. Further, given systems S 1 = (x i ) i and S 2 = (y i ) i of elements of M, we will say

Recovering function fields from their decomposition graphs 15 that S 1 and S 2 are l-adically equivalent if there exists an l-adic unit ε Z l such that x i = ε y i (all i). (5) We define correspondingly the l-adic N-equivalence of N-lattices, etc. Construction 5 Let G be an abstract decomposition graph on G which is level-δ complete curve-like and ample up to level δ for some given δ > 0. Recall the last exact sequence from point (4) from Definition/Remark 3: 0 Û G Λ G j G Div G can Ĉl G 0. The aim of this subsection is to describe the l-adic equivalence class of a lattice Div G in Div G, in case it exists, which will be called an abstract divisor group of G. In case the lattice Div G Div G exists, it satisfies Div G Z l = v Λ Tv. Further, the existence (of the equivalence class) of the lattice Div G will turn out to be equivalent to the existence (of the equivalence class) of a Û G -lattice Λ G in Λ G, which will turn out to be the preimage of Div G in Λ G. In particular, if Λ G exists, it satisfies Λ G Z l = Λ G,fin. The case δ = 1, i.e., G is complete curve-like and ample. In the notation from Definition/Remark 3 (5) above, let T = (τ v ) v be a distinguished system of inertia generators. Further, let F T be the abelian pro-l free group on the system T (written multiplicatively). Then one has a canonical exact sequence of pro-l groups 1 τ Z l F T T 1, where τ = v τ v in F T is the pro-l product of the generators τ v (all v). Observing that Hom ( F T,Z l ) = DivG in a canonical way, and taking l-adically continuous Homs, we get an exact sequence 0 Λ T = Hom ( T,Z l ) DivG = Hom ( F T,Z l ) Zl = Hom ( τ Z l,z l ) 0, where the last homomorphism maps each ϕ to its trace : ϕ ( τ v ϕ(τ v ) ). Thus Λ T consists of all the homomorphisms ϕ Hom ( F T,Z l ) with trivial trace. Consider the system B = (ϕ v ) v of all the functionals ϕ v Hom ( F T,Z l ) = DivG defined by ϕ v (τ w ) = 1 if v = w, and ϕ v (τ w ) = 0 for all v w. We denote by Div T = B (l) Div G the Z (l) -submodule of Hom ( ) F T,Z l = DivG generated by B. Then Div T is a lattice in Div G, and B is an l-adic basis of Div G. We next set Div 0 T := { v a v ϕ v Div T v a v = 0} = Div T Λ T.

16 Florian Pop Clearly, Div 0 T is a lattice in Λ T. And moreover, the system (e w = ϕ w ϕ v ) w v is an l-adic Z (l) -basis of Div 0 T for every fixed v. The dependence of Div T on T = (τ v ) v is as follows. Let T = (τ v) v = T ε with ε Z l be another distinguished system of inertia generators. If B = (ϕ v) v is the dual basis to T, then ε B = B. Thus B and B are l-adically equivalent, and we have Div T = ε Div T and Div 0 T = ε Div0 T. Therefore, all the subgroups of Div G of the form Div T, respectively Div 0 T, are l-adically equivalent (for all distinguished T). Hence the l-adic equivalence classes of Div T and Div 0 T do not depend on T, but only on G. Fact 6 In the above context, denote by Λ T the preimage of Div 0 T, thus of Div T, in Λ G. Consider all the finite-corank submodules Λ G,fin with Û G. Then the following hold: (i) Λ T is a Û G -lattice in Λ G, and Λ T Λ G,fin. (ii) Λ T is a Û G -lattice in (all as above). Moreover, j v (Λ T ) = Z (l) ϕ v (all v). Proof. Clear. Definition 7 In the context of Fact 6 above, we define objects as follows: (1) A lattice of the form Div T Div G will be called an abstract divisor group of G. We will further say that Div 0 T is the abstract divisor group of degree 0 in Div T. (2) The Û G -lattice Λ T is called a divisorial Û G -lattice for G in Λ G. And we will say that Λ T and Div T correspond to each other, and that T defines them. Note that Λ G Λ G,fin and Λ G Z l = Λ G,fin. Indeed, if x Λ G, then j v (x) = 0 for almost all v, etc. The case δ > 1. We begin by mimicking the construction from the case δ = 1, and then conclude the construction by induction on δ. Thus let T = (τ v ) v be any system of generators for the inertia groups T v (all 1-edges v). Further let F T be the abelian pro-l free group on the system T (written multiplicatively). Then T is a quotient F T T 1 in a canonical way. Observing that Hom ( F T,Z l ) = DivG in a canonical way, by taking l-adic Homs we get an exact sequence 0 Hom ( T,Z l ) Hom ( FT,Z l ) = DivG. Next let B = (ϕ v ) v be the system of all the functionals ϕ v Hom ( F T,Z l ) defined by ϕ v (τ w ) = 1 if v = w, and ϕ v (τ w ) = 0 for all v w. We denote by Div T = B (l) Hom ( F T,Z l ) the Z (l) -submodule generated by B. Then B is an l-adic basis of Hom ( F T,Z l ), i.e., Div T is l-adically dense in Div G = Hom ( F T,Z l ), and there are no non-trivial

Recovering function fields from their decomposition graphs 17 l-adic relations between the elements of B. We will call B = (ϕ v ) v the dual basis to T, and remark that Div T is a lattice in Hom ( T,Z l ). Finally, let T = (τ v) v be another system of inertia generators, and suppose that T = T ε for some ε Z l. If B = (ϕ v) v is the dual basis to T, then ε ϕ v = ϕ v inside Hom ( T,Z l ). Thus ε B = B. In other words, B and B are l-adically equivalent, and we have Div T = ε Div T. Fact 8 In the notations from above let a Û Gv -lattice Λ Gv Λ Gv with Û Gv Λ Gv be given for every valuation v of G. Then the following hold: (1) Up to l-adic equivalence, there exists at most one Û G -lattice Λ G in Λ G such that first, Û G Λ G Λ G,fin, and second, for every finite-corank submodule of Λ G,fin with Û G and the corresponding v := j v ( Û v ) +Û Gv Λ Gv,fin the following hold: (i) Λ := Λ G is a Û G -lattice in. (ii) j v (Λ Û v ) +Û Gv is a Û Gv -lattice in v, which is l-adically Û Gv -equivalent to Λ Gv v. Moreover, if the Û G -lattice Λ G exists, then its l-adic equivalence class depends only on the l-adic equivalence classes of the Û Gv -lattices Λ Gv (all v). (2) In the above context, suppose that G is ample, and that the Û G -lattice Λ G satisfying the conditions (i), (ii), exists. Then Û Gv + j v (Λ G Û v ) is a Û Gv -lattice, which moreover is l-adically Û Gv -equivalent to Λ Gv (all v). Proof. To (1): Let Λ G,Λ G be Û G -lattices in Λ G satisfying the conditions from (1) above. Let Λ G,fin be have finite non-zero corank, and satisfy Û G. By the ampleness of G, it follows that there exists v such that, first, Û v, and second, and v := j v ( )+Û Gv have equal coranks. Therefore, j v defines an isomorphism of /Û G onto v /Û Gv, and one has ( ) ker( j v ) Û G, j v ( ) Û Gv j v (Û G ). For as above, set Λ = Λ G. Then by hypothesis (i), it follows that Λ and Λ are both Û G -lattices in. Further, by hypothesis (ii), both Λ v := Û Gv + j v (Λ ) and Λ v := Û Gv + j v (Λ ) are Û Gv lattices in v, which are both equivalent to the Û Gv -lattice Λ Gv v. Therefore, there exists ε Z l such that Λ v = ε Λ v. Claim. Λ = ε Λ. Indeed, Λ v = ε Λ v implies that j v (Λ ) ε j v(λ )+Û Gv. Hence for every e Λ there exist e Λ and u v Û Gv such that j v (e ) = ε j v (e) + u v. Therefore we have u v = j v (e ε e) j( ), and hence u v j v ( ) Û Gv. Hence by assertion ( ) above, there exists u Û G such that j v (u) = u v ; thus j v (u) = j v (e ε e). But then we have e (ε e + u) ker( j v ), thus e (ε e + u) Û G by assertion ( ). We conclude that e ε e + Û G. Since e Λ was arbitrary, we have Λ ε Λ + Û G. On the other hand, by hypothesis we have Û G Λ and Û G Λ. Hence the above

18 Florian Pop inclusion is actually equivalent to Λ ε Λ. By symmetry, the other inclusion also holds, and we finally get Λ = ε Λ. We also observe that ε is unique up to multiplication by rational l-adic units, because Λ /Û G = ε Λ /Û G are l-adically equivalent lattices in the non-trivial Z l - module /Û G. Hence recalling that Λ G = Λ and Λ G = Λ, and taking into account the uniqueness of ε, one immediately gets that Λ G = ε Λ G, as claimed. To (2): First, since Λ G = Λ as mentioned above, it follows from hypotheses (i), (ii), that Û Gv + j v (Λ G Û v ) is l-adically equivalent to some Û Gv -sublattice of Λ Gv, as this is the case for all the Û Gv + j v (Λ Û v ). After replacing Λ Gv by some properly chosen l-adic multiple, say ε Λ Gv with ε Z l, without loss of generality, we can suppose that j v (Λ G Û v ) Λ Gv, and thus Û Gv + j v (Λ G Û v ) Λ Gv. For the converse inclusion, let Γ Λ Gv be a finite-corank submodule. Then by the ampleness of G, see Definition/Remark 3 (7) (ii), there exists a finite-corank submodule Λ G such that Γ Û Gv + j v ( Û v ). But then by properties (i), (ii), we get Γ Λ Gv Û Gv + j v (Λ Û v ) Û Gv + j v (Λ G Û v ). Since Γ was arbitrary and Λ Gv = Û Gv + Γ (Γ Λ Gv ), the converse inclusion follows. Let G be an abstract decomposition graph which is both level-δ complete curvelike and ample up to level δ for some δ > 1. In particular, all residual abstract decomposition graphs G v to non-trivial indices v of length δ v < δ are both level- (δ δ v ) complete curve-like and ample up to level (δ δ v ); and if δ v = δ 1, then G v is complete curve-like and ample. Hence if δ v = δ 1, then G v has an abstract divisor group Div Gv as defined/introduced in Definition 7. In the above context, let us fix notation as follows: Definition 9 In the above context, we define an abstract divisor group of G (if it exists) to be the lattice defined by any system T of inertia generators as above, Div G := Div T Div G, which together with its preimage Λ G in Λ G satisfies inductively on δ the following: (i) Abstract divisor groups Div Gv exist for all residual abstract decomposition graphs G v. Let Λ Gv be the preimage of Div Gv in Λ Gv (all v). (ii) Λ G satisfies conditions (i), (ii) from Fact 8 for all finite corank submodules Λ G with respect to the preimages Λ Gv defined at (i) above. Note that if Λ G exists, then Λ G Λ G,fin and Λ G Z l = Λ G,fin. Indeed, if x Λ G, then j v (x) = 0 for almost all v, etc. Remarks 10 Let G be an abstract decomposition graph which is level-δ complete curve-like and ample up to level δ for some δ > 0. Suppose that an abstract divisor group Div G := Div T for G exists, and let Λ G be its preimage in Λ G. Then one has: (1) The homomorphism j v : Λ G = Hom ( ) resv ( ) G,Z l Hom Tv,Z l = Zl ϕ v gives rise by restriction to a surjective homomorphism j v : Λ G Z (l) ϕ v.