FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS. 1. Introduction

Transcription

1 FLUCTUATIONS IN THE NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN 1 Introuction In this note, we stuy the fluctuations in the number of points on smooth projective plane curves over a finite fiel F as is fixe an the genus varies More precisely, we show that these fluctuations are preicte by a natural probabilistic moel, in which the points of the projective plane impose inepenent conitions on the curve The main tool we use is a geometric sieving process introuce by Poonen [8] Let S be the set of homogeneous polynomials F X, Y, Z) of egree over F, an let S ns S be the subset of polynomials corresponing to smooth or non-singular) curves C F : F X, Y, Z) 0 The genus of C F is 1) 2)/2 By running over all polynomials F S ns, one woul expect the average number of points of C F F ) to be We show that this is true, an that the ifference between #C F F ) an properly normalize) tens to a stanar Gaussian, N0, 1), when an ten to infinity in a certain range As inicate before, our main tool is a sieving process ue to Poonen [8] which allows us to count the number of polynomials in S which give rise to smooth curves C F, an the number of smooth curves C F which pass through a fixe set of points of P 2 F ) We enote by p the characteristic of F Theorem 11 Let X 1,, X be 2 + ii ranom variables taking the value 1 with probability )/ 2 + ) an the value 0 with probability 2 / 2 + ) Then, for 0 t 2 +, # {F S ns ; #C F F ) t} #S ns Prob X X t ) where enotes the integer part t 1/3 + 1) 2 min p +1, 3 ) + ))) 1 p 1, We now explain why these ranom variables moel the point count for smooth curves Intuitively, if F is any polynomial in S, then the set of F -points of the curve C F is a subset of P 2 F ), which has 2 + elements Heuristically, these points impose inepenent conitions on F Let us look at one of those conitions, say at the point [0 : 0 : 1] Put fx, y) F X, Y, 1) the ehomogenization of F an write fx, y) a 0,0 + a 1,0 x + a 0,1 y + Since we insist that C F is smooth, we cannot have a 0,0, a 1,0, a 0,1 ) 0, 0, 0), so there are 3 1 possibilities for this triple Of these triplets, the ones that correspon to the case where [0 : 0 : 1] 1

2 2 ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN is on the curve C F are those where a 0,0 0, of which there are 2 1 So the probability that [0 : 0 : 1] lies on C F is The argument works the same for any point in the plane, an in particular the expecte number of points in C F F ) is This explains the ranom variables of Theorem 11 Namely, the probability that X 1 respectively X 0) is the probability that a point P P 2 F ) belongs respectively oes not belong) to a smooth curve F X, Y, Z) 0 Remark 12 One coul take the iterate limit lim lim in Theorem 11 Or we coul invert the orer an take the lim lim provie that goes to infinity in such a way that > )+ε By stuying the moments we can substantially weaken this conition an compute the ouble limit lim, in a larger range It woul be ieal to be able to take the ouble limit with no conitions on an, but at present our error terms are not goo enough for that Since the average value of each of the ranom variables X i is +1)/ 2 ++1), an the stanar eviation is / 2 + ), it follows from the Triangular Central Limit Theorem [1] that X X 2 ++1) ) N0, 1) as tens to infinity We can show that this also hols for #C F F ) for F S ns, as an ten to infinity with > 1+ε, by showing that, uner these conitions, the integral moments of #C F F ) ) converge to the integral moments of X X 2 ++1) ) Theorem 13 Let k be a positive integer, an let Then, M k, ) E 1 M k, ) 1 #S ns i1 #CF F ) ) ) k k X i ) mink,2 ++1) k 1/3 + 1) 2 min p +1, 3 ) + ))) 1 p 1 Corollary 14 When an ten to infinity an > 1+ε, #C F F ) ) N0, 1) Finally, we point out that Theorem 11 implies that the average number of points on a smooth plane curve is, but this is not true anymore if one looks at all plane curves Our heuristic above shows that for a ranom polynomial F S the probability that a point P P 2 F ) actually lies on C F is 1/ This can also be proven easily see Section 21 for the proof); we recor the result here

3 NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS 3 Proposition 15 Let Y 1,, Y be ii ranom variables taking the value 1 with probability 1/ an the value 0 with probability 1)/ Then, for 2 +, #{F S ; #C F F )) t} #S Prob Y Y t ) Proposition 15 is an exact result without an error term) as there is no sieving involve For smooth curves, one has to sieve over primes of arbitrarily large egree, since a smooth curve is reuire not to have any singular points over F, not only over F This introuces the error term In particular, the average number of points on a plane curve F X, Y, Z) 0 without any smoothness conition is + 1 Some relate work Brock an Granville [2] calculate the average number of points in families of curves of given genus g over finite fiels, / N r C) 1 N r g, ) AutC/F ) AutC/F ) C/F,genusC)g C/F,genusC)g where N r C) enotes the number of F r-rational points of C It turns out that, epening on the value of r, N r g, ) shows very ifferent behavior as Inee, N r g, ) r + o r/2 ) unless r is even an r 2g, in which case N r g, ) r + r/2 + o r/2 ) This excess phenomenon has a natural explanation in terms of Deligne s euiistribution theorem for Frobenius conjugacy classes of the l-aic sheaf naturally attache to this family, as pointe out by Katz [5] Using Deligne s theorem, Katz showe that as, N r g, ) can be expresse in terms of the integral I r G) G trar )A, where G USp2g) in this case) is a compact form of the geometric monoromy group of that sheaf; the occurrence of the excess phenomenon epens on the values of I r G), which are compute using the representation theory of G This approach, which is escribe in a more general form in [6], has the avantage of being applicable in other situations in which the geometric monoromy group has been ientifie, for instance, when calculating the average number of points in the family of smooth egree hypersurfaces in P n over finite fiels In particular, for n 2, one obtains the average number of points of smooth planes curves of egree, which are the subject of the present investigation, but from a ifferent point of view Namely, while both [2] an [5] are concerne with curves of fixe genus as the number of points in the base fiel varies, we consier the complementary situation of working over a fixe fiel an allowing the genus to vary We also consier the uestion of the ouble limit as both the genus an the number of points in the base fiel grow Similar uestions were investigate in [7] for hyperelliptic curves, an in [3, 4, 9] for cyclic trigonal curves an general cyclic p-covers 2 Poonen s sieve We will aapt the results from Section 2 of [8] to our case, which is simpler as we take n 2 an X P 2 P 2 But, unlike Poonen, we nee to keep track of the error terms First let us o this in general in his setup, namely take Z X a finite subscheme Then U P 2 \Z will automatically be smooth an of imension 2 We will nee to choose Z an T H 0 Z, O Z ) in a way that imposes the appropriate local conitions for our curves at finitely many points

4 4 ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN The strategy is to check the smoothness separately at points of low, meium, an high egree, an then combine the conitions at the en The main term will come from imposing conitions on the values taken by both a ranom polynomial F S an its first orer erivatives at the points in U of relatively small egree for large enough) The error term will come from smoothness conitions at primes P of large egree compare to ) Following Poonen, enote A F [x 1, x 2 ] an A the set of polynomials in A of egree at most Denote by U <r the close points of egree < r an by U >r the close points of U of egree > r Set P,r {F S ; C F U is smooth of imension 1 at all P U <r, F Z T }, Q r, {F S ; P U st r eg P /3, C F U is not smooth of imension 1 at P }, Q high {F S ; P U >/3 st C F U is not smooth of imension 1 at P } 21 Points of low egree All the results of this section epen on the following lemma proven in [8] using classical results from algebraic geometry Lemma 21 For any subscheme Y P 2, the map φ : S H 0 P 2, O P 2)) H 0 Y, O Y )) is surjective for im H 0 Y, O Y ) 1 Proof Take n 2 in Lemma 21 in [8] Lemma 22 Let U <r {P 1,, P s } Then for 3rs + im H 0 Z, O Z ) 1, we have #P,r #T #S #H 0 Z, O Z ) s 1 3 eg Pi ) Proof The result follows from Lemma 22 in [8], as long as we ensure that + 1 is bigger than the imension of H 0 Z, O Z ) s i1 H0 Y i, O Yi ), where Y i is the close subscheme corresponing to P i in the manner escribe by Poonen Thus im H 0 Y i, O Yi ) 3 eg P i < 3r i1 Proof of Proposition 15 We can use this last result to compute the average number of points on the curves C F associate to the polynomials F S without any smoothness conition We pick P 1,, P an enumeration of the points of P 2 F ), an we take Z to be a m P -neighborhoo for each point P P 2 F ) this means that we look at the value of F at that point; for smoothness, we will also look at the value of its first orer erivatives) Thus H 0 Z, O Z ) O P /m P P P 2 F ) Each space has imension 1, so im H 0 Z, O Z ) 2 ++1, an #H 0 Z, O Z ) Let 0 t 2 + We want to count all curves C F such that P 1,, P t C F F ) an P t+1,, P C F F ) We then choose T {a i ) 1 i 2 ++1; a 1,, a t 0, a t+1,, a F },

5 NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS 5 an #T 1) t It follows by taking r 0 in Lemma 22 that, when 2 +, # { F S ; P 1,, P t C F F ), P t+1,, P C F F ) } Then, #P,0 #S #T #H 0 Z, O Z ) Prob #C F F ) t) #S 1) t ε 1,,ε {0,1} ε 1 + +ε t 1 ) t 1 ) t 1 1 ) t ) t Prob Y Y t ), an this proves Proposition 15 Now we want to use Lemma 22 to sieve out non-smooth curves We remark that Lemma 22 gives an exact formula without error term, but we nee to choose r as a function of an the prouct will contribute to the error term In aition, s itself epens on r As the number of close points of egree e in U is boune by the number of close points of egree e in P 2, which is 2e + e + 1 < 2 2e, the prouct 1 z eg P ) 1 ζ U z) P close point of U converges for Rz) > 2 For the same reason, we get that s 1) 1 3 eg Pi ) ζ U 3) 1 i1 Inee, in orer to show 1), we write s 1 3 eg Pi ) ζ U 3) 1 i1 eg P r For any seuence of numbers {x i ; 0 x i < 1}, we know that 1 1 x i ) x i i1 r r 1 3 eg P ) 1 Taking the seuence in uestion to be { 3 eg P } eg P r, it means that we nee an upper boun for 3 eg P 3j #{close points of U of egree j} eg P r jr All the P s until now have been close points of U, but U is a subset of P 2, so it has at most #P 2 F j ) 2j + j j close points of egree j Hence 3 eg P 2 j 2 r 1 1, an now we get eg P r 1 eg P r jr 1 3 eg P ) 1 1, 1 2 r 1 1 ))

6 6 ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN which proves 1) Substituting 1) in Lemma 22, we obtain #P,r 2) ζ U 3) 1 #T #S #H 0 Z, O Z ) 22 Points of meium egree Lemma 23 For a close point P U of egree e /3, we have r r )) #{F S ; C F U is not smooth of imension 1 at P } #S 3e Proof Take m 2 in Lemma 23 of [8] This also follows from Lemma 22 by taking r 0 an Z to be a m 2 P -neighborhoo of P which mean that we look at F an its first orer erivatives) Then, H 0 Z, O Z ) O P /m 2 P, an im H 0 Z, O Z ) 3 eg P We also choose T {0, 0, 0)}, as we want F an its first orer erivatives to vanish at P Then, #{F S ; C F U is not smooth of imension 1 at P } #S Lemma 24 #Q r, #S 2 r 1 1 Proof We follow the proof of Lemma 24 of [8] We have that #Q r, #S /3 P U eg P r #T #H 0 Z, O Z ) 3 eg P #{F S ; C F U is not smooth of imension 1 at P } #S, an #UF e) #P 2 F e) 2e + e e Then, using Lemma 23, we have /3 #Q r, 2 e 2 e 2 r #S 1 1 er er 23 Points of high egree Lemma 25 For P A 2 F ) of egree e, we have #{f A ; fp ) 0} #A min+1,e) Proof Take n 2 in Lemma 25 of [8] Lemma 26 #Q high #S 3 1) 2 min p +1, 3 ) p 1

7 NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS 7 Proof If we get a boun for all affine U A 2, the same boun multiplie by 3 will hol for any U P 2, since it can be covere by three affine charts So we can reuce the problem to affine sets We follow the proof of Lemma 26 of [8], while keeping track of the constants appearing in the error terms, which is not one in [8] as only the main term is neee for his application In our case the coorinates are simply x 1 an x 2, which have egree 1 an D i x i, i 1, 2 are alreay global erivations This allows us to work globally on the set U an there is no nee to work locally as in [8] Now we can work with ehomogenizations of polynomials in S, so we nee to fin the polynomials f A for which C f U fails to be smooth at some P U This happens if an only if fp ) D 1 f)p ) D 2 f)p ) 0 Any polynomial f A can be written as f g 0 + g p 1 x 1 + g p 2 x 2 + h p with g 0 A, g 1, g 2 A γ an h A η, where γ an η 1 p p The trick use by Poonen is base on the observation that selecting f uniformly at ranom amounts to selecting g 0, g 1, g 2 an h inepenently an uniformly at ranom The avantage is that D i f D i g 0 + g p i, so each erivative epens only on g 0 an one of the g 1, g 2 Set W 0 U, W 1 U {D 1 f 0}, W 2 U {D 1 f D 2 f 0} Claim 1 For i 0, 1 an for each choice of g 0,, g i, such that im W i 2 i, #{g i+1,, g 2, h); im W i+1 1 i} #{g i+1,, g 2, h)} 1) i 1 p 1 Bézout s theorem tells us that the number of 2 i)-imensional components of W i ) re is boune above by 1) i, since eg D i f 1, for each i, an eg U 1 The rest of the argument follows from Poonen s computation Claim 2 For any choice of g 0,, g 2, #{h; C f W 2 U >/3 } {all h} 1) 2 min p +1, 3 ) This follows from the fact that #W 2 1) 2 from Bézout s theorem as before) an the coset argument in [8] shows that for each P W 2, the set of ba h s at P is either empty or has ensity at most min p +1, 3 ) in the set of all h because eg P > /3) To finish the proof of the lemma, we put the two claims together an we get that #Q high #S 3 1) 2 min p +1, 3 ) p 1 Combining the points of small, meium an high egree, we get that 3) {F S ; C F U is smooth of imension 1 an F Z T } #T r )) ζ U 3)#H 0 Z, O Z ) r r +O )2 min ) p +1, 3 ) + 1 p 1

8 8 ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN We nee to choose an appropriate value for r Accoring to Lemma 22, we must have 3rs+im H 0 Z, O Z ) 1 an 1 r 2r + r +1) < s < 2r + r +1 When using 3) in Section 3, we will always have Z P 2 F ), thus im H 0 Z, O Z ) < 6 2 We take r 3B+log 3 for any fixe constant B 0 With this choice of r, the error term of 3) coming from points of meium an high egree is therefore B 1/3 4) O ) 2 min ) p +1, 3 ) + 1 p 1 3 Number of points We apply the results in Section 2 twice The first time to evaluate the fraction of homogeneous polynomials of egree that efine smooth plane curves, an the secon time to evaluate the fraction of homogeneous polynomials of egree that efine smooth plane curves with preetermine F - points By taking the uotient we then obtain an asymptotic formula for the fraction of smooth plane curves that have preetermine F -points For the first evaluation, we take Z an T {0} in 3) to get #{F S ns} ζ P 23) 1 B 1/3 )) 5) #S B 1/3 B 1/3 + O ) 2 min ) p +1, 3 ) + 1 p 1 Pick P 1, P an enumeration of the points of P 2 F ), an let 0 t 2 + We want to compute #{F S ns ; P 1,, P t C F F ), P t+1,, P C F F )} #S This is achieve by taking Z to be an m 2 P -neighborhoo for each point P P2 F ) this means that we look at the value of F an its first orer erivatives at each point) Thus 6) H 0 Z, O Z ) O P /m 2 P P P 2 F ) Each space has imension 3, so im H 0 Z, O Z ) ), an #H 0 Z, O Z ) ) Then we want T to be the set of a i, b i, c i )) 1 i such that a 1,, a t 0, a t+1,, a F, an a i, b i, c i ) 0, 0, 0) for 1 i 2 + This gives that 7) #T 2 1) t 1) t t) Using 3) with this choice of Z an T, we obtain #{F S ns; P 1,, P t C F F ), P t+1,, P C F F )} #S ζ U 3) 1 2 1) t 1) t t) ) B 1/3 +O ) 2 min ) p +1, 3 ) + 1 p 1 B 1/ B 1/3 ))

9 NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS 9 Here, by multiplicativity of zeta functions, ζ P 2z) ζ U z) ζ 1 Zz) 1 z ) Then, by taking the uotient of 7) an 5), we get that for U P 2 \ Z #{F S ns ; P 1,, P t C F F ), P t+1,, P C F F )} #S ns ) ) t 1) t t) ) t B 1/3 + 1) 2 min p +1, 3 ) + 1 p 1))) ) t 2 ) t t B 1/3 + 1) 2 min p +1, 3 ) + 1 1))) p Theorem 11 follows by taking B 0 above an noting that for any ε 1,, ε {0, 1} with ε ε t, Prob ) ) t X 1 ε 1,, X ε Moments ) ) t By Theorem 11 the number of points of smooth plane curves over F is istribute as X X 2 ++1, an the trace of the Frobenius as X X ) The mean of X X is ) Applying the triangular central limit theorem to the ranom variables X 1,, X 2 ++1, we have that X X ))/ is istribute as N0, 1) when We woul like to say the same thing about the istribution of the trace of Frobenius in our family when an go to infinity, which amounts to the computation of the moments We will first compute N k, ) 1 #S ns #CF F ) ) k, an then euce the result for M k, ) By using an exponential sum to count the number of points in the curve, we can write N k, ) 1 ) k 1 #S ns k S F P ), P P 2 F ) where S F P ) 1 t F e tf P ) )

10 10 ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN Thus, expaning the k-th power, where N k, ) 1 #S ns 1 ) k P 1,,P k P 2 F ) mink, ) #S ns hl, k) ) k/2 l1 S F P 1 ) S F P k ) P,b) P l,k S F P 1 ) b1 S F P l ) b l, P l,k { P, b); P P1,, P l ) with P i istinct points of P 2 F ), b b 1,, b l ) with b i positive integers such that b b l k} Notice that k hl, k) l1 Now let us fix P, b) P l,k Then, where 1 #S ns S F P 1 ) b1 S F P l ) b l Sa) 1 P,b) P l,k ) k t F e 1 #S ns a 1,,a l F ) { ta 1 a 0, 0 a 0 F P j )a j l Sa j ) bj, The b j have no influence on the result an we obtain nonzero terms only when a j 0 for 1 j l, an in this case 1 #S ns S F P 1 ) b1 S F P l ) b # {F Sns l ; F P i) 0 for 1 i l} 8) #S ns We remark that this can also be compute in a similar way as in Section 3 Choose the constant B k, Z as in 6) an T to be the set of a i, b i, c i )) 1 i such that a 1 a l 0, a l+1,, a F an a i, b i, c i ) 0, 0, 0) for 1 i 2 + Then, an #T 2 1) l 3 1) l, # {F S ns ; F P i) 0 for 1 i l} 2 + #S ns Now we sum over all the elements in P l,k : ) l k 1/3 l + 1) 2 l min p +1, 3 ) + l 1 p 1)) j1 N k, ) 1 ) k/2 mink, 2 ++1) l1 hl, k) P,b) P l,k ) l 2 + mink,2 ++1) k 1/3 + 1) 2 min p +1, 3 ) + 1 p 1)))

11 NUMBER OF POINTS ON SMOOTH PLANE CURVES OVER FINITE FIELDS 11 On the other han, we have E i1 X i k ) k k 1 hl, k) l1 i,b) A l,k E where A l,k i, b); i i 1,, i l ), 1 i j 2 + istinct, b b 1,, b l ) Since EX b1 1 Xb l l ) 2 + an #P l,k #A l,k, we conclue that k N k, ) E ) 10) i1 X i ) l ) X b1 i 1 X b l i l, l b j k mink,2 ++1) k 1/3 + 1) 2 min p +1, 3 ) + 1 1))) p Now using 9) an the binomial theorem, we get that k ) k M k, ) N j, ) ) k j j j0 k j0 ) k E j 1 E i1 i1 X i j k X i ) with the same error term as 10) This completes the proof of Theorem 13 Acknowlegments The authors wish to thank Pierre Deligne an Zeév Runick for suggesting the problem that we consier in this paper The authors are grateful to both of them as well as Pär Kurlberg for helpful iscussions This work was supporte by the Natural Sciences an Engineering Research Council of Canaa [BF, Discovery Grant to CD, to ML] an the National Science Founation of US [DMS an DMS to AB] ML is also supporte by a Faculty of Science Startup grant from the University of Alberta, an CD is also supporte by a grant to the Institute for Avance Stuy from the Minerva Research Founation References [1] P Billingsley, Probability an measure, thir eition, Wiley Series in Probability an Mathematical Statistics, John Wiley & Sons, Inc, New York, 1995 xiv+593 pp [2] B W Brock an A Granville, More points than expecte on curves over finite fiel extensions, Finite Fiels Appl ), no 1, [3] A Bucur, C Davi, B Feigon an M Lalín, Statistics for traces of cyclic trigonal curves over finite fiels, Int Math Res Not IMRN, Avance Access publishe on October 27, 2009, oi:101093/imrn/rnp162 ) k j j1

12 12 ALINA BUCUR, CHANTAL DAVID, BROOKE FEIGON, MATILDE LALÍN [4] A Bucur, C Davi, B Feigon an M Lalín, Biase statistics for traces of cyclic p-fol covers over finite fiels, submitte, 2009 [5] N M Katz, Frobenius-Schur inicator an the ubiuity of Brock-Granville uaratic excess, Finite Fiels Appl ), no 1, [6] N M Katz an P Sarnak, Ranom matrices, Frobenius eigenvalues, an monoromy American Mathematical Society Collouium Publications, 45 American Mathematical Society, Provience, RI, 1999 xii+419 pp [7] P Kurlberg an Z Runick, The fluctuations in the number of points on a hyperelliptic curve over a finite fiel, J Number Theory ), no 3, [8] B Poonen, Bertini theorems over finite fiels, Ann of Math 2) ), no 3, [9] M Xiong, The fluctuation in the number of points on a family of curves over a finite fiel, preprint, 2009 Institute for Avance Stuy an University of California at San Diego aress: alina@mathiaseu Concoria University an Institute for Avance Stuy aress: cavi@mathstatconcoriaca University of Toronto aress: bfeigon@mathtorontoeu University of Alberta aress: mlalin@mathualbertaca