Discrete Dynamics over Finite Fields

Transcription

1 Clemsn University TigerPrints All Dissertatins Dissertatins Discrete Dynamics ver Finite Fields Jang-w Park Clemsn University, Fllw this and additinal wrks at: Part f the Applied Mathematics Cmmns Recmmended Citatin Park, Jang-w, "Discrete Dynamics ver Finite Fields" (2009). All Dissertatins This Dissertatin is brught t yu fr free and pen access by the Dissertatins at TigerPrints. It has been accepted fr inclusin in All Dissertatins by an authrized administratr f TigerPrints. Fr mre infrmatin, please cntact kkeefe@clemsn.edu.

2 Discrete Dynamics ver Finite Fields A Dissertatin Presented t the Graduate Schl f Clemsn University In Partial Fulfillment f the Requirements fr the Degree Dctr f Philsphy Mathematics by Jang-W Park August 2009 Accepted by: Dr. Shuhng Ga, Cmmittee Chair Dr. Neil J. Calkin Dr. Kevin L. James Dr. Hiren Maharaj Dr. Gretchen L. Matthews

3 Abstract A dynamical system cnsists f a set V and a map f : V V. The primary gal is t characterize pints in V accrding t their limiting behavirs under iteratin f the map f. Especially understanding dynamics f nnlinear maps is an imprtant but difficult prblem, and there are nt many methds available. This wrk cncentrates n dynamics f certain nnlinear maps ver finite fields. First we study mnmial dynamics ver finite fields. We shw that determining the number f fixed pints f a blean mnmial dynamics is #P cmplete prblem and cnsider varius cases in which the dynamics can be explained efficiently. We als extend the result t the mnmial dynamics ver general finite fields. Then we study the dynamics f a simple nnlinear map, f(x) = x + x 1, n fields f characteristic tw. The main idea is t lift the map f t a prper finite cvering map whse dynamics is easier t understand. We lift the map f f t an isgeny g n an elliptic curve where the dynamics f g can be further reduced t that f a linear map n Z mdule. As an applicatin f finite cvering, we cnstruct a new family f permutatin maps ver finite fields frm the knwn permutatin maps. ii

4 Dedicatin T my mther and father, whse cnstant and uncnditinal lve has made me wh I am. iii

5 Acknwledgments The writing f a dissertatin can be a lnely and islating experience, yet it is bviusly nt pssible withut the supprt and encuragement f numerus peple. First f all, I am very grateful t my dctral advisr, Dr. Shuhng Ga fr his encuragement, advice, mentring, and research supprt thrughut my dctral study. I als truly appreciate his patience and tlerance during my numerus mishaps. I als thank my cmmittee members, Dr. Neil Calkin, Dr. Kevin James, Dr. Hiren Maharaj, and Dr. Gretchen Matthews. I am frtunate t have received their time and suggestins fr my research ver the years. I am especially grateful t Dr. Kevin James fr his timely crrectins and suggestins that helped make this dissertatin better. I wuld like t thank Dr. Judith Cttingham and Dr. Timthy Teitlff fr writing me recmmendatin letters regarding my teaching skills. I als thank Prfessr Eric Bach wh shwed me hw wnderful mathematics is and led me where I am nw. I am als grateful t Prfessr Hendrik W. Lenstra, Jr. fr his valuable insight which has been crucial t the main chapter f this wrk. I am thankful t numerus friends wh supprted and nurished me in many different ways. Amng them, I am especially thankful t Sundeep Samsn, my cinstructr and cffee buddy wh made this lng prcedure enjyable and encuraged me during a hard time. I als thank Ethan and Andrea Smith fr their helps and iv

6 kindness. I als thank my fficemates thrugh the years, especially Ray Heindl and Mingfu Zhu fr cuntless hurs f gd cnversatin. I thank Mr. W-Yung Ryu, Mr. Sang-Ouk Wee, and Prfessr Jeng-Han Kim wh have been great friends thrughut my graduate studies. I am especially grateful t my tw best friends, Tae-Hee Lee and Wn-Jin Lee wh have encuraged me with unfading friendship fr decades. I thank Jhann Sebastian Bach fr Gldberg Variatins which has been bth musical and mathematical inspiratin t me fr lng time, Glenn Guld and Dng- Hyek Lim whse interpretatins f Gldberg Variatins have nurished my sul, and Frédèric Chpin whse brilliant wrk, Plnaise, Op. 53, has widen my perspective n life. I als thank my favrite guitarists, Pat Metheny, Paul Gilbert, Guthrie Gvan and Billy McLaughlin wh prvided wnderful music with the instrument that I lve the mst. Finally, I wuld like t thank the mst imprtant peple in my life, my family. I am grateful t my sisters, Yung-W Park, Jung-W Park, Eun-W Park fr the supprt and the encuragement. I wuld als like t thank my aunts and uncles fr their prayers. I am deeply indebted t my parents, Chan-Ky Park and Kum-Sn Kim wh have trusted and supprted me fr my whle life with their uncnditinal lve. v

7 Table f Cntents Title Page Abstract Dedicatin i ii iii Acknwledgments iv List f Figures vii 1 Intrductin Mnmial Dynamics ver Finite Fields Intrductin Fixed Pints ver F Cycles f Lengths Greater than One ver F Mnmial Dynamics ver General Finite Fields Finite Cverings Intrductin A Dynamical System and its Assciated Elliptic Curve Prperties f g n E Grup Structure f E(F 2 n) Tree Structure f g n E(F 2 n) Cycle Structure f g n E(F 2 n) Dynamics f x x + x 1 n F 2 n { } Permutatin Maps ver Finite Fields Intrductin Prf Cnclusins Bibligraphy vi

8 List f Figures 1.1 Orbit f v under a map f Dynamics f f n F Dependency Graph χ f f f and its Strngly Cnnected Cmpnents Pset f the Dependency Graph χ f Pset f the Strngly Cnnected Cmpnents f χ f Pset G G 1 and G 2 fr the Vertex 3 f G Cmplete Tertiary Tree f height Special Quadripartite Graph Dependency Graphs f f and g Dependency Graphs f f 2 and g Cmpnent C Dependency Graph χ f f f Dynamics f f(x) = x + x 1 n F 2 4 { } Dynamics f f(x) = x + x 1 n F 2 5 { } Dynamics f f(x) = x + x 1 n F 2 6 { } Dynamics f g n E(F 2 10) and that f f(x) = x + x 1 n F 2 5 { }.. 76 vii

9 Chapter 1 Intrductin Dynamical systems are ubiquitus in science and engineering. They may represent the mtins f stars in the sky in astrnmy, the fluctuatin f stck markets in business, the heart beat in medical science, gene evlutin in genetics, r traffic in a highway system r in a city. Dynamical systems have lng been studied by many schlars in science, engineering, and mathematics, and they are active areas f research full f unknwns and challenges. We write In simple terms, a dynamical system cnsists f a set V and a map f : V V. f i = f f f = i }{{} th iteratin f f, i and f 0 dentes the identity map n V by cnventin. Fr a given pint v V, the rbit f v under f is the set f f i (v) s fr all i 0. A pint v V is called peridic r cyclic if there exists m 1 such that f m (v) = v. In this case, the rbit f v under f is a cycle and the smallest such m is called the cycle length f v. A pint v V is called preperidic if there exist 0 i < j such that f i (v) = f j (v). In this case, the rbit f v is depicted as in Figure 1.1. The number i in Figure 1.1 is called tail 1

10 length and the number j i is called the cycle length f v. In applicatins, it is desirable t understand cycle lengths, tail lengths, and their distributins. v v i 1 i = v j v i+1 v j 1 v 3 v i+2 v j 2 v 2 v 1 v = v 0 Figure 1.1: Orbit f v under a map f. In a classical dynamical system, V is a tplgical and metric space. A pint v V is called stable if, whenever u V is clse t v, the rbit f u stays clse t that f v. The Fatu set f f cnsists f all the stable pints f V. The Julia set f f is the cmplement f the Fatu set. S pints in Julia set tend t mve away frm each ther under iteratin f f and they behave chatically. The imprtant subjects in a classical dynamical system are the limiting behavirs f tw clse pints and finding the Julia set. Fr mre n classical dynamical system, we recmmend [Devaney, 2003] and [Rbinsn, 1998]. Understanding the discrete dynamics n finite sets requires different techniques. When V is finite, every pint is preperidic. S the stability and chas in classical dynamical systems are irrelevant in finite dynamical systems. We view the discrete dynamics f f n a finite set V as a directed graph. The graph has V as a vertex set and, fr any pair f v,w V, there is an edge frm v t w if and nly if f(v) = w. Figure 1.2 shws a dynamical system ver a finite field. 2

11 Example Let f : F 5 2 F 5 2 be f(x 1,x 2,x 3,x 4,x 5 ) = (x 2 x 3,x 1 x 4,x 3,x 4,x 4 ). The dynamics f f is shwn in Figure Figure 1.2: Dynamics f f n F 5 2 Ntice that the dynamics f f cnsists f disjint cycles with trees attached t them. As can be seen in Figure 1.2, it has five fixed pints, ne cycle f length 2, the maximum tail length is 2, and the maximum in-degree is 8. It is als nticeable that there are regularities in the tree structure. We are interested in dynamics ver finite sets, especially ver finite fields. We are interested in the fllwing questins: Hw many cycles are in the dynamics f f n V? What are the cycle lengths? What are the heights f trees? What are the in-degrees? 3

12 Althugh we can get answers fr all the questins abve by enumerating all pints, we are interested in the underlying mathematical thery. The gal is t analyze the dynamics withut actually enumerating all state transitins, since enumerating has expnential cmplexity in the number f mdel variables. In this wrk, we are particularly interested in mnmial dynamics and using finite cvering t investigate the dynamics f nnlinear maps ver finite fields. The fllwing is a brief survey f sme knwn results n varius discrete dynamics. Fr linear finite dynamical systems, Elspas [1959] examined the dynamics f linear systems ver prime fields and shwed that cycle structure can be determined by the elementary divisr f the matrix, and Hernandez-Tled [2005] generalized Elspas s results t arbitrary finite fields and als shwed that tree structure can be determined by the nilptent part f the map. Based n these results, Jarrah et al. [2006] presented an algrithms which describes the phase spaces. Xua and Zub [2009] have presented an efficient algrithm t analyze cycle structure f the dynamics f linear systems ver finite cmmutative rings. Studying dynamics f nnlinear maps is very challenging task. Only a few cases have been well understd. Zieve [1996] investigated the cycle lengths f plynmial maps ver varius rings. Even dynamics f quadratics plynmials ver finite fields are still pen except f(x) = x 2 and f(x) = x 2 2. The square map ver prime fields was studied in [Rgers, 1996] and the dynamics f f(x) = x 2 2 ver prime fields was analyzed in [Gilbert et al., 2001], [Park, 2003], and [Vasiga and Shallit, 2004]. Fr mnmial dynamics, Jarrah et al. [2008] prvided an analysis f blean mnmial dynamical systems and Clón-Reyes et al. [2006] shwed the structure f fixed pints f mnmial dynamics ver general finite fields can be reduced t blean mnmial dynamics. A map is called a permutatin map if it is bijective n V. Permutatin maps have applicatins in diverse areas such as cding thery, cmbinatrics, and cryp- 4

13 tgraphy. If V is finite, then the dynamics f permutatin maps cnsist f nly cycles. Especially fr permutatin maps ver finite fields, due t the fact that every map ver a finite field can be expressed by a plynmial, it was natural t fcus n maps defined by plynmials. Since Hermite [1863] investigated permutatin plynmials ver finite prime fields and Dicksn [1897] studied them ver general finite fields, numerus mathematicians and engineers have shwn their interests in permutatin plynmials. Fr mre backgrund material n permutatin plynmials, we refer the readers t Chapter 7 f [Lidl and Niederreiter, 1997] and, fr a detailed survey and sme pen prblems, t [Lidl and Mullen, 1988, 1993]. Tw wellknwn classes f permutatin plynmials are mnmials x k ver F q with k 1 and gcd(k,q 1) = 1 and Dicksn s plynmials ver F q with degrees relatively prime t q 2 1. Binmial plynmials f certain frms have been studied by several schlars; see [Akbary and Wang, 2006], [Masuda et al., 2006], [Masuda and Zieve, 2007, 2009], [Turnwald, 1998], and [Wang, 2002]. Fr permutatin plynmials in mre general frms, see [Akbary and Wang, 2005], [Park and Lee, 1998], and [Wan and Lidl, 1991]. In this wrk we fcus n studying dynamics f special nnlinear maps ver finite fields and its theretical applicatin. In Chapter 2, we study mnmial dynamics ver finite fields. A map f is called a mnmial map if f = (f 1,f 2,...,f m ) where each f i is a mnmial. Clón-Reyes et al. [2004] studied fixed pint structure f f ver F 2 by assciating the dynamics f f with its dependency graph χ f. They als intrduced a lp number f a strngly cnnected cmpnent which plays imprtant rle in their investigatin f cycle structure f mnmial dynamics in [Jarrah et al., 2008]. Jarrah et al. [2008] prved that a cmpnent with the lp number t in χ f wuld decmpse int t/d cmpnents in χ f d fr d dividing t. They shwed pssible lengths f cycles and their distributins when χ f has nly ne cmpnent. Frm this, 5

14 they presented lwer and upper bund fr the number f cycles f a given length fr general blean mnmial dynamics. When χ f has mre than ne cmpnent, the bstacle in studying the exact cycle structure f f is that structure f cycles f length d 1 depends n nt nly hw cmpnents decmpse in χ f d but als n hw cmpnents are cnnected in χ f. It is even difficult t determine the number f fixed pints f blean mnmial dynamics. We shw that the prblem f cunting fixed pints f a mnmial dynamics ver F 2 is #P cmplete, fr which n efficient algrithm is knwn. This is prved by a 1 1 crrespndence between fixed pints f f and antichains f the pset f strngly cnnected cmpnents f χ f. We als extend the results f blean mnmial dynamics t mnmial dynamics ver general finite fields. T determine fixed pints f a mnmial map f ver F q, we wrk n zer cmpnent and nnzer cmpnents separately. We find the zer cmpnents by examining the dependency graph f f as dne in blean mnmial dynamics. We shw hw nnzer cmpnents f f can be reduced t a linear map ver Z q 1 by using lgarithmic representatin f f. Hence deciding the values f nnzer cmpnents f fixed pints is equivalent t slving linear systems ver Z q 1. In Chapter 3, we apply finite cvering t analyze dynamics f nnlinear maps ver finite fields. We are particularly interested in the dynamics f f(x) = x+x 1 ver F 2 n { }. We lift f t an isgeny g = I +σ n the elliptic curve E : y 2 +xy = x 3 +1 where I is an identity map and σ is the Frbenius map. Fr a psitive integer n, let E(F 2 n) be the set f F 2 n-ratinal pints f E and E p (F 2 n) a p subgrup f E(F 2 n) where p is a prime. Since E(F 2 n) is a finite abelian grup, E(F 2 n) can be decmpsed as direct sum f E p (F 2 n) s where p is a prime dividing #E(F 2 n). We shw that all the tails f g cme frm the the dynamics g n E 2 (F 2 n). It is knwn that E 2 (F 2 n) is ismrphic t Z/(2 h 2 ) fr sme h 2. We prve that h 2 is equal t ν 2 (n) + 2 and every tree attached t a peridic pints is a cmplete binary tree f 6

15 height ν 2 (n) + 1. Fr an dd prime p, g is an autmrphism n E p (F 2 n). Hence all cycle lengths are explained by the dynamics f g n E p (F 2 n). Nte that E p (F 2 n) is ismrphic t Z/(p ap ) Z/(p bp ) where a p and b p depend n the factrizatin f σ n 1 in Z[σ]. We shw that the dynamics f g n E p (F 2 n) can be reduced t that f a linear map M = ( ) n a Z mdule. We distinguish three cases: (a) Fr p = 7, we shw that #E(F 2 n) is divisible by 7 if and nly 6 n and, fr that n = 6 7 e w with e 0 and 7 w, E 7 (F 2 n) is ismrphic t Z/(7 e ) Z/(7 e+1 ). We shw that all the cycle lengths f g n E 7 (F 2 n) can be btained frm the multiplicative rder f M mdul 7 c where c runs frm 1 t e + 1. (b) Fr dd prime p s with ( p 7) = 1, we shw that Ep (F 2 n) is ismrphic t Z/(p e ) Z/(p e ) with e = ν p (#E(F 2 n))/2 and the dynamics f g n E p (F 2 n) is identical t that f M ver Z/(p e ) Z/(p e ). (c) Fr dd prime p s with ( p 7) = 1, it is difficult t analyze the cycle lengths f g n E p (F 2 n) because the structure f E p (F 2 n) can be arbitrary. But we shw that when E p (F 2 n) is ismrphic t Z/(p e ), g n E p (F 2 n) can be reduced t a multiplicatin map n Z/(p e ), and when E p (F 2 n) is ismrphic t Z/(p e ) Z/(p e ), the dynamics f g n E p (F 2 n) is identical t that f M n Z/(p e ) Z/(p e ). Using this infrmatin, we shw that, in the dynamics f f n F 2 n { }, the length f a cycle prjected frm an even cycle in the dynamics f g n E(F 2 2n) is the half f the cycle length and the length f a cycle prjected frm an dd cycle has the same cycle length. We als shw that there are three different tail structures in the dynamics f f n F 2 n { }: (a) The tree structure attached t is as fllws: a cmplete binary tree f height 7

16 ν 2 (n) is attached t 0 and 0 is attached t. (b) Structure f a tree prjected frm a tree attached t a peridic pint P = (x,y) E(F 2 2n) with x F 2 n, y / F 2 n is a tree f height 0. (c) Structure f a tree prjected frm a tree attached t a peridic pint P E(F 2 n) \ {O} is a cmplete binary tree f height ν 2 (n) + 1. In Chapter 4, we present an interesting applicatin f finite cverings. We cnstruct a new family f permutatin maps ver finite fields with dd characteristic frm the knwn family f permutatin maps using finite cvering. The key idea is that we prject n th pwer map g using a prper prjectin map π which is different frm ne used t cnstruct Dicksn s plynmials and btain a new family f maps h satisfying π g = h π. We shw the exact cnditin fr new maps t be permutatin maps. Finally in Chapter 5, we recapitulate the results given in this wrk and cnsider the pssible questins fr future research. 8

17 Chapter 2 Mnmial Dynamics ver Finite Fields 2.1 Intrductin is defined by Fr this chapter, we fcus n the case when V = F n q and the map f : F n q F n q f = (f 1,f 2,...,f n ) where f i = c i x m i1 1 x m i2 2 x m in n, 1 i n, with c i F q and m ij N. Then f is called a mnmial map ver F q and the dynamics f a mnmial dynamics. Since ur wrk extends that f Clón-Reyes et al. [2004]; Jarrah et al. [2008], we will use their definitins and basic setup in mst f cases. We assciate f with a digraph χ f, called the dependency graph f f which has vertex set {1, 2,...,n}, and there is a directed edge frm j t i if and nly if c i 0 and x j f i. Nte that j is 9

18 adjacent t i if and nly if the value f x j affects f i and we allw self-lps in χ f. Example Let f be defined ver F 2 as f = (x 2,x 3 x 4,x 2,x 5 x 12,x 6,c,x 8 x 11,x 3 x 9,x 10,x 6,x 9,x 12 ) where c in F 2. The dependency graph χ f f f is as fllws: ={2, 3} C 1 ={9,10,11} C 2 C 3 ={12} 6 12 Figure 2.1: Dependency Graph χ f f f and its Strngly Cnnected Cmpnents When c = 1, the fixed pints f f are : (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), (0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1), (0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0), (0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1). 10

19 When c = 0, the fixed pints f f are : (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1). Let χ be any digraph. Fr any tw vertices i,j χ, if there is a directed path, r dipath fr shrt, frm i t j and a dipath frm j t i then we say i and j are strngly cnnected. A subset f vertices is called strngly cnnected if each pair f vertices in the subset is strngly cnnected. Any maximal strngly cnnected subset f vertices f χ is called a strngly cnnected cmpnent f χ, r simply a cmpnent f χ. Nte that a vertex itself is a cmpnent if and nly if it has a self-lp. Nte that different cmpnents f χ have disjint vertices, and there may be vertices in χ that d nt lie n any cmpnent. Fr any vertex i nt n any cmpnent, either there is a dipath frm i t sme cmpnent r there is a dipath frm sme cmpnent t i, but nt bth. Similary, fr any tw cmpnents, if there are paths fr ne cmpnent t the ther, then there is n path ging t the ppsite directin. We say a cmpnent C 1 is abve, r greater than, anther cmpnent C 2 if there is a dipath frm C 2 t C 1. This makes the set f all the cmpnents f χ int a partially rdered set, i.e., a pset. Example (revisited). Suppse that we have the dependency graph χ f as in Figure Then, fr c = 1, the pset is as in Figure

20 1 7 C 1 8 C 1 C C 2 C 3 6 C 3 Figure 2.2: Pset f the Dependency Graph χ f Let G be a set. A partial rder is a binary relatin ver G which satisfies reflexive, antisymmetric, and transitive. With a partial rder, G is called a partially rdered set. A pair f elements x and y in G are cmparable if x y r y x. A subset A f G is called an antichain if n tw elements in A are cmparable. Nte that the empty subset is an antichain and any singletn subset is an antichain as well. Example Suppse that G is as belw: C 1 C 2 C 3 Figure 2.3: Pset f the Strngly Cnnected Cmpnents f χ f Then all the pssible antichains f G are:, {C 1 }, {C 2 }, {C 3 }, and {C 1,C 2 }. 12

21 Nte that G in Figure is btained frm the pset f the dependency graph χ f in Figure 2.1 by cnsidering nly cmpnents. Fr a given dependency graph χ f f f, we define G f as the pset f strngly cnnected cmpnents in χ f and we call G f the cmpnent pset f χ f. Let A be a subset f a partially rdered set G. A is upper clsed if fr any x A and y G, x y implies that y A t. Similarly, A is lwer clsed if fr any x A and y G, x y implies that y A t. Let k be an arbitrary field. Fr any pint P = (a 1,a 2,...,a n ) k n, we define subsets S 0 (P) and S 1 (P) f χ f as S 0 (P) = {1 i n : a i = 0}, S 1 (P) = {1 i n : a i 0}. Then fixed pints f mnmial dynamics have the fllwing unique prperty. Prpsitin Let k be an arbitrary field and f : k n k n be a mnmial map. Suppse P = (a 1,a 2,...,a n ) k n is a fixed pint f f. Then S 0 (P) is upper clsed and S 1 (P) is lwer clsed. Prf. Since P = f(p), fr each j in the dependency graph χ f, we have a j = f j (P). Fr any vertex i that has an edge t j, if a i = 0 then a j = 0. Als, if a j 0 then a i 0 fr all vertices i adjacent t j. The prpsitin fllws by chasing the dipaths in χ f. This prperty gives us a different way t recgnize fixed pints f mnmial dynamics and we will investigate the structure f fixed pints using this prperty. 13

22 2.2 Fixed Pints ver F 2 In this sectin, we study hw t find all fixed pints f the dynamics f a given map f ver F 2 and delve int the related cmbinatrial prblems. Therem Let f = (f 1,f 2,...,f n ) : F2 n F2 n and let χ f be the dependency graph f f. Assume that n f i s are cnstant. Then there exists a 1 1 crrespndence between the set f fixed pints f f and the set f antichains f the cmpnent pset G f f χ f. Prf. Suppse P is a fixed pint f f. Then, by Prpsitin 2.1.1, S 1 (P) is lwer clsed. S the set f maximal strngly cnnected cmpnents amng the strngly cnnected cmpnents cntained in S 1 (P) frms an antichain. Nw, suppse A is an antichain f the cmpnent pset. Then, fr all 1 i n, set j i = 0 if j i C fr sme C A and set j i = 1 therwise. Let P A = (j 1,j 2,...,j n ). Nte that if j i = 0, then since j = 0 fr all j j i, f i (P A ) = 0. Als, if j i = 1, then since j = 1 fr all j j i, f i (P A ) = 1. This implies that f(p A ) = P A, i.e. P A is a fixed pint. Example (revisited). Suppse that f is defined in Example Recall that we have already seen the cmpnent pset G f f χ f in Figure and the crrespnding antichains. Frm this, we can find all the fixed pints f f: (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), {C 1 } (0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1), {C 2 } (1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1), {C 3 } (0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0), {C 1,C 2 } (0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1). 14

23 S, if we can cmpute the number f antichain f the cmpnent pset, then we knw the number f fixed pints f given blean mnmial dynamics. Definitin (Valiant [1979]). #P is the class f functins that can be cmputed by cunting Turing machines f plynmial time cmplexity. A prblem is #P cmplete if and nly if it is in #P, and every prblem in #P can be reduced t it by a plynmial-time cunting reductin. There is n knwn algrithms t slve #P cmplete prblem efficiently. Prvan and Ball [1983] shwed that cmputing the number f antichains f given pset is a #P cmplete prblem and Knuth and Ruskey [2003] studied sme special cases where the cunting can be dne efficiently. In the fllwing, we present a simple algrithm t cunt the number f antichains f a given pset Cunting the Number f Antichains f a Pset Let G be a pset and τ(g) be the number f antichains f G. Nte that any subset f a pset is a pset t. Then there are tw basic prperties f the number f antichains. First, if G is a disjint unin f G 1 and G 2, then τ(g) = τ(g 1 ) τ(g 2 ). Suppse v G. Then τ(g) = τ(g 1 ) + τ(g 2 ) where G 1 and G 2 are defined as fllwing: G 1 = G \ {u G : u cmparable t v} and G 2 = G \ {v}, but keeps the cnnectins. 15

24 The fllwing example will clarify the definitins f G 1 and G 2. Example Suppse that a pset G is as fllwing: 1 2 G = Figure 2.4: Pset G If we pick the vertex 3, then the crrespnding G 1 and G 2 are as in Figure 2.5: G = 1 4 and G = Figure 2.5: G 1 and G 2 fr the Vertex 3 f G Nte that if G is a tree f height 1 with n leaves, then τ(g) = n. Thus, with these prperties, we can develp a recursive algrithm fr cunting the number f antichains in any pset. - Algrithm 1 Input : a pset G. Output : τ(g) ( = the number f antichains ). 16

25 ALG 1 (G) : 1. If G is a tree f height 1 with n leaves, then return (1 + 2 n ). 2. Pick any maximal(r minimal) element v G. Define G 1 and G 2 as fllws: G 1 := G \ {u G : u < v}(r G \ {u G u > v}, respectively). G 2 := G \ {v}. 3. return (ALG 1 (G 1 ) + ALG 1 (G 2 )). End f ALG 1 (G) Nte that using maximal r minimal element in the abve algrithm des nt change the result r the perfrmance f the algrithm. Althugh cunting the number f antichains is generally knwn as a difficult prblem, there are certain cases in which we can cunt it efficiently [Knuth and Ruskey, 2003]. Here, we list sme f thse special types f psets. 1. Suppse that T is a tree. T is called a cmplete n ary tree f height h if if every nde f T except leaves has the same in-degree, n, and every leaf has the same depth, h. Let T(n,h) be a cmplete n ary tree f height h. Then the prperties abve gives us the linear time algrithm t cunt the number f antichains f the tree T(n,h). Let u T(n,h) be the rt. Then since it is n ary cmplete tree, there are n T(n, h 1) s attached t u. Thus τ(t(n,h)) = 1 + (τ(t(n,h 1))) n. Same reasning wrks fr a inverted cmplete n ary tree f height h. Here is an example f a cmplete n ary tree. Fr instance, cnsider T(3, 3), a cmplete tertiary tree f height 3 in Figure 2.6. Using the abve recurrence 17

26 Figure 2.6: Cmplete Tertiary Tree f height 3 relatin, we have τ(t(3, 3)) = 1 + (τ(t(3, 2))) 3 = 1 + ( 1 + (τ(t(3, 1))) 3) 3 ( = ( ) ) 3 3 = Fr psitive integers m and n, define M(m,n) by a m partite graph where each level has n vertices and each level is cmpletely cnnected nly with adjacent levels. M(4, 3) is shwn in Figure 2.7. Then, by chsing a vertex in the highest(r lwest level), we have τ(m(m,n)) = 2 n 1 +2 n τ(M(m 1,n)) = 2 n 1+τ(M(m 1,n)). Thus τ(m(m,n)) = (m 1)(2 n 1) + τ(m(1,n)). Nte that G(1,n) is just a pset with n singletns. S τ(m(m,n)) = (m 1)(2 n 1) + 2 n = m 2 n m + 1. Then using the abve frmula, we knw 18

27 Figure 2.7: Special Quadripartite Graph τ(m(4, 3)) = (4 1) =

28 2.3 Cycles f Lengths Greater than One ver F 2 In this sectin, we want t discuss hw t determine cycles f length greater than ne in blean mnmial dynamics. Nte that if f has a cycle f length m, then f m has m new fixed pints which are nt fixed pints f f. This implies that the dependency graph f f m has mre cmpnents than that f f. T be precise, we need t study when cmpnents f χ f can be decmpsed int smaller cmpnents f f m. The dependency graph f f m will be dented by χ f m. Nte that if a vertex is nt n any cmpnent f χ f, then it is nt n any cmpnent f χ f m. Hence all cmpnents f χ f m cme frm thse f χ f. The fllwing example shws the difficulty f studying cycle structure f the dynamics f f when χ f has mre than ne cmpnent. Example Suppse we have the fllwing dependency graphs χ f and χ g. χ f Gf χ g G g C 1 C 1 C 2 C 2 Figure 2.8: Dependency Graphs f f and g As we see in Figure 2.3.1, the cmpnent psets G f and G g are identical. This implies that, in this example, the set f fixed pints are same in bth dynamics. Nw cnsider the dependency graphs f f 2 and g 2. 20

29 χ 2 2 f G χ f g 2 Gg 2 C 11 C 12 C 11 C 12 C 21 C 22 C 21 C 22 Figure 2.9: Dependency Graphs f f 2 and g 2 Althugh f and g have the same fixed pints, the cycle structure f f and g are different since the cmpnent psets G f 2 and G g 2 are different. Example shws that t find ut the cmpnent pset G f m, we need precise infrmatin n hw vertices are cnnected t thers in χ f. In the rest f this chapter, we fcus n the case when χ f has nly ne cmpnent. T investigate further, we need the fllwing definitin. Definitin (Lp Number [Clón-Reyes et al., 2004]). The lp number f a vertex v χ f is the minimum f all numbers t 1 where t = p q fr all clsed walks p,q : v v. If there is n clsed walk frm v t v then we set the lp number t be zer. It was als shwn in [Clón-Reyes et al., 2004] that all elements in a cmpnent have the same lp number, which implies that the lp number is a prperty f the cmpnent. Example Suppse a cmpnent C is as fllws: 21

30 Figure 2.10: Cmpnent C By the definitin, the lp number f C is 6. Prpsitin Suppse a χ f and its lp number is t. Then, fr any tw lps l 1 and l 2, passing thrugh a, l 1 l 2 (md t). Prf. Withut lss f generality, assume that l 1 > l 2. Since the lp number is t, there exists clsed walks c 1 and c 2 such that c 1 c 2 = t. Suppse that l 1 l 2 = kt + α where 0 α < t. Then we have tw clsed walks l 1 + kc 2 and l 2 + kc 1 frm a t a, l 1 + kc 2 l 2 + kc 1 = l 1 + k c 2 l 2 k c 1 = ( l 1 l 2 ) k( c 1 c 2 ) = kt + α kt = α. By the minimality f the lp number, we have α = 0, hence l 1 l 2 0 (md t). 22

31 Definitin Let a,b χ f. Then directed distance frm a t b, dented by d(a,b), is the length f the shrtest path frm a t b. We define d(a,b) = if there is n path frm a t b. Lemma Let C be a cmpnent f χ f with lp number t. We define a relatin between any tw vertices a,b C by a b if d(a,b) 0 (md t). Then is the equivalence relatin n C. Prf. Let c 1 and c 2 be clsed walks frm a t a with c 1 c 2 = t thrughut the prf. Fr any lp p : a a, suppse that p = d(a,a) = kt + α where 0 α < t. Then we have clsed walks p + kc 2 and kc 1 with p + kc 2 kc 1 = p k( c 1 c 2 ) = kt + α kt = α. By the minimality f the lp number, we have α = 0. Hence every lp passing thrugh a has length 0 mdul t. In particular, d(a,a) = 0. Suppse that a b. Then, by the definitin, d(a,b) 0 (md t). We want t shw that d(b,a) 0 (md t). Let p 1 : a b with p 1 = k 1 t and p 2 : b a with p 2 = d(b,a). Then p 1 + p 2 is a lp frm a t a. Recall that p 1 + p 2 0 (md t), which implies p 2 0 (md t). Hence d(b,a) 0 (md t). Nw, suppse a b and b c fr a,b,c C, i.e., there exist a path p 1 frm a t b with p 1 = d(a,b) = k 1 t and a path p 2 frm b t c with p 2 = d(b,c) = k 2 t. Let p be any path frm c t a. Then we have a lp p 1 + p 2 + p 3 frm a t a and p 1 + p 2 + p = p 1 + p 2 + p 0 (md t). Hence, d(c,a) 0 (md t). 23

32 Nw partitin C accrding t its lp number t. Pick c C and let C i be C i = {a C : d(a,c) i (md t)}. It is easy t see that fr any a 1,a 2 C, a 1 a 2 if and nly if a 1,a 2 C i fr sme i. Thus C can be decmpsed as C = C 0 C 1 C 2... C t 1. Frm the definitin f C i s, we can see that the first t steps f the walk frm c t itself decides the decmpsitin and these subcmpnents C i s will nt change with the different chice f c. Mrever, fr any a C i, there exists b C i+1 such that d(a,b) = 1. Example (revisited). Recall that the lp number f C is 6. We can decmpse C int tw classes as fllwing: C = C 0 C 1 = {1, 3, 5, 7, 11, 14, 16} {2, 4, 6, 8, 10, 12, 13, 15, 17}. Therem Each cmpnent f the dependency graph χ f with lp number t decmpses int d cmpnents in χ f m with lp number t/d where d = gcd(m,t) and the lp number f newly generated cmpnents is t/d. Prf. This cmes directly by the prperties f the equivalence classes and the abve decmpsitin. Therem says that t find ut cycles f length greater than 1, it is enugh t lk at χ m f where gcd(m,t) > 1. Thus, t find ut cycles f length > 1, we d nt have t lk χ f m fr all m > 1. It is enugh t cnsider χ f m and the pset f χ f m fr 24

33 m such that m t and gcd(m,t) > 1 where t is the lp number f sme cmpnent in χ f m. Therem 3.8. in [Jarrah et al., 2008] shwed the number f pints f certain perid, equivalently the number f cycles f certain length. Here we present simpler prf fr the number f cycles f a given length dividing the lp number. Therem Suppse χ f has nly ne cmpnent C with the lp number t and k is a psitive integer which divides t. Let l(k) be the number f cycles whse lengths are exactly k. Then l(k) = 1 k d k µ(d)2 dt k. Prf. Fr any d,k that divide t, let A(d) be the set f peridic pints f perid f d as defined in [Jarrah et al., 2008]. Since the set f cycles f length d is pairwise disjint, A(d) = d l(d). Frm [Jarrah et al., 2008, Lemma 3.6.], we knw d l(d) = A(d) = 2 k. d k d k Then Móbius inversin gives us k l(k) = d t µ(d)2 k d. Hence l(k) = 1 k µ(d)2 k d. d k 25

34 2.4 Mnmial Dynamics ver General Finite Fields In this sectin, we study mnmial dynamics ver general finite fields. Even thugh we can apply many techniques that we discussed in the previus chapters, there is limitatin t them. Thus, we will study what the difficulties fr general finite fields are and hw we apprach this prblem. Nte that fr cycles f lengths greater than 1, we can cnvert the prblem t finding fixed pints f f m fr m > 1, we will fcus n finding fixed pints f f. Let q be a pwer f an dd prime and f = (f 1,f 2,...,f n ) : Fq n Fq n be a mnmial map. Then, fr each i = 1,...,n, f i = c i x m i1 1 x m i2 2 x m in n = γ bi x m 1i 1 x m 2i 2 x m ni n where γ is a primitive element f F q. Withut lss f generality, we assume that f i 0 fr all 1 i n. Since any nn-zer element in F q can be represented as a certain pwer f γ, we can take lg n bth sides and btain n lg α f i b i + m ij lg γ x j (md q 1). j=1 ) Let A = (m ij N n n. Nw we can express the mnmial map f = (f 1,f 2,...,f n ) as a matrix representatin. ( lgγ f 1, lg γ f 2,...,lg γ f n ) = (b1,b 2,...,b n ) + ( lg γ x 1, lg γ x 2,...,lg γ x n ) A. We als write this as lg γ f = b + lg γ x A 26

35 where b = (b 1,b 2,...,b n ), lg γ x = (lg γ x 1, lg γ x 2,...,lg γ x n ). Observe that f b + lg γ x A, f 2 b(i + A) + lg γ x A 2, f 3 b(i + A + A 2 ) + lg γ x A 3,. f k b(i + A + A A k 1 ) + lg γ x A k. Next we shw hw this can be used t find the fixed pints f f. Nte that we can still find which crdinates are zer in a fixed pint by examining the pset f χ f just as we did ver F 2 by using Prpsitin After chsing the nnzer cmpnents, we need t shw hw t find the actual values. Withut lss f generality, we demnstrate belw hw t find fixed pints x that are nnzer n all cmpnents. We call such fixed pints nntrivial fixed pints. Then a pint x is a nntrivial fixed pint f f if and nly if it satisfies lg γ x b + lg γ x A (md q 1), i.e., lg γ x (I A) b (md q 1). (2.1) Let M = I A. Then since Z/(q 1) is a principal ideal dmain, we knw that there 27

36 exists invertible matrices P,Q Z n n q 1 such that d 1... P M Q D d l (md q 1) where d i q 1 and d i d j fr all i j. The matrix D is called a Smith-Nrmal Frm. Thus (2.1) implies that lg γ x P 1 D b Q (md q 1). Let y = lg γ x P 1 and b = b Q. Then, since P is invertible, x is a fixed pint f f if and nly the crrespnding y is the slutin t y D = b (md q 1). The system has slutin if and nly if d i b j fr 1 i r, 0 b j (md q 1) fr r + 1 i n. Thus finding nntrivial fixed pints is equivalent t slving the linear equatin ver a ring prvided discrete lg prblem is relatively easy n given finite field, i.e., this apprach wrks efficiently when F q is small. Here is an example which explains hw t find nn-trivial fixed pints with this apprach. 28

37 Example Let a mnmial dynamical system f : F 3 5 F 3 5 be f(x 1,x 2,x 3 ) = (x 2,x 2 1x 3 2x 2 3, 3x 3 1x 2 2). Then the dependency graph χ f f f has nly ne cmpnent s that there is nly ne trivial fixed pint, (0, 0, 0) Figure 2.11: Dependency Graph χ f f f. Nw we want t find nntrivial fixed pints f f. Nte that F 5 =< 2 >. Thus, using the abve idea, this mnmial system can be represented as the fllwing linear equatin: lg 2 f = (0, 0, 3) + lg 2 x 1 3 2, i.e., lg 2 f = b + lg 2 x A. Fixed pints f f satisfy lg 2 x = b + lg 2 x A. 29

38 Let M = I A = Then P M Q D (md 4) where P = 1 2 2,Q = 0 0 1, and D = Nte that b = b Q (1, 0, 2) (md 4). Since y D (y 1,y 2, 2y 3 ) (md 4), the slutins t y D b (md 4) are y = (1, 0, 1) and (1, 0, 3). Then lg 2 x = (1, 1, 0) and (3, 3, 2), i.e., the nntrivial fixed pints are x = (2, 2, 1) and (3, 3, 4). Nte that ring Z/(q 1) has zer-divisrs. Thus it is pssible that the dynamics f a given map des nt have nntrivial fixed pints. The fllwing tw examples will shw such cases with different reasns. Example Let a mnmial dynamical system f : F 3 5 F 3 5 be f(x 1,x 2,x 3 ) = (x 2,x 2 1x 2 x 2 3, 3x 3 1x 2 2). 30

39 Nte that this map is btained by small mdificatin f the map in Example and, indeed, the dependency graph f f and its cmpnent pset is the same with that given in Example With the same apprach, this mnmial system can be represented as the fllwing linear equatin: lg 2 f = (0, 0, 3) + lg 2 x 1 1 2, i.e., lg 2 f b + lg 2 x A (md 4). Fixed pints satisfy lg 2 x b + lg 2 x A (md 4). Let M = I A = Then P M Q D (md 4) where P = 1 2 2,Q = 0 0 1, and D = Nte that b = b Q (1, 0, 2) (md 4). Since y D (y 1,y 2, 0) (md 4), there is n 31

40 y Z 3 4 satisfying y D (1, 0, 2) (md 4). Hence there is n nntrivial fixed pints. the scalar. Here is the example f dynamics which has n nn-trivial fixed pints due t Example Let a mnmial dynamical system f : F 3 5 F 3 5 be f(x 1,x 2,x 3 ) = (x 2, 3x 2 1x 3 2x 2 3, 3x 3 1x 2 2). T btain this map, we altered the cnstant cefficient f f 2 f the map in Example Thus the linear equatin representing the given systems will be identical except b. S we have lg 2 f = (0, 3, 3) + lg 2 x 1 3 2, i.e., lg 2 f = b + lg 2 x A. Frm Example 2.4.1, we already knw that b = b Q (1, 0, 1) (md 4) and y D (y 1,y 2, 2y 3 ) (md 4). Since 2 is nt invertible in Z 4, there is n y Z 3 4 such that y D (1, 0, 1) (md 4). Hence there is n nn-trivial fixed pints. 32

41 Chapter 3 Finite Cverings 3.1 Intrductin The idea f finite cverings riginated frm the hlmrphic dynamics literature. In this sectin, we wuld like t give a brief explanatin f it and present examples f dynamical systems ver finite fields which can be explained by it. Fr mre general infrmatin, it is recmmended t read On the Lattè s Map by J. Milnr in [Hjrth and Petersen, 2006]. Let f be a ratinal map f degree tw r mre frm the Riemann sphere Ĉ = C { } t itself and E f be the set f all pints with finite grand rbit under f. f is called a finite qutient f an affine map if there exists a discrete additive subgrup Λ f C, an affine map L : C/Λ C/Λ, and a finite-t-ne hlmrphic map Θ : C/Λ Ĉ \ E f such that the fllwing diagram is 33

42 cmmutative: C/Λ L C/Λ Θ Θ Ĉ \ E f f Ĉ \ E f The cmmutativity f the diagram implies that each rbit f a pint in C/Λ is prjected by Θ t an rbit f a pint in Ĉ \ E f and since L is an affine map, its dynamics are simpler than that f f. Especially, when every pint in Ĉ \ E f has preimages in C/Λ, we say the dynamics f L n C/Λ cvers that f f n Ĉ \ E f. This prvides a great tl t study dynamics f ratinal maps. Finite qutients f affine maps can be classified as pwermaps, Chebyshev maps, and Lattè s maps accrding t their Julia sets. These three classes can be well extended t ver finite fields and the dynamics f them n finite fields can be explained easily. Fr example, let F q be a finite field f q elements where q is a pwer f prime. It is easy t see that the dynamics f n-th pwer map n F q is cvered by that f the multiplicatin by n n Z/(q 1) which can be analyzed effrtlessly. Fr Chebyshev maps ver finite fields, suppse L : Z/(q 2 1) Z/(q 2 1) by L(y) = ny fr y Z/(q 2 1) and π(y) = α y + α y where α is a generatr f F q. Then we have the fllwing cmmutative diagram: 2 y L ny π π α y + α y f α ny + α ny Ntice that the image f π cntains F q and f is the n-th degree Chebyshev plynmial. As π is a 2 cver, which is the reasn t use the quadratic extensin, any dd cycle 34

43 f L prjects via π t a cycle f f f the same length and any even cycle f L prjects t a cycle f half length. The cycle lengths f L are the rders f n mdul m where m (q 2 1). Therefre, the cycle lengths f the n-th degree Chebyshev plynmial n F q are determined by the rders f n mdul m with m running thrugh the divisrs f q 2 1. When we restrict V t a finite field, hlmrphicity is nt defined. There is a pssibility that we have maps which are nt finite qutients f affine maps but whse dynamics can be explained by this idea. We can generalize this idea as fllws: Let V and W be algebraic varieties and f : V V and g : W W be mrphisms. Then g is called an n-cvering f f if there exists a map π : W V where fr any x V, π 1 (x) = n (cunting multiplicity) such that the fllwing diagram is cmmutative: W g W π π V f V Thus ur main cncern is t study the dynamics f f ver V by explring a prper cvering space W, a cvering mrphism g, and the prjectin map π and studying the dynamics f g ver W. In the fllwing, we present a map which is nt a finite qutient f an affine map ver C, but whse dynamics can be analyzed by finite cvering. 35

44 3.2 A Dynamical System and its Assciated Elliptic Curve Let f : F 2 n { } F 2 n { } be a map defined as x + x 1 if x F 2 f(x) = n, if x = 0 r. Figure 3.1, Figure 3.2, and Figure 3.3 shw the dynamics f f n F 2 n { } fr different values f n. Figure 3.1: Dynamics f f(x) = x + x 1 n F 2 4 { }. As we see in the figures, the dynamics f f shw regularities in structures f cycles and trees. H.W. Lenstra, Jr. bserved that f can be cvered by dynamics f a certain isgeny n a Kblitz curve [Kblitz, 1991]. Mre precisely, let E be the elliptic curve grup ver the algebraic clsure F 2 defined by E : y 2 + xy = x (3.1) 36

45 Figure 3.2: Dynamics f f(x) = x + x 1 n F 2 5 { }. Then, with the pint O at the infinity, E frms an abelian grup with respect t the additin f pints defined as fllwing [Silverman, 1986, Grup Law Algrithm 2.3.]: O is an identity in E. Let P = (x 1,y 1 ) and Q = (x 2,y 2 ). If P O, then P = (x 1,y 1 + x 1 ). Suppse Q P. Then P + Q = (x 3,y 3 ) where ( y 1+y 2 x 1 +x 2 ) 2 + y 1+y 2 x 1 +x 2 + x 1 + x 2 if P Q, x 3 = x if P = Q. x 2 1 and ( ) y 1 +y 2 x 1 +x 2 (x 1 + x 3 ) + x 3 + y 1 if P Q, y 3 = x (x 1 + y 1 x 1 )x 3 + x 3 if P = Q. Let σ : E E be the Frbenius mrphism, that is, fr P = (x,y) O, σ(x,y) = (x 2,y 2 ). Define a map g : E E by g(p) = P + σ(p) where + is the additin f pints n the curve. Nte that, fr P = (x,y) O, g(x,y) = (I + σ)(x,y) = (x,y) + (x 2,y 2 ) = (x,y ), 37

46 Figure 3.3: Dynamics f f(x) = x + x 1 n F 2 6 { }. where x = x + x 1 and x y + y if (x,y) (x 2,y 2 ), y x = 2 x 2 x y + y if (x,y) = (x 2,y 2 ). x x (3.2) Thus we have the fllwing cmmutative diagram: E g E π π F 2 { } f F 2 { } where the prjectin map π is defined as x if P = (x,y) O, π(p) = if P = O. 38

47 Let E(F 2 2n) be the set f F 2 2n-ratinal pints f E. Since fr any x F 2 n { }, π 1 (x) E(F 2 2n), the dynamics f g n E(F 2 2n) cvers that f f n F 2 n { }. Mrever, g is an isgeny f E, i.e., a grup hmmrphism n E. This enables us t fcus n the dynamics f g n E(F 2 2n) t understand that f f n F 2 n { }. Thrughut this chapter, E will dente the elliptic curve as defined in (3.1), End(E) dentes the ring f grup endmrphism, m trsin grup f E ver algebraic clsure is dente by E[m], and, fr a field k, E(k)[m] dentes E[m] E(k). Fr a ratinal prime p, E p (k) dentes p subgrup f E(k), i.e., the rder f any elements in E p (k) is a pwer f p. 39

48 3.3 Prperties f g n E Since I and σ are endmrphisms f E, s is g. Thus g(p +Q) = g(p)+g(q). One can check that the minimum plynmial m σ (T) f σ is m σ (T) = T 2 + T + 2 Z[T]. S the minimum plynmial g is m g (T) = T 2 T + 2 Z[T], i.e., g 2 g + 2 = 0 (3.3) as a map n E. Since (0, 1) is the nly pint f rder 2 and O is the nly fixed pint f g, ne can shw that kerg = {O, (0, 1)}. Then we have the fllwing recurrence relatin: fr any n 1, gn g n+1 = gn 1 g n. Let M = Then, fr any n 0 and P E, gn (P) = M n g n+1 (P) P. (3.4) g(p) Thus the dynamics f g depends n the behavir f M and the subgrups < P > and < g(p) > f E. The fllwing prpsitins shw the basic prperties f g. 40

49 Prpsitin Fr any pint P E with dd rder, g(p) has the same rder. Prf. Suppse the rder f P is m which is dd. Since mp = O, g(mp) = mg(p) = O, the rder l f g(p) divides m. Als, g(lp) = lg(p) = O, s lp is in the kernel f g. If lp = (0, 1), then 2lP = O, i.e., 2l m, which cntradicts that m is dd. Thus lp = O. Hence, l = m. Prpsitin Suppse P E and P = m with m even. Then g(p) = m 2. Prf. Let m = 2l. Then mp = 2(lP) = O. Since (0, 1) is the nly pint f rder 2, lp = (0, 1). Thus lg(p) = g(lp) = g((0, 1)) = O, i.e., l divides g(p). Nte that g(p) < l implies that P < 2l. Hence, the rder f g(p) = l. Prpsitin and Prpsitin tell us that E[m] is g invariant fr any psitive integer m. Althugh it is enugh t cnsider the grup structure f E(F 2 2n) fr ur purpse, we investigate the grup structure f E(F 2 n) fr any n 1. 41

50 3.4 Grup Structure f E(F 2 n) Suppse #E(F 2 n) = p p hp where p s are ratinal primes and h p 1. Since E(F 2 n) is a finite abelian grup, E(F 2 n) is decmpsed as E(F 2 n) = E 2 (F 2 n) + p 2 E p (F 2 n) where E p (F 2 n) is p subgrup f E(F 2 n). As prved in Sectin 3.3, E p (F 2 n) is g invariant. Thus we study the structure f E p (F 2 n) fr each prime divisr p f #E(F 2 n). By Therem 3 in [Rück, 1987], E 2 (F 2 n) = Z/(2 h 2 ) where h 2 = ν 2 (n) + 2, i.e., E 2 (F 2 n) is a cyclic grup f rder 2 h 2. We will explre the size f E 2 (F 2 n) in depth in Sectin 3.5. Nw we fcus n E p (F 2 n) fr p 2 ratinal prime dividing #E(F 2 n). Therem 3 in [Rück, 1987] als says E p (F 2 n) = Z/(p ap ) Z/(p hp ap ) where 0 a p h p. Recall that σ 2 + σ + 2 = 0 as a map n E. S Q(σ) = Q( 7). Mrever, since Z[σ] is the ring f integers fr Q(σ), End(E) = Z[σ]. Z[σ] is, in fact, a PID. Lemma ([Rück, 1987]). Let m be a psitive dd integer. Then E[m] E(F 2 n) if and nly if σ n 1 = m w End(E) where w End(E). 42

51 Prf. Suppse E[m] E(F 2 n). Then the kernel f multiplicatin by m is cntained in the kernel f σ n 1. Since multiplicatin by m is separable [Silverman, 1986, Crallary 5.4.], the universal mapping prperty fr Abelian varieties [Weil, 1948, Prpsitin 10.] shws that σ n 1 = m w where w End(E). Suppse σ n 1 = w m End(E). Fr any pint P E[m], (σ n 1)(P) = w(mp) = wo = O, which implies that P E(F 2 n). Thus E[m] E(F 2 n). The factrizatin f σ n 1 gives us infrmatin n the structure f E(F 2 n). In this sectin, we analyze the structure f E(F 2 n) by studying the factrizatin f σ n 1 in Z[σ]. Fr ur purpse, we dente ν p ( ) the valuatin crrespnding t a prime p in Z[σ]. Fr a ratinal prime p and fr any α + βσ Z[σ] with α,β Z, we define ν p (α + βσ) by ν p (α + βσ) = min(ν p (α),ν p (β)) where ν p ( ) is the valuatin f Z crrespnding t p. Lemma Let p Z be a ratinal prime with p 2. Suppse σ n 1 = p t w Z[σ] where p w. Then E p (F 2 n) = Z/(p ap ) Z/(p bp ) with a p = t and b p = t + ν p (w w) where w is the cnjugate f w in Z[σ]. Prf. Suppse σ n 1 = p t w Z[σ] where p w. Then Lemma implies that E[p t ] E(F 2 n), but E[p t+1 ] E(F 2 n). Frm Crllary 6.4.(b) in Silverman [1986], we knw that E[p t ] = Z/(p t ) Z/(p t ). 43

52 Nte that E(F 2 n) = ker(σ n 1) by definitin. S #E(F 2 n) = # ker(σ n 1) (by [Silverman, 1986, III.5.5. and III.4.10.c.]) = deg(σ n 1) (by [Silverman, 1986, III.6.1.]) = (σ n 1)(σ n 1) where σ is the dual isgeny f σ. Thus #E(F 2 n) = (σ n 1)(σ n 1) = (p t w)(p t w) = p 2t ww. This implies that ν p (#E(F 2 n)) = 2t + ν p (ww). Since E p (F 2 n) cntains E[p t ] but nt E[p t+1 ], E p (F 2 n) = Z/(p ap ) Z/(p bp ) where a p = t and b p = t + ν(ww). in Z[σ]. Thus, t determine a p fr each p, we need t knw the factrizatin f σ n 1 Lemma Suppse p Z[σ] is prime and n 0 is the smallest natural number such that ν p (σ n 0 1) = e with e 1. Then ν p (σ n 1) e if and nly if n 0 n. Prf. Write n as n = an 0 + r where 0 r n 0 1. Since σ n 0 1 (md p e ), we have σ n = σ an 0+r = (σ n 0 ) a σ r σ r (md p e ). Thus σ n 1 (md p) if and nly if σ r 1 (md p). Since n 0 is the smallest such that σ n 0 1 (md p), r = 0. Hence, n 0 n. 44