The distribution of the number of points on trigonal curves over $\F_q$
aa r X i v : . [ m a t h . N T ] A ug THE DISTRIBUTION OF THE NUMBER OF POINTS ON TRIGONALCURVES OVER F q MELANIE MATCHETT WOOD
Abstract.
We give a short determination of the distribution of the number of F q -rationalpoints on a random trigonal curve over F q , in the limit as the genus of the curve goesto infinity. In particular, the expected number of points is q + 2 − q + q +1 , contrastingwith recent analogous results for cyclic p -fold covers of P and plane curves which have anexpected number of points of q + 1 (by work of Kurlberg, Rudnick, Bucur, David, Feigonand Lal´ın) and curves which are complete intersections which have an expected number ofpoints < q + 1 (by work of Bucur and Kedlaya). We also give a conjecture for the expectednumber of points on a random n -gonal curve with full S n monodromy based on functionfield analogs of Bhargava’s heuristics for counting number fields. Introduction
If we fix a finite field F q , we can ask about the distribution of the number of ( F q -rational)points on a random curve over F q . There has been a surge of recent activity on this questionincluding definitive answers of Kurlberg and Rudnick for hyperelliptic curves [11], of Bucur,David, Feigon and Lal´ın for cyclic p -fold covers of P [4, 5], of Bucur, David, Feigon and Lal´ınfor plane curves [6], and of Bucur and Kedlaya on curves that are complete intersections insmooth quasiprojective subschemes of P n [7]. In the first three cases, the average number ofpoints on a curve in the family is q + 1. In contrast, for curves that are complete intersectionsin P n , the average number of points is < q + 1, despite, as pointed out by Bucur and Kedlaya,the abundance of F q points lying around in P n . In this paper, we give the distribution of thenumber of points on trigonal curves over F q (i.e. curves with a degree 3 map to P ), and inparticular show that the average number of points is greater that q + 1.Let T g := { π : C → P | C is a smooth, geometrically integral, genus g curve with π degree 3 } . Our main theorem is the following.
Theorem 1.1.
Let F q have characteristic ≥ . We have lim g →∞ |{ C ∈ T g ( F q ) | | C ( F q ) | = k }| | T g ( F q ) | = Prob( X + · · · + X q +1 = k ) , where the X i are independent identically distributed random variables and X i = with probability q q +6 q +6 with probability q +66 q +6 q +6 with probability q q +6 q +6 with probability q q +6 q +6 . oreover, we give a rigorous explanation of the random variables X i . Theorem 1.1 is acorollary of the following, which gives the distribution of the number of points in the fiberover a given F q -rational point of P . Theorem 1.2.
Let F q have characteristic ≥ . Given a point z ∈ P ( F q ) , lim g →∞ |{ ( C, π ) ∈ T g ( F q ) | | π − ( x )( F q ) | = k }| | T g ( F q ) | = q q +6 q +6 for k = 0 q +66 q +6 q +6 for k = 1 q q +6 q +6 for k = 2 q q +6 q +6 for k = 3 . Moreover, these probabilities (of various size fibers) are independent at the F q points z , . . . , z q +1 of P . Corollary 1.3.
The average number of points of a random trigonal curve over F q (in the g → ∞ limit as above) is q + 2 − q + q +1 . The method used in this paper will be to relate trigonal curves to cubic extensions offunction fields, and then to use the work of Datskovsky and Wright [8] to count cubicextensions with every possible fiberwise behavior above each rational point of the base curve.In fact, our methods work with any smooth curve E over F q replacing P . Let T E,g be themoduli space of genus g curves with a specified degree 3 map to E . Theorem 1.4.
Let F q have characteristic ≥ . Given a point z ∈ E ( F q ) , lim g →∞ |{ ( C, π ) ∈ T E,g ( F q ) | | π − ( x )( F q ) | = k }| | T g ( F q ) | = q q +6 q +6 for k = 0 q +66 q +6 q +6 for k = 1 q q +6 q +6 for k = 2 q q +6 q +6 for k = 3 . Moreover, these probabilities (of various size fibers) are independent at the F q points of E .In particular, the average number of points of a random curve over F q with a degree mapto E (in the g → ∞ limit) is | E ( F q ) | (1 + qq + q +1 ) . Related Work.
Studying the distribution of the number of points on a curve is equiv-alent to studying the distribution of the trace of Frobenius on the ℓ -adic cohomology group H . Bucur, David, Feigon and Lal´ın, for cyclic p -fold covers of P , in fact give the finerinformation of the distribution of the trace of Frobenius on each subspace of H invariantunder the cyclic action. Their methods, under appropriate interpretation, use Kummer the-ory to enumerate the cyclic p -fold covers of P , and thus require the hypothesis that q ≡ p ). The hardest part of the method is sieving for p -power free polynomials.The work of Bucur, David, Feigon and Lal´ın for plane curves [6] and of Bucur and Kedlayaon curves that are complete intersections [7] is also given “fiberwise,” (as in Theorems 1.2and 1.4 of this paper) in that it computes the probability that any point is the ambientspace is in a random smooth curve in the specified family and shows these probabilities areindependent. (Here we think of the embedding of a curve i : C → X , and [6, 7] computes foreach point z ∈ X ( F q ) the distribution of the size of the fiber i − ( z ) for random C . Note thatsince i is an embedding, in this case the fiber has either 0 or 1 points, so these are Bernoullirandom variables.) urlberg and Wigman [12] have also studied the distribution of the number of F q pointsin families of curves in which the average number of points is infinite.In Section 3 of this paper, we discuss distribution of points on other families of n -gonalcurves, and in particular give a conjecture for the expected number of points on a random n -gonal curve with full S n monodromy, based on function field analogs of Bhargava’s heuristicsfor counting number fields. The conjecture in fact predicts the expected number of pointsin a fixed fiber over P .When one instead fixes a genus g and lets q tend to infinity, the philosophy of Katz andSarnark [10] predicts how the average number of points behaves, and in particular it shouldbe governed by statistics of random matrices in a group depending on the monodromy ofthe moduli space of curves under consideration.2. Proof of Theorem 1.4
Let F q have characteristic >
3. Let E be a smooth curve over F q , and let k be the functionfield of E . Then smooth, integral curves over F q with finite, degree 3 maps to P F q are inone-to-one correspondence with cubic extensions k ′ of k . However, these smooth, integralcurves may not be geometrically integral. In this case, the only such possibility is given bya cubic extension of the constant field of E , and since we will take a g → ∞ limit, we canignore this extension altogether.We now consider all the possible completions of a cubic extension k ′ of k . For a place v of k , the absolute tame Galois group of the local field k v is topologically generated by x, y with the relation xyx − = y q , where y is a generator of the inertia subgroup and x is a Frobenius element. For a place v of k , the following is a chart of all possible cubic ´etale k v algebras L (and thus of allpossible isomorphism types for k ′ ⊗ k k v ). To each isomorphism type, we note the imagesof x, y from the absolute Galois group of k v in the associated representation to S (givenby the action of Galois on the three homomorphisms L → ¯ k v ), the number of F q rationalpoints in the fiber above v , and a constant c L that will be important later. (It turns outthat c L = | Aut( L ) || D L/kv | ), but this fact will not be used.) of F q L x y points c L in fiber k ⊕ v () () 3 1 / K ⊕ k v , K/k v degree 2 unram. field extn. (12) () 3 1 / K , K/k v degree 3 unram. field extn. (123) () 0 1 / K ⊕ k v , K/k v degree 2 ram. field extn. () (12) 2 1 / qK ⊕ k v , K/k v degree 2 ram. field extn. (12) (12) 2 1 / qK , K/k v degree 3 ram. field extn. (123) (123) 1 1 / q if q ≡ q ≡ K , K/k v degree 3 ram. field extn. (132) (123) 1 1 / q if q ≡ q ≡ K , K/k v degree 3 ram. field extn. () (123) 1 1 / q if q ≡ q ≡ K , K/k v degree 3 ram. field extn. (12) (123) 1 1 /q if q ≡ q ≡ S be the set of places of k corresponding to the F q -rational points of E . Let Σ be achoice Σ v of a cubic ´etale k v algebra for each v ∈ S . We define c Σ = Q v ∈ S c Σ v , where theconstants c Σ v are defined by the chart above. We also choose a Σ ′ similarly. Let N Σ ( q n )be the number of isomorphism classes of cubic extensions k ′ of k such that for all v ∈ S , wehave k ′ ⊗ k k v ∼ = Σ v , and such that the norm of the relative discriminant | D k ′ /k | = q n . Nowwe can state the result of Datskovsky and Wright and cubic extensions of function fields.
Theorem 2.1 (Corollary of Theorem 4.3 of [8]) . In the above notation lim n →∞ N Σ ( q n ) N Σ ′ ( q n ) = c Σ c ′ Σ . Theorem 1.4 now follows because the above chart gives all the possibilities for the com-pletions of cubic extensions k ′ of k at each F q -rational place of k (and the thus determinednumber of F q -rational points in that fiber), and Theorem 2.1 gives their relative probabilities.For a triple cover ( C, π ) ∈ T E,g ( F q ) corresponding to an extension k ′ /k , we have | D k ′ /k | = q n ,where g = n − − χ ( E )2 , and thus we can replace the limit in n with a limit in g .3. Further Directions and Conjectures
The above computations of the distribution of points of trigonal curves suggests thatsimilar questions could be attacked for n -fold covers of P (or an arbitrary curve) usingmethods from the study of counting extensions of global fields by their discriminants, which as been more heavily studied in the case of number fields. In analog to this work, onewould naturally only study n -gonal curves with a specified monodromy group (i.e. theGalois group of the Galois closure of the field extension). However, for this method to apply,one needs information on the densities of local behaviors of those number fields. In thenumber field case, such results are only currently available for cubic extensions (the work ofDatskovsky and Wright [8] used above, or going back to Davenport and Heilbronn [9] over Q ), for ( Z /p Z ) k Galois extensions ([14, 13]), and for degree n extensions with Galois closuregroup S n and n = 4 ,
5) (by work of Bhargava [1, 2]). The first two cases above also haveresults in the function field case, which have been applied in this paper and forthcoming workof the author, respectively. It is intriguing to ask if a function field version of Bhargava’scounting results [1, 2] could give the distribution of the number of points of 4-gonal and5-gonal curves. In analogy with the number field case, one expects such methods wouldonly count 4-gonal curves with S monodromy, and that D -monodromy curves would bea non-negligible portion of all 4-gonal curves. Since the methods for counting D quarticextensions have not given any results on densities of local behaviors or independence of thosebehaviors, understanding the distribution of points of 4-gonal curves with D monodromywould be very interesting.In the case of n -gonal curves with full S n -monodromy, we can at least predict what thedistribution of the number of points in each fiber should be using the heuristics of Bhargava[3, Conjecture 5.1]. In particular, we give the prediction for the average. Conjecture 3.1.
Let E be a smooth curve over a finite field F q of characteristic > n andfix z ∈ E ( F q ) . The average number of points in the fiber over z of a random curve over F q with a degree n map to E and full monodromy (in the g → ∞ limit) is P k k P n − ℓ =0 p ( n,n − ℓ,k ) q ℓ P n − ℓ =0 p ( n,n − ℓ ) q ℓ = 1 + 1 q + O ( 1 q ) , where p ( n, m, k ) is the number of partitions of n into m parts such that the parts take exactly k values and p ( n, m ) is the number of partitions of n into m parts. It seems within reach to prove this conjecture for n = 4 , n = 4 the expected fiber size is (conjecturally)1 + q + qq + q + 2 q + 1and for n = 5 the expected fiber size is (conjecturally)1 + q + 2 q + 2 qq + q + 2 q + 2 q + 1 . In particular, we now show that Conjecture 3.1 follows from the function field analog of [3,Conjecture 5.1].Let k be the function field of E and v place of E corresponding to a k v -rational point. Then[3, Conjecture 5.1].gives conjectures for local densties for k v algebras among k ′ ⊗ k k v while k ′ ranges over degree n S n -extensions of k . In particular, [3, Conjecture 5.1] predicts, as werange over degree n ´etale k v algebras L , that L appears with relative density | Aut( L ) || D L/kv | recall | D L/k v | is the absolute norm of the relative discriminant of L/k v ). Recall that isomor-phism classes of degree n ´etale k v algebras exactly correspond to continuous homomorphismsGal( ¯ k v /k v ) → S n , and we can rephrase the above to say that L appears with relative density | χ : Gal( ¯ k v /k v ) → S n corr. to L || D L/k v | . Continuous homomorphisms Gal( ¯ k v /k v ) → S n correspond to choices x, y ∈ S n such that xyx − = y q , and | D L/k v | = q n − y ) . The number of F q rational points above v inthe curve corresponding to the extension k ′ of k is given by the number of h x, y i -orbits on { , . . . , n } that are also h y i -orbits.Let y ∈ S n have a i orbits of size b i , for i = 1 , . . . k . Then the number of x ∈ S n suchthat xyx − = y q is Q i a i ! b a i i . If we consider a single cycle σ of y of length b i , then thenumber of x ∈ S n such that σ remains a h x, y i -orbit is a i Q i a i ! b a i i . So given the choice of y , the expected contribution of σ to rational points is a i . Thus, given a choice of y , theexpected number of rational points is k , the number of distinct cycle lengths of y . Note that Q i a i ! b a i i is the size of the centralizer of y , so choosing a permutation y ∈ S n with relativeprobability Q i a i ! b aii q n − y ) is the same as choosing it with relative probability | Cent( y ) | q n − y ) or | ConjClass( y ) | q n − y ) . This is equivalent to choosing a conjugacy class in S n , i.e. a partitionof n , with relative probability q n − . Since a partition with k distinct part sizes, gives anexpected k parts, Conjecture 3.1 follows from the function field analog of [3, Conjecture 5.1]. References [1] M. Bhargava, The density of discriminants of quartic rings and fields, Ann. of Math. (2) (2005),1031–1063.[2] M. Bhargava, The density of discriminants of quintic rings and fields, Ann. of Math., (3) (2010),1559–1591.[3] M. Bhargava, Mass formulae for extensions of local fields, and conjectures on the density of numberfield discriminants, Int. Math. Res. Not. IMRN , no. 17, Art. ID rnm052, 20 pp.[4] A. Bucur, C. David, B. Feigon and M. Lal´ın, Statistics for traces of cyclic trigonal curves over finite fields,
Int. Math. Res. Not. IMRN , Advance Access published on October 27, 2009, doi:10.1093/imrn/rnp162.[5] A. Bucur, C. David, B. Feigon and M. Lal´ın, Biased statistics for traces of cyclic p-fold covers over finitefields, to appear in Proceedings of 2008 Banff Workshop ”Women In Numbers” (in the Fields InstituteCommunications Series).[6] A. Bucur, C. David, B. Feigon and M. Lal´ın, Fluctuations in the number of points on smooth planecurves over finite fields,
Journal of Number Theory (2010), 2528–2541.[7] A. Bucur, and K. Kedlaya, The probability that a complete intersection is smooth, arXiv:1003.5222v1[8] B. Datskovsky and D. J. Wright, The adelic zeta function associated to the space of binary cubic forms.II. Local theory, J. Reine Angew. Math. (1986), 27–75.[9] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II, Proc. Roy. Soc.London Ser. A (1971), no. 1551, 405–420.[10] N. M. Katz and P. Sarnak,
Random matrices, Frobenius eigenvalues, and monodromy . American Math-ematical Society Colloquium Publications, 45. American Mathematical Society, Providence, RI, 1999.xii+419 pp.[11] P. Kurlberg and Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over afinite field,
J. Number Theory (2009), no. 3, 580–587.[12] P. Kurlberg and I. Wigman. Gaussian point count statistics for families of curves over a fixed finitefield.
Int. Math. Res. Not. IMRN , to appear.
13] M. M. Wood, On the probabilities of local behaviors in abelian field extensions,
Compos. Math. (2010), no. 1, 102–128.[14] D. J. Wright, Distribution of discriminants of abelian extensions, Proc. London Math. Soc. (3) (1989), no. 1, 17–50. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI53705 USA, and American Institute of Mathematics, 360 Portage Ave, Palo Alto, CA 94306-2244 USA
E-mail address : [email protected]@math.wisc.edu