THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM

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1 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM JOEL SPECTER Abstract. Let be a continuous ring endomorphism o Z p x /Z p o degree p. We prove that i acts on the tangent space at 0 by a uniormizer and commutes with an automorphism o ininite order, then it is necessarily an endomorphism o a ormal group over Z p. The proo relies on inding a stable embedding o Z p x in Fontaine s crystalline period ring with the property that appears in the monoid o endomorphisms generated by the Galois group o Q p and crystalline Frobenius. Our result veriies, over Z p, the height one case o a conjecture by Lubin. 1. Introduction Given a one-dimensional ormal group law F/Z p, the endomorphism ring o F provides an example o a nontrivial amily o power series over Z p which commute under composition. This paper investigates the question o the converse: under what conditions do amilies (in our case pairs) o commuting power series arise as endomorphisms o an integral ormal group? Let p be a prime number, C p be the completion o an algebraic closure o Q p, and m Cp C p be the open unit disk. Deine S 0 (Z p ) to be the set o ormal power series over Z p without constant term. 1 Substitution deines a composition law on S 0 (Z p ) under which it is isomorphic to the monoid o endomorphisms o the ormal p-adic unit disk which ix 0. Main Theorem. Let and u be a commuting pair o elements in S 0 (Z p ). I (0) is prime in Z p and has exactly p roots in m Cp, u is invertible and has ininite order, then there exists a unique ormal group law F/Z p such that, u End Zp (F ). The ormal group law F is isomorphic to Ĝm over the ring o integers o the maximal unramiied extension o Q p. The study o analytic endomorphisms o the p-adic disk was initiated by Lubin in [Lub94]. There he showed that i S 0 (Z p ) is a non-invertible transormation which is nonzero modulo p and u S 0 (Z p ) is an invertible, non-torsion transormation, then assuming and u commute: (1) The number o roots o in m Cp is a power o p (c. [Lub94, Main Theorem 6.3, p. 343]) Mathematics Subject Classiication. 11S20, 11S31, 11S82 (Primary), 14L05, 13F25, 14F30 (Secondary). Key words and phrases. p-adic dynamical system, p-adic Hodge Theory, Lubin-Tate Group. The author was supported in part by National Science Foundation Grant DMS and by a National Science Foundation Graduate Research Fellowship under Grant No. DGE In general, or any ring R, we deine S0(R) to be the set ormal o power series over R without constant term. 1

2 2 JOEL SPECTER (2) The reduction o modulo p is o the orm a(x ph ) where a S 0 (F p ) is invertible (c. [Lub94, Corollary 6.2.1, p. 343]). 2 Both (1) and (2) are well known properties o a non-invertible endomorphism o a ormal group law over Z p. In light o this, and knowing: (3) I a power series u S 0 (Q p ) acts on the tangent space at 0 as an automorphism with ininite order, then it is an automorphism o a unique one dimensional ormal group law F u /Q p. I S 0 (Q p ) commutes with u then End Qp (F u ) (c. [Lub94, Theorem 1.3, p. 327]), one might preliminarily conjecture that the ormal group F u associated to u is deined over Z p. This is alse and counterexamples have been constructed by Lubin [Lub94, p. 344] and Li [Li02, p ]. Instead, Lubin hypothesizes that or an invertible series to commute with a non-invertible series, there must be a ormal group somehow in the background [Lub94, p. 341]. Interestingly, all known counterexamples are constructed rom the initial data o a ormal group deined over a inite extension o Z p and thereore conorm to Lubin s philosophy. Despite this, the somehow in Lubin s statement has yet to be made precise. Result (2) is reminiscent o a hypothesis in the ollowing integrality criterion o Lubin and Tate: (4) Let S 0 (Z p ). Assume (0) is prime in Z p and (x) x ph mod p, then there is a unique ormal group law F /Z p such that End Zp (F ) (c. [LT65, Lemma 1, p ]). Perhaps based on this and the counterexamples o [Li02] and [Lub94], Lubin has oered the ollowing conjecture as to when our preliminary intuition should hold true: Lubin s Conjecture. [Sar05, p. 131] Suppose that and u are a non-invertible and a nontorsion invertible series, respectively, deined over the ring o integers O o a inite extension o Q p. Suppose urther that the roots o and all o its iterates are simple, and that (0) is a uniormizer in Z p. I u = u then, u End O (F ) or some ormal group law F/O. For each integral extension o Z p, Lubin s conjecture is naturally divided into cases: one or each possible height o the commuting pair and u. This note serves as a complete solution to the height one case o Lubin s conjecture over Z p. Previous results towards Lubin s conjecture ([LMS02], [Sar10], [SS13]) have used the ield o norms equivalence to prove some special classes o height one commuting pairs over Z p arise as endomorphisms o integral ormal groups. All other cases o Lubin s conjecture, i.e. those over proper integral extensions o Z p or over Z p with height greater than one, are completely open. We call a ormal group Lubin-Tate (over Z p ) i it admits an integral endomorphism satisying the hypotheses o (4). All height one ormal groups over Z p are Lubin-Tate. Thereore, given the height one commuting pair (, u), one strategy to prove that the ormal group F u is integral is to identiy the Lubin-Tate endomorphism o F u and appeal to result (4). This is the approach we employ in this paper. Beore we present an outline o the proo, we remind the reader o some structures which exist in the presence o a ormal group. Let F be a height one ormal group over Z p. Then F is Lubin-Tate, so there exists a unique endomorphism o F such that (x) x p mod p. Q p. 2 The analogous statements both hold when Zp is replaced by the ring o integers in a inite extension o

3 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 3 Consider the -adic Tate module T F. The absolute Galois group o Q p acts on T F through a character χ F : G Qp Z p. The character χ F is crystalline. We sketch a proo o this act ollowing [KR09]. Let O Cp be the integral closure o Z p in C p and Ẽ+ := lim O Cp /p, where the limit is taken with respect to the p-power Frobenius map. Because (x) x p mod p, elements o T are canonically identiied with elements o Ẽ+. Let v be a generator o T. There is a unique element v Ã+ := W (Ẽ+ ), the Witt vectors o Ẽ+, which lits v and with the property that the action o G Qp and the Frobenius endomorphism ϕ satisy: (A) g( v) = [χ F (g)] F ( v) or all g G Qp, (B) ϕ( v) = ( v) [KR09, Lemma 1.2]. The logarithm o F evaluated at v converges in Fontaine s crystalline period ring B cris. The resulting period log F ( v) spans a G Qp ϕ -stable Q p -line on which G Qp acts through χ F. The element v B cris is transcendental over Z p and thereore property (B) implies that the Lubin-Tate endomorphism can be recovered rom the action o Frobenius on v. Inspired by this observation, we construct in this note an appropriate substitute or v which is directly accessible rom the commuting pair (, u). The argument is as ollows. In section 2.1, we recall some preliminary acts concerning the commutative height one p-adic dynamical system (, u). In particular, we recall the logarithm, log, o the commuting pair. Next, in section 2.2, we attach to (, u) a character χ : G K u (0) Zp Z p, where K is some particular inite extension o Q p. The character χ is (a posteriori) a restriction o the character attached to the Tate module o the ormal group or which and u are endomorphisms. Then, in section 3, we show the character χ is crystalline o weight one. This is achieved by constructing an element x 0 Ã+ such that [χ (g)] (x 0 ) = g(x 0 ), where [χ (g)] is the unique Z p -iterate o u with linear term χ (g). We show the logarithm converges in A cris when evaluated at x 0 and the resulting period t := log (x 0 ) generates a G K -stable Q p -line V o exact iltration one. Multiplicity one guarantees V is stable under crystalline Frobenius. Taking π to denote the crystalline Frobenius eigenvalue on V, we show [π ], the multiplication by π endomorphism o F u, converges when evaluated at x 0 and satisies [π ] (x 0 ) = ϕ(x 0 ). The proo is concluded by showing [π ] is a Lubin-Tate endomorphism o F u and thereore F u is deined over Z p by (4). Remark 1.1. There is recent work by Berger [Ber14a] which uses methods similar to ours to study certain iterate extensions. Berger s work is related to the phenomena which occur when the hypotheses o Lubin and Tate s result (4) are weakened and one assumes merely that is a lit o Frobenius i.e. (0) is not assumed to be prime. This paper can be seen as an orthogonal generalization o Lubin and Tate s hypotheses. 2. The p-adic Dynamical System 2.1. The Logarithm and Roots. Throughout this paper, we will assume, u S 0 (Z p ) are a ixed pair o power series satisying: Assumption 2.1. Assume: u = u, u is invertible and has ininite order, and v p ( (0)) = 1 and has exactly p roots in m Cp.

4 4 JOEL SPECTER The goal o this section is to remind the reader o some o the properties o the ollowing two structures associated by Lubin (in [Lub94]) to the commuting pair (, u). The irst is the set o -preiterates o 0: Λ := {π m Cp : n (π) = 0 or some n Z + }. The second is the logarithm o ; this is the unique series log S 0 (Q p ) such that log (0) = 1 and log ((x)) = (0) log (x). In the context that and u are endomorphisms o a ormal group F/Z p, the set Λ is the set o p -torsion points o F and log is the logarithm o F. One essential tool or understanding a p-adic power series and its roots is its Newton polygon. Given a series, g(x) := a i x i C p x, i=0 the Newton polygon N (g) o g is the convex hull in the (v, w)-cartesian plane, R 2, o the set o vertical rays extending upwards rom the points (i, v p (a i )). Let pr 1 : R 2 R denote the projection map to the irst coordinate. The boundary o N (g) is the image o an almost everywhere deined piecewise linear unction B g : pr 1 (N (g)) R 2. The derivative o B g is almost everywhere deined and increasing. The points o the boundary o N (g) where the slope jumps are called the vertices o N (g), whereas the maximal connected components o the boundary o N (g) where the slope is constant are called the segments. The width o a segment is the length o its image under pr 1. The ollowing data can be read o rom the Newton polygon o g : Theorem 2.2. The radius o the maximal open disk in C p centered at 0 on which g converges is equal to lim p dbg dv. v Theorem 2.3. The series g has a root in its maximal open disk o convergence o valuation λ i and only i the Newton polygon o g has a segment o slope λ o inite, positive width. The width o this slope is equal to the number o roots o g (counting multiplicity) o valuation λ. Theorem 2.4 (Weierstrass Preparation Theorem). I the series g is deined over a complete extension L/Q p contained in C p and the Newton polygon o g has a segment o slope λ o inite, positive width then g = g 1 g 2 where g 1 is a monic polynomial over L and g 2 L x, such that the roots o g 1 are exactly the roots o g (with equal multiplicity) o valuation λ. We call g 1 the Weierstrass polynomial corresponding to the slope λ. For proos o these statements we reer the reader to [Kob77]. In this paper, we will be concerned with roots o power series in m Cp and thereore ocus on segments o Newton polygons o negative slope. Proposition 2.5. The endpoints o the segments o the Newton polygon o n which have negative slope are (p k, n k) or each integer k with 0 k n. Proo. We prove the claim by induction on n. We begin by proving the claim in the case that n = 1. Because (0) = 0 and (0) 0, the Newton polygon o has a vertex at (1, v p ( (0))) = (1, 1). Since was assumed to have p roots in m Cp, the initial inite slope o N () must be negative. Furthermore, because is deined over Z p, the Newton polygon

5 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 5 o is contained in the quadrant R 2 0 and all vertices have integer coordinates. Thereore, the end o the initial segment o N () must have a vertex on the v-axis. By the Weierstrass Preparation Theorem, the nonzero roots o corresponding to this segment satisy an Eisenstein polynomial and as such they are simple. Since has exactly p roots in m Cp, the Newton polygon o has a vertex at (p, 0) and all nonzero roots in m Cp are accounted or by the initial inite segment o N (). Let n 1. The base case o our inductive argument already tells us a lot about n+1. For example because N () has (p, 0) has a vertex on the v-axis, where a p F p. It ollows (x) a p x p mod (p, x p+1 ), n+1 (x) a n+1 p x pn+1 mod (p, x pn+1 +1 ). Thereore, N ( n+1 ) has a vertex at (p n+1, 0). Similarly, because N () has a vertex at (1, 1) and has a ixed point at 0, the Newton polygon N ( n+1 ) has a vertex at (1, n + 1). The v coordinate o a vertex abutting any inite, negatively sloped segment o N ( n+1 ) must occur between 1 and p n+1. Now assume we have shown the claim or n. The roots o n+1 in m Cp are either roots o n or roots o n (x) π where π is one o the p 1 nonzero roots o. For any nonzero root π o, the Newton polygon o n (x) π consists a single segment o slope 1/(p n+1 p n ) and width p n. Thus N ( n+1 ) contains a segment o slope 1/(p n+1 p n ). This slope is shallower than the slope o any segment o N ( n ) and thereore the segment o N ( n+1 ) o slope 1/(p n+1 p n ) must be the inal negatively sloped segment. For each nonzero root π o, the Weierstrass polynomial corresponding to this single negatively sloped segment o N ( n (x) π) is Eisenstein over Z p [π] and thereore its roots are distinct. Because the nonzero roots o are distinct, there are exactly p n (p 1) roots o n+1 o valuation 1/(p n+1 p n ) and thereore N ( n+1 ) has a vertex at (1, p n ). The remaining negative segments o N ( n+1 ) must correspond to roots o n+1 which are roots o n. By the inductive hypothesis, one can deduce the segments o N ( n ) o inite, negative slope have width p k p k 1 and slope 1/(p k p k 1 ) where k runs over the integers 1 k n. Hence, all Weierstrass polynomials o n are Eisenstein over Z p and thereore all roots o n in m Cp are simple. It ollows, since every root o n in m Cp is a root o n+1, the Newton polygon N ( n+1 ) contains or each segment o N ( n ) a segment o the same slope and at minimum the same width. These segments span between the vertices (1, n + 1) and (1, p n ). The only way or this to occur is i the boundary o the Newton polygon o N ( n+1 ) in this range is equal to the boundary Newton polygon o N ( n ) translated up by one in the positive w direction. The claim ollows by induction. Proposition 2.5 gives us a good understanding o the position o Λ in the p-adic unit disc. Unpacking the data contained in the Newton polygon o n or each n, we see that Λ is the disjoint union o the sets 1 C n := {π Λ : v p (π) = p n p n 1 } or each n 1 and C 0 := {0}. For n 1, the set C n has cardinality p n p n 1 and its elements are the roots o a single Eisenstein polynomial over Z p. Each o these are expected properties o the p -torsion o a height one ormal group over Z p. In that case, the set C n is the set

6 6 JOEL SPECTER o elements o exact order p n. The same holds in the absence o an apparent ormal group. Comparing the Newton polygons o n and n+1, we note C n or n 1 can alternatively be described as: C n = {x m Cp : n (x) = 0 but n 1 (x) 0}. Because and u commute, the closed subgroup o S 0 (Z p ) topologically generated by u acts on Λ and preserves the subsets C n. Simultaneously, because and u are deined over Z p, the Galois group G Qp, the absolute Galois group o Q p, acts on Λ, preserves the subsets C n, and commutes with the action o u Zp. In the next section, we will explore these commuting actions in more detail. Next, we recall the deinition o the logarithm o the commuting pair (, u). Given any series g S 0 (Z p ) such that g (0) is nonzero nor equal to a root o unity, there exists a unique series log g S 0 (Q p ) such that, (1) log g(0) = 1, and (2) log g (g(x)) = g (0) log g (x) [Lub94, Proposition 1.2]. Lubin shows that two series g 1, g 2 S 0 (Z p ) such that g 1 (0) and g 2 (0) satisy the above condition have equal logarithms, i and only i, they commute under composition [Lub94, Proposition 1.3]. Consider the sequence o series in Q p x whose n-th term is: n (x) (0) n. Lubin shows that this sequence converges coeicient-wise to a series log satisying: (1) log (0) = 1 (2) log ((x)) = (0) log (x) [Lub94, Proposition 2.2]. 3 The limit is the logarithm o (, u). It is the unique series over Q p with these properties [Lub94, Proposition 1.2]. From proposition 2.5, one obtains that the vertices o Newton polygon o log are (p k, k) where k ranges over the nonnegative integers. Thereore by Theorem 2.2, log converges on m Cp. Furthermore, the Newton polygon o log displays that log has simple roots (as each o its Weierstrass polynomials is Eisenstein over Z p ) and the root set o log and Λ have the same number o points on any circle in m Cp. One guesses that the root set o log (counting multiplicity) is Λ. This is true and proved by Lubin [Lub94, Proposition 2.2]. The logarithm o is invertible in S 0 (Q p ). Conjugation by log deines a continuous, isomorphism rom the centralizer o in S 0 (Q p ) to End Qp (Ĝa). This map sends the power series e to e (0)x. I a Z p, we denote by [a] (X) := log 1 (a(log (X))). The series [a] (X) is the unique element o S 0 (Q p ), which commutes with and acts on the tangent space at 0 by a. Consider the subgroup u Zp S 0 (Z p ). By the previous paragraph, conjugation by log identiies this group with the closed subgroup u (0) Zp o Z p. By assumption 2.1, this group is ininite. Replacing u with an appropriate inite iterate, we may assume without loss o generality that: 3 One can place a topology on Zp X induced by a set o valuations arising rom Newton copolygons. This topology is strictly iner than the topology o coeicient-wise convergence. Lubin proves that the limit deining log converges in the closure o Z p X under this topology.

7 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 7 Assumption 2.6. The group u Zp or all m Z p. is topologically isomorphic to Z p and v p (1 u (0) m ) = v p (1 u (0)) + v p (m) 2.2. A Character Arising rom a Commuting Pair. Let F be a height one ormal group over Z p. A undamental invariant attached to F is its Tate module, T p F. Given that F is height one, T p F is a rank one Z p -module on which G Qp acts through a character χ F : G Qp Z p. This map is surjective. Simultaneously, the automorphism group, Aut Zp (F ), acts on T p F and commutes with the action o G Qp. This action is through the character which sends an automorphism to multiplication by its derivative at 0. Because the automorphism group Aut Zp (F ) is isomorphic to Z p via this character, given any g G Qp there is a unique s g Aut Zp (F ) such that or all nonzero v T p F the equality g.v = s g.v holds. The value χ F (g) can be recovered rom s g as χ F (g) = s g(0). This act should be considered remarkable or it allows one to recover the character χ F without any explicit knowledge o the additive structure o T p F. Rather χ F can be completely deduced rom the structure o the G Qp -orbit o any nonzero element v as a G Qp Aut Zp (G)- set. In this section, guided by this observation, we will attach to our commuting pair (, u) a character χ rom the Galois group o a certain inite extension K o Q p to Z p. The character χ arises rom the Tate module o the latent ormal group or which and u are endomorphisms. We begin by deining sequences o elements o m Cp which will substitute or the rôle o elements o the Tate module o F. Deinition 2.7. [Lub94, p. 329] An -consistent sequence is a sequence o elements (s 1, s 2, s 3,... ) o m Cp such that (s 1 ) = 0 and or i > 1, (s i ) = s i 1. Denote the set o all -consistent sequences whose irst entry is nonzero by T 0. Proposition 2.8. The Galois group G Qp acts transitively on T 0. Proo. The n-th coordinate o any -consistent sequence in T 0 is a root o n which is not a root o n 1. It is enough to show that G Qp acts transitively on these elements. Comparing the Newton polygons o n and n 1, we deduce that the set o such elements satisy a degree p n p n 1 Eisenstein polynomial over Q p. This polynomial is irreducible, and hence G Qp acts transitively on its roots. Choose an -consistent sequence Π = (π 1, π 2, π 3,... ) such that π 1 0. The sequence Π will be ixed throughout the remainder o this paper. Because u S 0 (Z p ) is nontrivial, it can have only initely many ixed points in m Cp. Let k be the largest integer such that u(π k ) = π k. As u Zp = Z p is a p-group and there are p 1 nonzero roots o, the integer k is at least 1. Let T πk denote the set o all -consistent sequences (s 1, s 2, s 3,... ) such that s i = π i or i k. Proposition 2.9. T πk is a torsor or the closed subgroup o S 0 (Z p ) generated by u. Proo. The subgroup o S 0 (Z p ) topologically generated by u is isomorphic to Z p. Thereore, as T πk is ininite, i u Zp acts transitively on T πk it must also act reely. To prove this action is transitive, we invoke the counting powers o the orbit stabilizer theorem.

8 8 JOEL SPECTER We begin by calculating the ixed points o iterates o u. Consider the set Λ u := {π m Cp : u n (π) = π or some n Z + } o u-periodic points in m Cp. Lubin proves Λ u = Λ [Lub94, Proposition 4.2.1]. Let e be a inite iterate o u. Consider the series e(x) x. The roots o this series are the ixed points o e, and hence each root is an element o Λ u. Since e is deined over Z p, the set o roots o e in m Cp is a inite union o G Qp -orbits in Λ u = Λ. The G Qp -orbits in Λ are the collection o sets C n = {x m Cp : n (x) = 0 but n 1 (x) 0}, where n ranges over the positive integers and C 0 = {0}. Evaluation under is an e-equivariant surjection rom C n to C n 1. Hence the ixed points o e are contained in i n C i or some n. By [Lub94, Proposition 4.5.2] and [Lub94, Proposition 4.3.1], respectively, the roots o e(x) x are simple and e(x) x 0 mod p. Thus, the negative slopes o the Newton polygon o e(x) x must span between (1, v p (1 e (0))) and the v-axis. Since when n 1 the set C n is the ull set o roots o an Eisenstein polynomial Z p, each o the sets C n account or a segment in the Newton polygon o e(x) x with a decline o one unit. We conclude that the roots o e(x) x and hence the ixed points o e are simple and equal to i v p(1 e (0)) C i. From this calculation, we deduce that v p (1 u (0)) = k, and the point-wise stabilizer o π C n is the set {e u Zp : v p (1 e (0)) n}. By assumption 2.6, it ollows the stabilizer o any π C n or n k is the group u pn k Z p. We now show that u Zp acts transitively on T πk. As in the proo o proposition 2.8, it is enough to show that u Zp acts transitively on the set o possible values or the n-th coordinate o a sequence in T πk. Let n k and consider the set W n (π k ) = {π m Cp : n k (π) = π k }. The set W n (π k ) are the possible values o an n-th coordinate o a sequence in T πk. Because π k 0, the Newton polygon o n k π k has exactly one segment o positive slope. The length o this segment is p n k and its slope is 1 p n 1 (p 1). Hence, W n(π k ) has order p n k and is contained in C n. It ollows that the u Zp -orbit o a point π W n (π k ) has order u Zp /u pn k Z p = p n k = W n (π k ). We conclude the action o u Zp on W n (π k ) is transitive. Let K := Q p (π k ). The transitive G Qp action on T 0 restricts to a transitive G K -action on T πk. The ixed ield o the kernel o this action is K := Q p (π i i > 0). Let σ G K. Because T πk is a torsor or u Zp there is a unique element u σ u Zp such that u σ Π = σπ. We set χ (σ) := u σ(0). The power series u σ can be recovered rom χ (σ) as u σ = [χ (σ)]. Proposition The map [χ ] : G K u Zp is a surjective group homomorphism satisying [χ (σ)] (π i ) = σ(π i ) or all i > 0. The ixed ield o the kernel o [χ ] is K.

9 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 9 Proo. Outside o the act that [χ ] is a homomorphism, this proposition ollows directly rom the discussion above. We show [χ ] is a homomorphism. Let σ 1, σ 2 G K. Then [χ (σ 2 )] [χ (σ 1 )] (Π) = [χ (σ 1 )] [χ (σ 2 )] (Π) = [χ (σ 1 )] σ 2 (Π) = σ 2 ([χ (σ 1 )] Π) = σ 2 σ 1 (Π). By uniqueness, it ollows [χ (σ 2 )] [χ (σ 1 )] = [χ (σ 2 σ 1 )] and [χ ] is a homomorphism. The character χ comes via a geometric construction and thereore should have good behavior rom the viewpoint o p-adic Hodge theory. In the next section we show this is the case. Speciically, we show χ is crystalline o weight The Hodge Theory o χ 3.1. Fontaine s Period Rings. To show that χ is crystalline, we must show that χ occurs (by deinition) as a G K -sub-representation in Fontaine s period ring B cris. In this section, we will recall the construction o this ring as well as several other o Fontaine s rings o periods (or which the original constructions can be ound in [Fon94]). We will ollow the naming conventions o Berger [Ber04]. The construction o the period rings begin in characteristic p. Let Ẽ+ := lim O x x p C p /p. The ring Ẽ+ inherits an action o the Galois group G Qp rom the action o G Qp on O Cp. Additionally, by virtue o being a ring o characteristic p, the ring Ẽ+ comes equipped with a Frobenius endomorphism F robẽ+. The map F robẽ+ commutes with the action o G Qp. The map F robẽ+ is an isomorphism and thereore Ẽ+ is perect. Set Ã+ = W (Ẽ+ ), the Witt vectors o Ẽ+. The ormality o the Witt vectors implies that the commuting actions o F robẽ+ and G Qp on Ẽ+ lit, respectively, to commuting actions o F robẽ+ and G Qp on Ã+ as ring endomorphisms. The lit o F robẽ+ is denoted by ϕ. Let B + := Ã+ [ 1 p ]. Fontaine constructs a surjective G Q p -equivariant ring homomorphism θ : B+ C p which can be deined as ollows. The map θ is the unique homomorphism which is continuous with respect to the p-adic topologies on B + and C p, and which maps the Teichmüller representative {x} to lim n x n where x n is any lit o x 1/pn to O Cp. The period ring B + dr is deined as the ker(θ)-adic completion o B +. The ring B + dr is topologized so that its inherits the topology generated by the intersection o the ker(θ)-adic and p-adic topologies o the dense subring B +. The map θ extends to a continuous G Qp - equivariant surjection θ : B + dr C p and ker(θ) is principal. The ring B + dr is abstractly isomorphic to C p t. The map θ induces a N-graded iltration on B + dr where Fili B + dr := ker(θ) i. The de Rham ring o periods, B dr, is deined as the raction ield o B + dr. The iltration on B + dr extends to a Z-graded iltration on B dr. Let V/Q p be a inite dimensional representation o the absolute Galois group o a p-adic ield E. We say V is de Rham i the E-dimension o D dr (V ) := Hom Q p G E (V, B dr ) is equal to the dimension o V. The iltration on B dr induces a iltration D dr (V ). The nonzero graded pieces o this iltration are called the weights o V. The weight o the cyclotomic character is 1.

10 10 JOEL SPECTER The ring B dr does not admit a natural extension o the Frobenius endomorphism, ϕ, o Ã+. To rectiy this, Fontaine deines a ring B cris o B dr on which the endomorphism ϕ extends. The ring B cris is deined as ollows: Let A cris be the PD-envelope o Ã+ with respect to the ideal ker(θ) Ã+. The ring A cris is a subring o B+ dr. We deine A cris to be the p-adic completion o A cris. One can show that the inclusion o A cris into B+ dr extends naturally to an inclusion o A cris into B + dr. Under this inclusion A cris is identiied with the set o elements o the orm i=0 a n tn n! where a n Ã+ such that lim n a n = 0 in the p-adic topology and t ker(θ) Ã+. The ring B + cris is deined as B+ cris := A cris[ 1 p ]. The ring B+ cris contains a unique G Qp -stable Q p -line on which G Qp acts by the cyclotomic character. The ring B cris is the deined rom B + cris by inverting any non-zero vector in this line. Let E 0 be the maximal unramiied extension o Q p contained in E. A inite dimensional representation V/Q p o the absolute Galois group o a p-adic ield E is called crystalline i the E 0 -dimension o D cris (V ) := Hom Q p G E (V, B cris ) is equal to the dimension o V. The endomorphism ϕ extends uniquely to an endomorphism o each o the rings A cris, A cris, B + cris, and B cris The Universal -Consistent Sequence. Let Z p x 1 be the ring o ormal power series over Z p in the indeterminate x 1. In this section we will deine a ring, A, containing Z p x 1, which parameterizes -consistent sequences in certain topological rings. The initial term o such a sequence will be parameterized by x 1. We will show that one can deine a continuous injection A Ã+ under which the image o A is G K -stable and satisies σ(x 1 ) = [χ (σ)] (x 1 ). We begin by deining A. For each positive integer i, set A i := Z p x i, the ring o ormal power series over Z p in the indeterminate x i. Denote, or each i, the homomorphism that maps x i (x i+1 ) by [ (0)] : A i A i+1. We deine A := lim A i to be the colimit o the rings A i with respect to the transition maps [ (0)]. Finally, the ring A is deined to be the p-adic completion o A. We view A as a topological ring under the adic topology induced by the ideal (p, x 1 ). The reader should be aware that while A is complete with respect to the iner p-adic topology, it is not complete with respect to the (p, x 1 )-adic topology. Under the topology on A the elements x i are topologically nilpotent and together topologically generate A as a Z p -algebra. The ring A is a natural object to consider rom the viewpoint o nonarchemedian dynamics or it is universal or -consistent sequences in the ollowing sense: given any p-adically complete, ind-complete adic Z p -algebra S and any -consistent sequence s := (s 1, s 2, s 3,... ) o topologically nilpotent elements in S there exists a unique homomorphism φ s : A S such that φ s (x i ) = s i. Conversely, any homomorphism rom A to S arises as φ s or some sequence s. In other words, A represents the unctor rom p-adically complete, ind-complete adic Z p -algebras to sets which sends an algebra S to the set o -consistent sequences o topologically nilpotent elements in S. Alternatively, we claim that A is canonically isomorphic to W (A /pa ), the Witt vectors o the residue ring A /pa, and hence A is closely connected to constructions in p-adic Hodge theory. The act that A = W (A /pa ) ollows rom the observation that A is a strict p-ring, i.e. that A is complete and Hausdor or the p-adic topology, p is not a zero divisor in A, and A /pa is perect [Haz09]. O these three criteria only the third is not immediately obvious or A. To see that A /pa is perect, we recall:

11 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 11 Theorem 3.1. [Lub94, Corollary 6.2.1, p. 343] Let k be a inite ield, and let u, 0 be invertible and non-invertible, respectively, in S 0 (k), commuting with each other. Then either u is a torsion element o S 0 (k) or has the orm (x) = a(x ph ) with h Z + and a S 0 (k) is invertible. In our case, theorem 3.1 implies (x) a(x p ) mod p or some invertible series a S 0 (F p ). It ollows: A /pa = A /pa = lim A i /pa i = lim F p x i x i a(x p i+1 ) = lim F p x i x i a(x i+1 ) p = lim F p a i (x i ) a i (x i ) (a i+1 (x i+1 )) p = lim y i y p i+1 F p y i, where y i := a i (x i ). As this inal ring is perect the claim ollows. Our goal is to ind an injection A Ã+ with particularly nice properties. Since A is a strict p-ring, any injection A /pa Ẽ+ lits canonically to an injection A Ã+. Let Π := (π 1, π 2, π 3,... ) be the -consistent sequence o elements in m Cp ixed in section 2.2. Then, as (x) a(x p ) mod p, we observe or each i Z + the series ˆπ i := a k i (π k ) mod p k i Ẽ+. Deine ˆΠ := (ˆπ 1, ˆπ 2, ˆπ 3,...). The sequence ˆΠ is -consistent and consists o topologically nilpotent elements o Ẽ+. Let φ ˆΠ : A Ẽ+ be the homomorphism associated to ˆΠ. Proposition 3.2. The kernel o the map φ ˆΠ : A Ẽ+ is pa. The induced injection φ ˆΠ : A /pa Ẽ+ identiies A /pa with a G K -stable subring o Ẽ+ such that or all σ G K and positive integers i the equality σ(φ ˆΠ(x i )) = φ ˆΠ([χ (σ)] (x i )) holds. Proo. The ring Ẽ+ has characteristic p and thereore φ ˆΠ actors through A /pa = A /pa = lim A i /pa i. To prove the proposition, we show that the restriction o the map induced by φ ˆΠ to A i /pa i is an injection and satisies σ(φ ˆΠ(x i )) = φ ˆΠ([χ (σ)] (x i )) or all σ G K. The ring A i /pa i is isomorphic to F p x i. Thereore, any homomorphism out o A i /pa i is either injective or has inite image. We claim the latter is alse or the map induced by φ ˆΠ. To see this, consider S 0 (F p ) acting on φ ˆΠ(A i /pa i ). We claim that the stabilizer o φ ˆΠ(x i ) is trivial and hence φ ˆΠ(x i ) has ininite orbit. Let z S 0 (F p ) be a nontrivial element and n = deg(z(x) x). Then z(φ ˆΠ(x i )) x i = z(a k i (π k )) π k mod p k i. For any positive integer k, the coordinate z(a k i (π k )) π k mod p is the image under the n reduction map O Cp O Cp /p o an element o p-adic valuation. Hence or k 0, we p k p k 1 have z(a k i (π k )) π k mod p is nonzero. It ollows z does not stabilize φ ˆΠ(x i ).

12 12 JOEL SPECTER The proo that G K acts in the desired way on φ ˆΠ(x i ) is by direct calculation. Let σ G K. Then σ(φ ˆΠ(x i )) = σ(ˆπ i ) = σ(a k i (π k )) mod p k i = a k i (σ(π k )) mod p k i = a k i ([χ (σ)] (π i )) mod p k i = [χ (σ)] (a k i (π i )) mod p k i = [χ (σ)] (ˆπ) = φ ˆΠ([χ (σ)] (x i )). Because A is a strict p-ring, the injection induced by φ ˆΠ : A /pa Ẽ+ lits canonically to an injection W (φ ˆΠ) : A Ã+. Henceorth, we will identiy A with its image under W (φ ˆΠ). The G K action on A /pa lits unctorially to a G K action on A ; urthermore, the map W (φ ˆΠ) is G K -equivariant and identiies A with a G K -stable subring o à +. Our next goal is to precisely describe the action o G K on A. The ollowing theorem provides the rigidity needed to understand these lits. Let F q be a inite extension o F p and consider an invertible series ω S 0 (F q ) such that ω (0) = 1. The absolute ramiication index o ω is deined to be the limit e(ω) := lim n (p 1)v x(ω pn (x) x)/p n+1. Theorem 3.3. [LMS02, Proposition 5.5]. Assume e(ω) <, then the separable normalizer o ω Zp in S 0 (F q ) is a inite extension ω Zp by a group o order dividing e(ω). Lemma 3.4. There exists a positive integer m such that a N u Zp mod p. Proo. Note as and u commute, so too do a and the reduction o u modulo p. One observes, upon considering the Newton polygon u n (x) x, that v x (u n (x) x) = 1 u (0) n p. By assumption 2.6, the absolute ramiication index o u is thereore inite. The result ollows by Theorem 3.3. Proposition 3.5. Let σ G K. Then σ(x i ) = [χ (σ)] (x i ). Proo. Consider the sequence [χ (σ)] (Π univ ) := ([χ (σ)] (x i )) i Z + o elements in A. Because and [χ (σ)] commute, the sequence [χ (σ)] (Π univ ) is - consistent. Hence, by the universal property o A, there exists an endomorphism [χ (σ)] := φ [χ (σ)] (Π univ ) o A which maps x i to [χ (σ)] (x i ). We wish to show that the Witt lit W (σ) is equal to [χ (σ)]. The ormer map is deined by its action mod p on Teichmüller representatives. Denote the Teichmüller mapping by { } : A /pa A. For α A /pa the element {α} can be deined as ollows: let α n be any lit o α 1/pn, then {α} := lim n αn pn. The map { } is well deined as this limit converges and is independent o the choice o lits α n. The automorphism W (σ) is the unique map such that W (σ){α} = {σ(α)}.

13 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 13 We claim the set o Teichmüller lits o x i mod p together topologically generate A (in the (p, x 1 )-adic topology) as a Z p -algebra. To see this, let A tm be the sub-algebra topologically generated by these elements. We show A tm = A. First note that the residue rings A tm /pa tm and A /pa are equal. Next observe that A tm is a strict p-ring. To see this, note A tm is closed in the p-adic topology on A and is thereore p-adically complete and Hausdor, A tm is a sub-algebra o A and hence p is not a zero divisor in A tm, and A tm /pa tm = A /pa and hence the residue ring A tm /pa tm is perect. It ollows the image o A tm /pa tm = A /pa under the Teichmüller map lies in A tm. But A (in the p-adic topology) is topologically generated as a Z p -algebra by the image o A /pa under the Teichmüller map. We conclude A tm = A. Thereore, to show W (σ) = [χ (σ)], it is enough to show W (σ({x i})) := {σ(x i )} is equal to [χ (σ)] ({x i}) or all i Z +. Fix a positive integer i. By lemma 3.4, there exists an integer N > 0 and a p-adic number k such that a N = u k. Hence, as (x) a(x p ) mod p, it ollows or every positive integer m the element u km (x Nm+i ) is a lit o x p1/nm i mod p. Similarly, u km ([χ (σ)] (x Nm+i )) is a lit o (σ(x i )) p1/nm mod p. Using these lits we will compare the action o W (σ) and [χ (σ)] on {x i}. Observe [χ (σ)] ({x i}) = lim m ([χ (σ)] (u km (x Nm+i ))) pnm = lim m (u km ([χ (σ)] (x Nm+i))) pnm = lim m (u km ([χ (σ)] (x Nm+i ))) pnm = W (σ)({x i }). Remark 3.6. The ring A is not an unamiliar object in the study o ormal groups (or more generally p-divisible groups). I is the endomorphism o a ormal group F := Sp(Z p x 1 ), the completion o A is the ring o global unctions on the universal cover o F (see [SW13, Section 3.1]) The p-adic Regularity o χ. In this section, we will examine the image o the sequence x i under log. To ensure convergence, we will appeal to the ollowing lemma o Berger: Lemma 3.7. [Ber14b, Lemma 3.2] Let E be a inite extension o Q p and take L(X) E X. I x B + dr, then the series L(x) converges in B+ dr i and only i L(θ(x)) converges in C p. By construction, the image o A lies in Ã+ and thereore θ(x i ) O Cp. The map rom A to Ã+ is a lit o the map φ ˆΠ : A Ẽ+ which sends x i to ˆπ i := a k i (π k ) mod p k i. Thereore, θ(x i ) π i mod p and hence θ(x i ) m Cp. Now the series log (X) converges on m Cp and so we see rom lemma 3.7 that log (x i ) converges in B + dr or all i Z+. Set x 0 := (x 1 ). Then θ(x 0 ) m Cp and hence log (x 0 ) converges in B + dr. We deine t := log (x 0 ). We note that ( (0)) n+1 log (x n ) = log ( n+1 (x n )) = log ((x 1 )) = t and thereore the values log (x n ) are Q p -multiples o t. We call t a undamental period o. The reader should be warned that t depends not only on but also our choice o -consistent

14 14 JOEL SPECTER sequence Π. I is an endomorphism o a ormal group deined over Z p any two undamental periods dier by a p-adic unit. How does G K act on t? Let σ G K, then σ(t ) = σ log ((x 1 )) = log ((σ(x 1 ))) = log ( [χ (σ)] (x 1 )) = (0)χ (σ) log (x 1 ) = χ (σ)t Thereore, assuming t 0, the period t generates a G K -stable Q p -line o B + dr on which G K acts through χ. From this it ollows χ is de Rham o some positive weight. Our irst proposition o this section shows that this is in act the case. Proposition 3.8. χ is de Rham o weight 1. Proo. The claim will ollow i we can show t Fil 1 B + dr \ Fil2 B + dr. We begin by showing t Fil 1 B + dr. Assume this were not the case. Then χ would be Rham o weight 0. However, all such representations are potentially unramiied and K, the ixed ield o ker χ, is an ininitely ramiied Z p -extension o K. It ollows that t Fil 1 B + dr. Next we show t Fil 2 B + dr. Since 0 = θ(t ) = log (θ(x 0 )), we observe θ(x 0 ) is a root o log. The quotient B + dr /Fil2 B + dr is a square zero extension o B+ dr /Fil1 B + dr = C p. Since all roots o log in m Cp are simple, we conclude t Fil 2 B + dr. Lemma 3.9. The derivative o the logarithm, log, is an element o Z p x. Proo. Since the series log converges on m Cp, so too does log. The Newton polygon o log contains the point (0, 0). Hence, log Z p x i and only i log has no roots in m C p. Assume or the sake o contradiction log had a root π in m C p. Then (3.1) log (u(π))u (π) = (log u) (π) = u (0) log (π) = 0. The power series u (x) is invertible in Z p x, so equation 3.1 implies log (u(π)) = 0. Hence, u Zp acts on the set o roots o log. Since u preserves valuation and log has only initely many roots o any ixed valuation, some power o u ixes π. From the equality Λ u = Λ, we conclude the element π is a root o log. But this is a contradiction as log has simple roots. Proposition The period t is an element o A cris, and χ is crystalline. Proo. By lemma 3.9: where a n Z p or all n Z +. Thereore, log (x) = t = i=1 i=1 a n n xn a n (n 1)! xn 0 n!.

15 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 15 It ollows i θ(x 0 ) = 0, then t A cris. To see θ(x 0 ) = 0, note that θ(x 0 ) is a root o log in m Cp and θ(x 0 ) θ((x 1 )) (π 1 ) 0 mod p. As 0 is the unique root o log o p-adic valuation greater than 1, the element θ(x 0 ) = 0 and the claim ollows. 4. The Main Theorem 4.1. Constructing the Formal Group. Let V be the Q p -line o B + cris generated by the period t. In section 3.3, we deduced that V is a G K -sub-representation o B + cris isomorphic to χ. Since K/Q p is totally ramiied, any G K -representation on a vector space V/Q p has multiplicity in B + cris at most one. Thereore, V is the unique Q p -line o B + cris isomorphic to χ. It ollows V is preserved by crystalline Frobenius. Let π Q p be the eigenvalue o crystalline Frobenius acting on V. Because V has weight one, the p-adic valuation o π is equal to 1. The next lemma is undamental. We will show that the equality ϕ(log (x 0 )) = π log (x 0 ) implies that ϕ(x 0 ) = [π ] (x 0 ). This will be enough to show [π ] S 0 (Z p ) and satisies [π ] (x) x p mod p, rom which we will deduce the that and u are endomorphisms o an integral ormal group by a lemma o Lubin and Tate. Lemma 4.1. The power series [π ] π x + x 2 Z p x and satisies [π ] (x) x p mod p. Proo. First note that by construction x 0 Ã+ B + cris and so x 0 is acted upon by crystalline Frobenius. Consider ϕ(x 0 ). We claim θ(ϕ(x 0 )) = 0. To see this, observe log (θ(ϕ(x 0 ))) = θ(ϕ(log (x 0 )) = θ(π t ) = 0. Hence, θ(ϕ(x 0 )) is a root o log (x). As x 0 Ã+ and θ(x 0 ) = 0, the image o crystalline Frobenius satisies θ(ϕ(x 0 )) po Cp. The unique root o log (x) in po Cp is 0, thereore we conclude θ(ϕ(x 0 )) = 0. Because θ(x 0 ) = 0 and θ(ϕ(x 0 )) = 0, any series e(x) Q p x converges in B + dr when evaluated at x 0 or ϕ(x 0 ). From this we deduce the equality log (ϕ(x 0 )) = ϕ(t ) = π t = log ([π ] (x 0 )). Additionally, we may apply log 1 to both sides, obtain convergent results and deduce ϕ(x 0 ) = [π ] (x 0 ). Write [π ] (x) = π x + a n x n, where a n Q p. We prove by induction on n that a n Z p. Let n 2 and assume a i Z p or all i < n. Set n 1 [π ] <n := π x + a i x i. Observe the equality n=2 i=2 [π ] <n (x 0 ) ϕ(x 0 ) x p 0 mod F il n B + dr Ã+ + pã+.

16 16 JOEL SPECTER Hence, as the homomorphism Z p x Ã+ given by mapping x x 0 is injective and (x) is the pullback o F il n B + dr Ã+, it must be the case that [π ] <n (x) x p mod p. Express (x) = (0)x + 2 (x) where 2 (x) x 2 Z p x. Note that [π ] (x) ([π ] <n (x)) + a n (0)x n mod x n+1 and [π ] ((x)) [π ] <n ((x)) + a n (0) n x n mod x n+1. As (x) and [π ](x) commute, (4.1) ([π ] <n (x)) [π ] <n ((x)) a n ( (0) (0) n )x n mod x n+1. The let hand side o (4.1) is a power series over Z p and reduces modulo p to ([π ] <n (x)) [π ] <n (x p ) (x) p 0 mod p. Thereore, the right hand side o (4.1) is a power series over pz p. Since v p ( (0) (0) n ) = 1, it ollows a n Z p. By induction we conclude [π ] is an element o π x + x 2 Z p x. From the equality ϕ(x 0 ) = [π ] (x 0 ), we observe [π ] (x) x p mod p. Theorem 4.2. Let and u be a commuting pair o elements in S 0 (Z p ). I (0) is prime in Z p and has exactly p roots in m Cp, u is invertible and has ininite order, then there exists a unique ormal group law F/Z p such that, u End Zp (F ). The ormal group law F is isomorphic to Ĝm over the ring o integers o the maximal unramiied extension o Q p. Proo. By [LT65, Theorem 1.1], the series [π ] is an endomorphism o a unique height one ormal group over F/Z p. Furthermore, and u are endomorphisms o F. The second hal o the claim ollows as all height one ormal groups over Z p are orms o Ĝm and are trivialized over the ring o integers o the maximal unramiied extension o Q p. 5. Acknowledgements I would like to thank Frank Calegari, Vlad Serban, Paul VanKoughnett, and the anonymous reeree or their helpul edits o earlier drats o this paper. Reerences [Ber04] Laurent Berger, An introduction to the theory o p-adic representations, Geometric aspects o Dwork theory 1 (2004), [Ber14a], Iterated extensions and relative lubin-tate groups, Annales mathématiques du Québec (2014), [Ber14b], Liting the ield o norms, J. École Polytechn. Math 1 (2014), [Fon94] Jean-Marc Fontaine, Le corps des périodes p-adiques, Astérisque 223 (1994), no. 59, 111. [Haz09] Michiel Hazewinkel, Witt vectors. part 1, Handbook o algebra 6 (2009), [Kob77] Neal Koblitz, p-adic numbers, p-adic analysis, and zeta-unctions, vol. 58, Springer, [KR09] Mark Kisin and Wei Ren, Galois representations and lubin-tate groups, Documenta Mathematica 14 (2009), [Li02] Hua-Chieh Li, p-typical dynamical systems and ormal groups, Compositio Mathematica 130 (2002), no. 01, [LMS02] François Laubie, Abbas Movahhedi, and Alain Salinier, Systèmes dynamiques non archimédiens et corps des norms, Compositio Mathematica 132 (2002), no. 1,

17 THE CRYSTALLINE PERIOD OF A HEIGHT ONE p-adic DYNAMICAL SYSTEM 17 [LT65] Jonathan Lubin and John Tate, Formal complex multiplication in local ields, Annals o Mathematics (1965), [Lub94] Jonathan Lubin, Nonarchimedean dynamical systems, Compositio Mathematica 94 (1994), no. 3, [Sar05] Ghassan Sarkis, On liting commutative dynamical systems, Journal o Algebra 293 (2005), no. 1, [Sar10], Height-one commuting power series over Z p, Bulletin o the London Mathematical Society [SS13] 42 (2010), no. 3, Ghassan Sarkis and Joel Specter, Galois extensions o height-one commuting dynamical systems, Journal de Théorie des Nombres de Bordeaux 25 (2013), no. 1, [SW13] Peter Scholze and Jared Weinstein, Moduli o p-divisble groups, Cambridge J. Math (2013), address: jspecter@math.northwestern.edu Joel Specter, Department o Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, United States