Biased statistics for traces of cyclic p-fold covers over finite fields

Transcription

1 Biased statistics for traces of cyclic p-fold covers over finite fields Alina Bucur, Chantal David, Brooke Feigon and Matilde Lalín February 25, 2010 Abstract In this paper, we discuss in ore detail soe of the results on the statistics of the trace of the Frobenius endoorphis associated to cyclic p-fold covers of the projective line that were presented in [1] We also show new findings regarding statistics associated to such curves where we fix the nuber of zeros in soe of the factors of the euation in the affine odel MSC: 11G20, 11T55, 11G25 1 Introduction To epsilon Let p be a prie and fix a prie power such that 1 od p In [1], we discussed the statistics for the distribution of the trace of the Frobenius endoorphis of curves C when C varies over irreducible coponents of the oduli space of cyclic p-fold covers of P 1 F To do so, we first consider all curves with affine odels where Y p F X, F F d1,,d p 1 F [X], 11 F d1,,d p 1 F F 1 F2 2 F p 1 p 1 : F 1,, F p 1 F [X] are onic, suare-free, pairwise coprie, and deg F i d i for 1 i p 1} Unless otherwise entioned, all polynoials will be onic Roughly speaking, varying over all curves of an irreducible coponent of the oduli space of cyclic p-fold covers eans to vary over odels 11 with F in certain unions of sets of the type F d1,,d p 1, and statistics for the trace of Frobenius over the coponents of the oduli space can be deduced fro the statistics associated to these sets see Section 5 Let µ p C be the set of pth roots of unity, let µ 0 p µ p 0}, let ξ p be a priitive pth root of unity, and let χ p be a non-trivial character of order p of F Let C be the cyclic p-fold cover with affine odel 11 Then, the nuber of affine points of C is x F 1 + χ p F x + S p F, Institute for Advanced Study, alina@athiasedu Concordia University, cdavid@athstatconcordiaca University of Toronto, bfeigon@athtorontoedu University of Alberta, lalin@athualbertaca 1

2 and the affine trace is S p F x F χ p F x Theore 11 [1, Theore 73] Let ε 1,, ε p 1 µ 0 p such that of the ε i are zero Then, as d 1,, d p 1, F F d1,,d p 1 : χ p F x i ε i, 1 i } p 1 + p 1 p + p 1 Fd1,,d p 1 Furtherore, let X 1,, X be iid rando variables taking the value 0 with probability p 1/ + p 1 and each of the values in µ p with eual probability /p + p 1 Then, as d 1,, d p 1, F F d1,,d p 1 : S p F t } Prob X i t for any t Z[ξ p ] C F d1,,d p 1 The case p 2 was proven in [2], and the case p 3 was proven in [1] The proof of the general case was sketched in [1], and we give ore details in Section 3 of this paper It ay not be clear a priori where the rando variables of Theore 11 coe fro, but they can be explained by a siple heuristic See [1] for ore details As an interediate step in the proof of Theore 11, we have to consider the sets of polynoials F k1,,kp 1 d 1,,d p 1 F F d1,,d p 1 : F i has k i roots over F, 1 i p 1 } Those spaces do not have a natural geoetric interpretation as the sets of polynoials F d1,,d p 1 which paraetrize irreducible coponents of oduli spaces, but they also lead to interesting results involving natural probabilities We present those results in this paper We first concentrate on the case p 3 where the results are easier to explain, and then ove on to the general case for any odd prie p Theore 12 Fix 0 k and let i1 F k d 1,d 2 F F 1 F 2 2 F d1,d 2 : F 2 has k roots over F } Let ε 1,, ε 0, 1, ξ 3, ξ3} 2 such that of the εi are zero Then, as d 1, d 2, F Fd k : χ 1,d 2 3F x i ε i, 1 i } k k 1 k F k d 1,d 2 Let X 1,, X be rando variables taking the value 0 with probability 1/ + 1 and any of the values 1, ξ 3, ξ3 2 with eual probability /3 + 1 together with a bias counting the nuber of k-tuples of roots taken by the variables as described in Section 2 Then, as d 1, d 2, F Fd k : S 1,d 2 3F t} Prob X i t for any t Z[ξ 3 ] C F k d 1,d 2 i1 2

3 The proof of Theore 12 is given in Section 2 We can interpret those results in the following way: Let F d be the set of onic suare-free polynoials F of degree d It follows by exactly the sae steps as Theore 11 for p 2 that, as d, F F d : S 3 F t} F d Prob X i t, 12 where X 1,, X are iid rando variables variables taking the value 0 with probability 1/ + 1 and any of the values 1, ξ 3, ξ3 2 with eual probability / Then, if F 2 has no roots over F k 0, the polynoials F Fd k lead to the sae probability as the suare-free polynoials If F 1,d 2 2 has k roots over F 1 k, then the rando variables X 1,, X of Theore 12 are distributed in such a way that the probability that X i ε i for 1 i depends on, the nuber of zeros aong ε 1,, ε and the X i are not iid in this case More precisely, ProbX i ε i, 1 i is k T 1 ties the probability associated with suare-free polynoials, where k is the nuber of k-tuples of zeros aong ε 1,, ε and T is a noralizing factor insuring that the su of the probabilities is 1 see Section 2 for ore details We then say the probability ProbX i ε i, 1 i in this case is the probability associated with the faily of suare-free polynoials biased by the nuber of zeros of ε 1,, ε We now study the general case, where we take polynoials F F 1 F2 2 F p 1 p 1 in F d 1,,d p 1 where soe of the F i have a prescribed nuber of roots Theore 13 Fix 1 v p 1, and let k v+1,, k p 1 be non-negative integers with k k v k p 1 We also suppose that 0 k Let F,kv+1,,kp 1 d 1,,d p 1 F F d1,,d p 1 : F i has k i roots over F, v + 1 i } Let ε 1,, ε µ 0 p such that of the ε i are zero Then, as d 1,, d p 1, F F,kv+1,,kp 1 d 1,,d p 1 : χ p F x i ε i, 1 i } F,kv+1,,kp 1 d 1,,d p 1 i1 k k k v + v p + v Let X 1,, X be rando variables taking the value 0 with probability v/ +v and any of the values in µ p with eual probability /p + v together with a bias counting the nuber of k-tuples of roots taken by the variables as described in Section 3 Then, as d 1,, d p 1, F F,kv+1,,kp 1 d 1,,d p 1 : S p F t} F,kv+1,,kp 1 Prob X i t for any t Z[ξ p ] C d 1,,d p 1 The proof of Theore 13 is given in Section 4 We can interpret the results as we did for the case of p 3 If v 1 and k 0, then the probability for the set of polynoials F 1 F2 2 F p 1 p 1 is the sae as for the suare-free polynoials If v 1 and 1 k, then the probability ProbX i ε i, 1 i in this case is the probability associated with the faily of suare-free polynoials biased by the nuber i1 1 The difference between 12 and Theore 11 with p 2 is that in the forer case, we consider the trace S 3 F and in the latter case, the trace S 2 F Both results follow directly fro a count on how any polynoials in F d take a prescribed set of values 3

4 of zeros of ε 1,, ε When v 2 and k 0, then the probability ProbX i ε i, 1 i in this case is the probability associated with the faily of polynoials F 1 F2 2 with F 1, F 2 onic, suare-free and co-prie If v 2 and 1 k, the probability ProbX i ε i, 1 i in this case is the probability associated with the faily of polynoials of the for F 1 F2 2 biased by the nuber of zeros of ε 1,, ε The general case follows siilarly We can give a ore geoetric version of Theore 11 in ters of the oduli space of cyclic p-fold covers For any prie p, let H g,p denote the oduli space of cyclic p-fold covers of genus g It breaks into a disjoint union of irreducible coponents H d1,,dp 1 indexed by the inertia type of the branch points of F X, and H g,p H d1,,dp 1, 13 d 1+2d 2+ +p 1d p 1 0 od p 2gp 1d 1+ +d p 1 2 where the union is disjoint and each coponent H d1,,dp 1 is irreducible see Section 5 for ore details We reark that for p 2, there is just one coponent in the above decoposition and the oduli space is irreducible Consider the first étale cohoology group with Q l coefficients, where l is a prie with l 1 od p so that Q l contains the pth roots of unity The group of order p generated by the cyclic autoorphis acts on the curve C and therefore on the first cohoology group, which gives us a representation of the aforeentioned cyclic group on this Q l -vector space Since the group is abelian and we have enough roots of unity in Q l, the representation splits into a direct su of 1-diensional representations Since these are 1-diensional representations, they correspond to ultiplication by soe scalar In order to find the scalar, one can use the Lefschetz Verdier fixed point forula fro which it follows that our cyclic autoorphis acts on these subspaces by ultiplication by different powers of χ p and the diensions of isotypical subspaces are eual For ore details, see [3] However, the Rieann Hurwitz forula shows that the trivial character appears with ultiplicity 0 This shows that the cyclic autoorphis of order p splits the first cohoology group of C into p 1 subspaces H 1 χ p, H 1 χ 2 p,, H 1 χ p 1 p the autoorphis acts by ultiplication by χ p, χ 2 p,, χ p 1 p on which respectively Futherore, the action of the cyclic autoorphis is defined over the base field F since this contains the pth roots of unity and Frobenius fixes F Thus the two actions of Frobenius and of the cyclic autoorphis coute and it suffices to study the trace of the Frobenius on one of these subspaces, say TrFrob C H 1 χp Moving to another subspace corresponds to a new choice of the character χ p Theore 14 [1, Theore 74] If is fixed and d 1,, d p 1, the distribution of the trace of the Frobenius endoorphis associated to C as C ranges over the coponent H d1,,dp 1 of cyclic p-fold covers of P 1 F is that of the su of + 1 iid rando variables X 1,, X +1, where each X i takes the value 0 with probability p 1/ + p 1 and each value in µ p with probability /p + p 1 More precisely, for any t Z[ξ p ] C with t + 1 and any 1 > ε > 0, we have, as d 1,, d p 1, } C H d1,,dp 1 : TrFrob C H 1 χp t H d1,,dp 1 Prob +1 X i t In the last theore, and in the rest of the paper, the notation, applied both to suation and cardinality, eans that curves C on the oduli spaces are counted with the usual weights 1/ AutC as in the ass forula The proof of Theore 14 is given in Section 5 i1 4

5 2 Cyclic trigonal curves and proof of Theore 12 For p 3, a cyclic p-fold cover of P 1 F is called a cyclic trigonal curve, and every cyclic trigonal curve has an affine odel of the for Y 3 F X where F X F 1 XF 2 X 2 F d1,d 2 Let 0 k We consider in this section the sets of polynoials F k d 1,d 2 F F 1 F 2 2 F d1,d 2 : F 2 has k roots over F } Fix 1 od 3 In all this section, ξ 3 denotes a non-trivial third root of unity in C, and χ 3 a non-trivial cubic character of F We will denote by ζ the incoplete zeta function of the rational function field F [X] given by ζ s F F s P 1 P s s 1 Proposition 21 [1, Proposition 43] Let 0 l, let x 1,, x l be distinct eleents of F, and a 1,, a l F Then for any 1 > ε > 0, we have F F d1,d 2 : F x i a i, 1 i l} Kd1+d2 ζ 2 2 l 1 + O 1 εd2+εl + d1/2+l, where K P 1 1 P + 1 2, 21 and the product runs over all onic irreducible polynoials of F [X] In particular, taking l 0, we have F d1,d 2 Kd1+d2 ζ O 1 εd2 + d1/2 22 The statistics for the nuber of polynoials F Fd k 1,d 2 taking prescribed values then follow easily fro the previous proposition Corollary 22 [1, Corollary 44] Let x 1,, x be an enueration of eleents in F Let a 1 a 0, and a +1,, a F Then, for 1 > ε > 0, F Fd k 1,d 2 } K d 1+d : F x 2 i a i, 1 i k ζ O εd2 k+ε + d1+k /2+ Corollary 23 Let x 1,, x be an enueration of eleents in F Let a 1 a 0, and a +1,, a F Then, for 1 > ε > 0, F Fd k : F x 1,d 2 i a i, 1 i } F k d 1,d 2 k k 1 k O 1 εd2 k+ε + d1+k 3/2, 5

6 and F Fd k : χf x 1,d 2 i ε i, 1 i } F k d 1,d 2 Proof We first use Corollary 22 to copute which are zero Then F k d 1,d 2 k a 1,,a F Ma 1,,a F k d 1,d 2 By Corollary 22, the ain ter of the above su euals K d1+d2 ζ k k k k 1 k O 1 εd2 k+ε + d1+k 3/2 Let Ma 1,, a be the nuber of values of a i F Fd k : F x 1,d 2 i a i, 1 i } Kd1+d2 ζ 2 2 k k + 2 k k k K d 1+d 2 k + 1 ζ k + 2 Taking the axial value of the error ter for between k and, this gives F k d 1,d 2 K d 1+d 2 k k ζ k The first assertion follows by dividing the result of Corollary 22 by O 1 εd2 k+ε + d1+k 3/ F k d 1,d 2, and the second assertion by rearking that for ε i 0, then χf x i ε i if and only if F x i 0, and for ε i 1, then χf x i ε i if and only if F x i is one of the 1/3 cubes in F, and siilarly for ε i ξ 3, ξ 2 3 Taking d 1, d 2 in Corollary 23, this proves the first stateent of Theore 12 We can also describe the asyptotic of Corollary 23 in ters of a natural probability Let X be the rando variable taking the value 0 with probability 1/+1, and any value } 1, ξ 3, ξ3 2 with probability /3 + 1, as in Theore 12 For each -tuple ε 1,, ε, 0, 1, ξ 3, ξ3} 2 let be the nuber of i such that ε i 0 Let X 1,, X be rando variables distributed as X with a bias counting the unordered k-tuples i 1,, i k } 1,, } such that ε i1 ε ik 0 More precisely, let 1 1 Prob X i ε i : 1 i, 24 k T

7 where T Using the value of T in 24, we get ε 1,,ε 0,1,ξ 3,ξ 2 3} 1 k k k k 1 k k k 1 k + 1 Prob X i ε i : 1 i k k k 1, which is the probability appearing in Corollary 23 The second stateent of Theore 12 follows by suing the probabilities for all tuples ε 1,, ε such that ε ε t 3 General p-fold covers and proof of Theore 11 We recall that F d1,,d p 1 } F F 1 F2 2 F p 1 p 1 : F i onic, suare-free, pairwise coprie, deg F i d i, 1 i p 1 The proof of the following proposition was sketched in [1] We give ore details here For F, G F [X], let gcdf, G denote their greatest coon divisor Proposition 31 [1, Proposition 71] Fix 0 l, x 1,, x l distinct points in F and a 1,, a l nonzero eleents of F Then, for each r 1 and ε > 0, F F d1,,d r : F x i a i, 1 i l } l L r 1 d1+ +dr ζ 2 r + r 1 r 1 + O εl εd h+ +d r d h + d1/2+l, where L r 1 : K 1 K r 1, with K j : j 1, j 1, P + 1 P + j P and K 0 L 0 1 Furtherore, taking l 0, this gives r Fd1,,d r L r 1 d1+ +dr ζ 2 r 1 + O εd h+ +d r d h + d1/2 31 h2 h2 7

8 Proof If r 1, F d1 is the set of suare-free polynoials, and this is Lea 5 in [2], where the epty su d2 + + dr of the error ter is understood to be 0 We then suppose that r 2 Let G d1,,d r F 1,, F r F [X] r : F i onic, suare-free, pairwise coprie, degf i d i, 1 i r} By Lea 42 in [1], F F d1,,d r : F x i a i, 1 i l } F 2,,Fr G d2,,dr Q rj2 F j x i 0,1 i l d1 l ζ 21 2 l d1 l ζ 21 2 l S F2Fr d 1 l F 2,,Fr G d2,,dr Q rj2 F j x i 0,1 i l deg F rd r where, following the notation fro Lea 42 in [1], and we define P F 2F r 1 + P deg F 2d 2 bf 2 F r + O F 2,,Fr G d2,,dr Q rj2 F j x i 0,1 i l S U d l F F d : F, U 1, F x i a i, 1 i l}, bf µ 2 F P F 1 + P 1 1 F x i 0, 1 i l, 0 otherwise, O d1/2 d1/2+d2++dr, 32 for any polynoial F F [X] Here the product is over all the onic irreducible polynoials P F [X] dividing F We now evaluate M r : bf 2 F r deg F rd r deg F 2d 2 We notice that b is ultiplicative and that bf 2 F r 0 if the F i are not relatively prie in pairs Then we have M r bf 2 bf r degf rd r degf r 1 d r 1 gcdf r 1,Fr 1 degf 2 d 2 gcdf 2,Fr 1,,gcdF 2,F 3 1 For any j 1, let c U j F µ 2 F P F 1 + j P 1 1 F x i 0, 1 i l, gcdf, U 1, 0 otherwise 33 Notice that in particular, bf c 1 1F Then we can write M r c 1 1F r c Fr 1 F r 1 degf rd r degf r 1d r 1 1 F 2 degf 2d 2 c F3Fr 8

9 Using Lea 32 with j 1, we have c F3Fr 1 F 2 K l 1 d2 + 1 ζ degf 2d 2 P F 3F r P O εd2+ +dr+l d2 34 P + 2 Using 34 in 32, this proves the theore for r 2 For r 3, we first have to evaluate 1 F 3 1 F 2 degf 3d 3 c F4Fr K 1 d2 ζ 2 K 1 d2 ζ 2 K 1 d2 ζ 2 degf 2d 2 c F3Fr l + 1 c F4Fr 1 F degf 3d 3 l l P F 4F r P F 4F r P + 1 P + 2 Using Lea 32 with j 2, this gives 1 F 3 1 F 2 degf 3d 3 c F4Fr K 1K 2 d2+d3 ζ 2 2 P F 3F r degf 3 d 3 F 3 x i 0,1 i l gcdf 3,F 4 Fr 1 P O εd2+ +dr+l d2 P + 2 µ 2 F 3 P 1 + O εd2+ +dr+l d2 P + 2 P F 3 P + 1 c F4Fr 2 F O εd2+ +dr+l d2 P + 2 degf 3d 3 degf 2d 2 c F3Fr l P F 4F r P O εd2+ +dr+l d2 + εd3+ +dr+l d3 P + 3 Using the last euality in 32, this proves the Proposition with r 3 In general, continuing in this way up to the last su deg F rd r, we obtain the result Lea 32 Assue the hypotheses of Proposition 31 and let U be a polynoial of degree u with Ux i 0 for 1 i l Let j 1, and let c U j F be defined as in 33 Then for any 1 > ε > 0, c U j F K j d l + j P + j 1 + O εd+u+l d ζ 2 + j + 1 P + j + 1 degf d P U Proof We will use the function field version of the Wiener-Ikehara Tauberian Theore This is Theore 171 in [5] For our application, it is iportant to get an error ter which is independent of U and and for this we need a ore precise stateent of the Tauberian Theore than in [5] so we will go through the proof here First, we consider the Dirichlet series G j s F c U j F F s ζ s ζ 2s H js P P x i 0,1 i l,p U s 1 + j P s l P U P P + j P s P, P + j 9

10 where H j s P j 1 P s + 1 P + j In the above, we have used the hypothesis that Ux i 0, and therefore the pries P U are different fro the pries X x i for 1 i l Notice that H j s converges absolutely for Res > 0, and G j s is eroorphic for Res > 0 with siple poles at the points s where ζ s 1 1 s 1 has poles, that is, s n 1 + i 2πn log, with n Z Notice that H j1 K j, and Res s1 ζ s 1 log Thus G js has a siple pole at s 1 with residue Define Zu by Z s G j s Thus Zu 1 u2 1 u l K j + j P + j 35 ζ 2 log + j + 1 P + j + 1 P U 1 + u + j l P j 1 1 u deg P + 1 deg P + udeg P deg P 1 + j deg P + j P U Fix any ε such that 1 > ε > 0 Then Zu is a eroorphic function on the disk u u ε } with a siple pole at u 1 Let C u C u ε }, oriented counterclockwise For any 0 < δ < 1, let C δ u C u δ}, oriented clockwise Notice that Zu is a eroorphic function between the u d+1 two circles, with a siple pole at u 1 with residue Thus by the Cauchy integral forula, Zu Res u 1 u d+1 li u 1u 1 Zu u d+1 K l j + j P + j d ζ 2 + j + 1 P + j + 1 P U 1 Zu 2πi C δ +C u d+1 du K l j + j P + j d 36 ζ 2 + j + 1 P + j + 1 P U Now we observe that Thus Zu 1 2πi C δ n0 Zu u deg F n d+1 du c U j F u n deg F d c U j F 37 Cobining 36 and 37 we see that deg F d c U j F K l j + j P + j d + 1 ζ 2 + j + 1 P + j + 1 2πi P U C Zu du ud+1 10

11 Let M be the axiu value of Zu on C Then clearly 1 Zu du 2πi ud+1 Mεd so degf d c U j F To conclude the proof we note that M ax H js 1 + s ε 1 ε l P U C K l j + j P + j d + OM εd 38 ζ 2 + j + 1 P + j + 1 P U l 1 s j P s P U 1 P ε 1 εl P U 1 P P + j P ε εl U ε εu+l 39 By 39, the absolute error ter in 38 is O εd+u+l and we get the result Proposition 31 will be used with r p 1 Denote F k1,,kp 1 d 1,,d p 1 F F 1 F p 1 p 1 F d 1,,d p 1 : F i has k i roots in F, 1 i p 1 Corollary 33 Fix 0 Choose x 1,, x an enueration of the points of F, and values a 1 a 0, a +1,, a F Pick a partition k k p 1 Then, for any ε > 0, F F k1,,kp 1 d : F x 1,,d p 1 i a i, 1 i } Lp 2 d1+ +dp 1 1 k 1,, k p 1 ζ 2 p 1 + p 1 + p 1 1 p O ε εd h+ +d p 1 k h + +k p 1 d h k h + d1 k1/2+ h2 } Proof Let F be a polynoial in F k1,,kp 1 d such that F x 1,,d p 1 i a i for 1 i Then, F can be written as F X X x j bj GX, j1 where the b j are positive integers with the property that for any 1 i p 1, the nuber of b j s such that b j i is k i and G F d1 k 1,,d p 1 k p 1 is such that Gx i 0 for 1 i There are then k 1,,k p 1 choices for the bj s After this choice of b j s, we choose the values α i Gx i for 1 i 11

12 This gives, F F k1,,kp 1 d : F x 1,,d p 1 i a i, 1 i } G F k 1,, k p 1 d1 k 1,d 2 k 2,,d p 1 k p 1 : Gx i a i α 1,,α F j1 x i x j bj for + 1 i, and Gx i α i for 1 i } The result is proved by using Proposition 31 with r p 1 in the last expression Corollary 34 Choose x 1,, x an enueration of the points of F Fix 0, a 1 a 0 and a +1,, a F Then for any ε > 0, F F d1,,d p 1 : F x i a i, 1 i } L p 2 d1++dp 1 p 1 ζ 2 p 1 + p 1 + p 1 1 p O ε+1 ε εd h+ +d p 1 d h + d1 /2+ h2 Let ε 1,, ε µ 0 p and let be the nuber of values of ε i which are 0 Then for any ε > 0, F F d1,,d p 1 : χ p F x i ε i, 1 i } L p 2 d1++dp 1 p 1 ζ 2 p 1 + p 1 p + p 1 p O ε+1 ε εd h+ +d p 1 d h + d1 /2+ h2 Proof Adding over all the p 1-partitions of and using Corollary 33 and the identity p 1, k 1,, k p 1 k 1++k p 1 which follows fro the Multinoial Theore, we get the first assertion For the second assertion, we note that if ε µ p, there are 1 p eleents α F such that χ p α ε Finally, Theore 11 follows by dividing F F d1,,d p 1 : χ p F x i ε i, 1 i } Corollary 34 by F d1,,d p 1 Proposition 31 This gives F F d1,,d p 1 : χ p F x i ε i, 1 i } p 1 Fd1,,d p 1 + p 1 p + p 1 p O ε+1 ε εd h+ +d p 1 d h + d1 /2+ h2 Notice that the order in which we perfor the nested su in euation 32 is arbitrary Thus, we can assue, without loss of generality, that d 2 d p 1 while proving Proposition 31 As a result, the error ters in all the stateents of this section go to zero for d 1,, d p 1 as long as ε is sall enough, for exaple, 0 < ε < 1/r in Proposition 31 and 0 < ε < 1/p 1 in the other stateents 12

13 4 General p-fold covers and proof of Theore 13 Let v be a fixed integer such that 0 v p 1 We now study the distribution of the traces S p F when F varies over polynoials in F,kv+1,,kp 1 d 1,,d p 1 F F d1,,d p 1 : F i has k i roots over F, v + 1 i p 1 } Then, if F F,kv+1,,kp 1 d 1,,d p 1, the nuber of roots is fixed for F v+1,, F p 1 only We will always write k k v k p 1 for the total nuber of fixed roots Notice that for v 0 we necessarily have k Using Corollary 33, we obtain the following three results Lea 41 Let 1 v p 1 Fix such that 0 k Choose x 1,, x an enueration of the points of F, and values a 1 a 0, a +1,, a F Then, for any ε > 0, we have F F,kv+1,,kp 1 d 1,,d p 1 : F x i a i, 1 i } v k L p 2 d1+ +dp 1 1 k, k v+1,, k p 1 ζ 2 p 1 + p 1 + p 1 1 p O ε+1 ε εd h+ +d p 1 d h + d1+k /2+ h2 Notice that the case v 0 was already covered in Corollary 33 Proof We first write F F,kv+1,,kp 1 d 1,,d p 1 : F x i a i, 1 i } F F k1,,kp 1 d : F x 1,,d p 1 i a i, 1 i } 41 k 1+ +k v k By Corollary 33, the ain ter of 41 is L p 2 d1+ +dp 1 1 ζ 2 p 1 + p 1 + p 1 1 k 1+ +k v k, k 1,, k p 1 and then we use the fact that k 1,, k p 1 k 1+ +k v k k k 1,, k v k, k v+1,, k p 1 k 1+ +k v k v k k, k v+1,, k p 1 By using a bound on the error ter of Corollary 33 which is valid for all k 1,, k v such that k 1 + +k v k, the result follows 13

14 Lea 42 As defined above, let 1 v p 1 and k k v k p 1 with 0 k Then, for any ε > 0, F,kv+1,,kp 1 Lp 2 d1+ +dp 1 + v d 1,,d p 1 k, k v+1,, k p 1 ζ 2 p 1 + v k + p 1 p O εd h+ +d p 1 d h + d1+k 3/2 h2 For v 0, we obtain F k1,,kp 1 Lp 2 d1+ +dp 1 k d 1,,d p 1 k, k 1,, k p 1 ζ 2 p 1 + p 1 p O ε εd h+ +d p 1 k h + +k p 1 d h k h + d1 k1/2+ h2 Proof We have F,kv+1,,kp 1 d 1,,d p 1 k 1 F F,kv+1,,kp 1 d 1,,d p 1 : F x i a i, 1 i } 42 For v 1 we use Lea 41 and the chicken, chicken, chicken identity k k, k v+1,, k p 1 k, k v+1,, k p 1 Then, the ain ter of F,kv+1,,kp 1 is d 1,,d p 1 L p 2 d1+ +dp 1 ζ 2 p 1 v k v + p 1 k, k v+1,, k p 1 k Lp 2 d1+ +dp 1 k k, k v+1,, k p 1 ζ 2 p 1 v k v + p 1 k Lp 2 d1+ +dp 1 + v k, k v+1,, k p 1 ζ 2 p 1 + v k + p 1 Replacing by the axial value in the error ter of Lea 41, the result follows for v 0 If v 0 we use Corollary 33 in euation 42 in this case, the su in euation 42 has one ter for k The ain ter of is F k1,,kp 1 d 1,,d p 1 k 1 k k Lp 2 d1+ +dp 1 1 k k 1,, k p 1 ζ 2 p 1 + p 1 + p 1 1 k k Lp 2 d1+ +dp 1 1 k, k 1,, k p 1 ζ 2 p 1, + p 1 + p 1 k 14

15 and the error ter is the sae as in Corollary 33 Proposition 43 As defined above, let 1 v p 1 and k k v k p 1 with 0 k Fix such that 0 k Choose x 1,, x an enueration of the points of F, and values a 1 a 0, a +1,, a F Then for any ε > 0, F F,kv+1,,kp 1 d 1,,d p 1 : F x i a i, 1 i } F,kv+1,,kp 1 k d 1,,d p 1 k p O εd h+ +d p 1 d h + d1+k 3/2, h2 v k + v 1 + v and F F,kv+1,,kp 1 d 1,,d p 1 : χ p F x i ε i, 1 i } F,kv+1,,kp 1 k d 1,,d p 1 k p O εd h+ +d p 1 d h + d1+k 3/2 h2 v k + v p + v If v 0, we need k, and in this case F F k1,,kp 1 d : F x 1,,d p 1 i a i, 1 i } 1 F k1,,kp 1 d 1,,d p 1 k 1 k p O ε εd h+ +d p 1 k h + +k p 1 d h k h + d1 k1/2+, h2 and F F k1,,kp 1 d : χ 1,,d p 1 pf x i ε i, 1 i } 1 F k1,,kp 1 d 1,,d p 1 k p k p O ε εd h+ +d p 1 k h + +k p 1 d h k h + d1 k1/2+ h2 Proof For v > 0, the first assertion follows by dividing the result of Lea 41 by the result of Lea 42, and the second assertion by observing that χ p F a i ξ p if and only if F a i takes 1/p possible values in F For v 0 we use Corollary 33 instead of Lea 41 The result of Theore 13 then follows by taking d 1,, d p 1 in Proposition 43 15

16 We now copare the result of Proposition 43 with a probabilistic odel Let X be the rando v variable taking value 0 with probability +v, and any value in µ p with probability p+v For each -tuple ε 1,, ε µ 0 p, let be the nuber of i such that ε i 0 Let X 1,, X be rando variables distributed as X with a bias counting the unordered k-sets i 1,, i k } where ε ij 0 for j 1,, k, ie, Prob X i ε i : 1 i 1 v, k T + v p + v where T ε 1,,ε µ 0 p v k + v p + v v p k + v p + v k k v k + v + v k k v k + v Thus, we obtain Prob X i ε i : 1 i k k k v, 43 + v p + v which are the probabilities in Proposition 43 for v 0 The second stateent of Theore 13 then follows fro Proposition 43 by suing the probabilities for all tuples ε 1,, ε such that ε ε t Finally, we reark that the probability for F d1,,d p 1 can be written as the ixed probability involving the probabilities for all the F,kv+1,,kp 1 d 1,,d p 1 as F F d1,,d p 1 : χf x i ε i, 1 i } Fd1,,d p 1 F F,kv+1,,kp 1 d 1,,d p 1 : χf x i ε i, 1 i } F,kv+1,,kp 1 d 1,,d p 1 F,kv+1,,kp 1 Fd1,,d p 1 0 k i d i v<i p 1 d 1,,d p 1 5 The geoetric point of view We prove in this section that Theore 14 is a conseuence of Theore 11 Let C be a cyclic p-fold cover of P 1 F Then, C has an affine odel of the for Y p F X, where F X is a polynoial in F [X] If GX HX p F X, then Y p F X and Y p GX are isoorphic over F, so it suffices to consider curves Y p F X where F X is a polynoial which is pth-power free 16

17 Let F F [X] be pth-power free and onic Recall that pth-power free over F is the sae as pth-power free over F So F factors in F [X] as d 1 d 2 F X X a 1,i X a 2,i 2 X a p 1,i p 1 i1 i1 where the a i,j are distinct eleents of F Let C F be the cyclic p-fold cover given by Y p F X F 1 XF 2 X 2 F p 1 X p 1 where the F i are suare-free, relatively prie, and deg F i d i for 1 i p 1 and d deg F d 1 + 2d p 1d p 1 The nuber of branch points on C F is R d d p 1 if d 0 od p or R d d p otherwise as the point at infinity is a branch point in the latter case, and the Rieann-Hurwitz forula iplies that the genus is g p 1R 2/2 Then, the curve C F has genus g if d d 1 + 2d p 1d p 1 0 od p and 2g p 1d d p 1 2, or if d d 1 + 2d p 1d p 1 0 od p and 2g p 1d d p 1 1 Over F, one can reparaetrize and choose an affine odel for any cyclic p-fold cover with d 1 + 2d p 1d p 1 0 od p Furtherore, the oduli space H g,p of cyclic p-fold covers of P 1 F of a fixed genus g splits into irreducible subspaces indexed by euivalence classes of p 1-tuples of nonnegative integers d 1,, d p 1 with the property that d 1 + 2d p 1d p 1 0 od p We will not expand on the euivalence relations here, but we would like the reader to note that in the p 3 case, they aount to interchanging d 1 and d 2 The oduli space can be written as a disjoint union over its connected coponents, d p 1 i1 H g,p H d1,,dp 1, 51 where each coponent H d1,,dp 1 is irreducible For ore details about these Hurwitz spaces, see [4] Fro now on, we assue that d 1 + 2d p 1d p 1 0 od p, and we define F j d 1,,d p 1 F F 1 F 2 2 F p 1 p 1 F d 1,,d j 1,d j 1,d j+1,,d p 1} for 1 j p 1, F 0 d 1,,d p 1 F d1,,d p 1, F [d1,,d p 1] p 1 j0 F j d 1,,d p 1 For any set F of onic polynoials in F [X], we denote by F the set of polynoials αf where α F and F F This defines the sets F d1,,d p 1, F j d 1,,d p 1 and F [d1,,d p 1] which are used in this section When we write a cyclic p-fold cover of P 1 as C F : Y p F X 52 where F X is pth-power free, we choose an affine odel of the curve To copute the statistics for the coponents H d1,,dp 1 of the oduli space H g,p, we need to work with failies of odels where we count each curve, seen as a projective variety of diension 1, up to isoorphis, with the sae ultiplicity To do so, we have to consider all pth-power free polynoials in F [X], and not only onic ones We fix a genus g, and a coponent H d1,,dp 1 for this genus as in the decoposition 51 For each curve of this coponent, we want to count its different affine odels C : Y p GX Since C is obtained fro an autoorphis of P 1 F, this eans that G F [d1,,d p 1] since G F d1,,d p 1 if the roots of F are sent to the roots of G, and G F j d 1,,d p 1 if a root of F j is sent to the point at infinity 17

18 Assue that C has genus g > p 1 2 and there are two ways of writing C as a cyclic p-fold cover, ie two aps φ 1,2 : C P 1 They induce a ap φ : C P 1 P 1, φ φ 1, φ 2 The iage of φ is a curve rationally euivalent to n 1 ties the horizontal fiber plus n 2 ties the vertical fiber, where n i 1, p} since it has to divide the degree of the projections The adjunction forula says that the arithetic genus of the iage is eual to n 1 1n 2 1 p 1 2 Hence the geoetric genus is at ost p 1 2, which is strictly less then the genus of C itself Hence the ap ψ fro C to the iage of φ cannot be an isoorphis, in fact it ust have degree > 1 But both φ i factor through ψ, hence the degree of ψ can be only either 1 or p Since we already excluded 1, it follows that the degree is p, which in turn iplies that n 1 n 2 1 Hence iφ ust be the graph of an autoorphis P 1 P 1, and φ 1 and φ 2 are therefore related by this autoorphis Hence all curves C isoorphic to C are obtained fro the autoorphiss of P 1 F, naely the 2 1 eleents of PGL 2 F By running over the eleents of PGL 2 F, we obtain 2 1/ AutC different odels C : Y p GX where G F [d1,,d p 1] This shows that H d1,,dp 1 C H d 1,,d p 1 1 AutC F [d1,,d p 1] 2, 53 1 where, as before, the notation eans that the curves C on the oduli space are counted with the usual weights 1/ AutC For 1 j p 1, we denote ŜpF j χ j pf x, x P 1 F where the value of F at the point at infinity is given by the value at zero of X d1+2d2+ +p 1dp 1 F 1/X Fix an enueration of the points on P 1 F, x 1,, x +1, such that x +1 denotes the point at infinity Then leading coefficient of F F Fd1,,d F x +1 p 1, 0 F p 1 F j j1 d 1,,d p 1 The nuber of points on the projective curve C F with affine odel 52 is given by 1 p χ j pf x ŜpF j and x P 1 F j1 TrFrob C H 1 Ŝj pf, 1 j p 1 54 χ j p It follows easily fro the definitions above that p 1 p 1 p 1 F F d1,,d p 1 and leading coefficient of F is a pth-power, ŜpF j SpF j + 1 F j1 j1 F d1,,d p 1 and leading coefficient of F is not a pth-power, 0 F p 1 F j j1 d 1,,d p 1 j1 18

19 As in 53, we write } C H d1,,dp 1 : TrFrob C H 1 χp t C H d 1,,d p 1 TrFrob C H 1 χp t 1 AutC 55 It then follows fro 53, 54 and 55 that } C H d1,,dp 1 : TrFrob C H 1 χp t H d1,,dp 1 F F [d1,,d p 1] : Ŝ p F t} F 56 [d1,,d p 1] We first copute F p 1 [d1,,d p 1] 1 j0 F j d 1,,d p p 1 L p 2 d1+ +dp 1 ζ 2 p 1 57 p O εd h+ +d p 1 d h +1 + d1 1/2 h2 by Proposition 31 Fix a + 1-tuple ε 1,, ε +1 where ε i µ 0 p for 1 i + 1 Denote by the nuber of i such that ε i 0 We want to evaluate the probability that the character χ p takes exactly these values at the points F x 1,, F x +1 where x +1 is the point at infinity of P 1 F, as F ranges over F [d1,,d p 1] Case 1: ε +1 0 In this case, only polynoials fro p 1 F j j1 d can have χ 1,,d p 1 pf x +1 ε +1 Also, the nuber of zeros aong ε 1,, ε is now 1 Thus, using Corollary 34, F F [d1,,d p 1] : χ p F x i ε i, 1 i + 1} α F 1p O F p 1 j1 F j d 1,,d p 1 : χ pf x i ε i χ 1 L p 2 d1+ +dp 1 ε+1 ε p 1 ζ 2 p 1 p p 1 p α, 1 i } p + p 1 εd h+ +d p 1 d h + d1 /2+ h2 +1 Case 2: ε +1 µ p In this case, only polynoials fro F d1,,d p 1 can have χ p F x +1 ε +1, and there are values

20 of ε 1,, ε which are zero Thus, F F [d1,,d p 1] : χ p F x i ε i, 1 i + 1} F F d1,,d p 1 : χ p F x i ε i ε 1 +1, 1 i } α F χpαε +1 1 L p 2 d1+ +dp 1 p 1 p ζ 2 p 1 + p 1 p + p 1 p O ε+1 ε εd h+ +d p 1 d h + d1 /2+, h2 510 which is the sae as 59 Then, it follows fro 57, 59 and 510 that F F [d1,,d p 1] : χ p F x i ε i, 1 i + 1} p 1 F [d1,,d p 1] + p 1 p + p 1 p O ε+1 ε+1 εd h+ +d p 1 d h + d1 /2+ Putting everything together, we obtain } C H d1,,dp 1 : TrFrob C H 1 χp t F F [d1,,d p 1] : Ŝ p F t} H d 1,,d p 1 F [d1,,d p 1] F F [d1,,d p 1] : χ p F x i ε i, 1 i + 1} F [d1,,d p 1] ε 1,,ε +1 ε 1 + +ε +1 t ε 1,,ε +1 ε 1 + +ε +1 t 1 + O Prob +1 p 1 + p 1 ε+1 ε+1 p 1 p + p 1 h2 +1 εd h+ +d p 1 d h + d1 /2+ h2 X i t 1 + O i1 +1 p 1 εd h+ +d p 1 d h + d1 3/2 h2 +1 where X 1,, X +1 are iid rando variables that take the value 0 with probability p 1/ + p 1 and any value in µ p with probability /p + p 1 Taking d 1,, d p 1, this proves Theore 14 20

21 Acknowledgents This work was initiated at the Banff workshop Woen in Nubers organized by Kristin Lauter, Rachel Pries, and Renate Scheidler in Noveber 2008, and the authors would like to thank the organizers and BIRS for a great workshop and excellent working conditions The authors also wish to thank Eduardo Dueñez, Nicholas Katz, Kiran Kedlaya, Pär Kurlberg, Lea Popovic, Rachel Pries, and Zeév Rudnick for helpful discussions related to this work This work was supported by the Natural Sciences and Engineering Research Council of Canada [BF, Discovery Grant to CD, to ML]; the National Science Foundation of US [DMS to AB]; and the University of Alberta [Faculty of Science Startup grant to ML] References [1] A Bucur, C David, B Feigon, and M Lalín; Statistics for traces of cyclic trigonal curves over finite fields, accepted for publication, Int Math Res Not IMRN [2] P Kurlberg and Z Rudnick; The fluctuations in the nuber of points on a hyperelliptic curve over a finite field J Nuber Theory , no 3, [3] M Raynaud; Caractéristiue d Euler-Poincaré d un faisceaux et cohoologie des variétés abeliennes, Seinaire Bourbaki 9, Exp , [4] M Roagny and S Wewers; Hurwitz Spaces, Groupes de Galois arithétiues et différentiels, , Séin Congr, 13, Soc Math France, Paris, 2006 [5] M Rosen; Nuber Theory in Function Fields, Graduate Texts in Matheatics, 210 Springer-Verlag, New York, 2002 xii+358 pp 21