Traces of High Powers of the Frobenius Class in the Moduli Space of Hyperelliptic Curves
TTRACES OF HIGH POWERS OF THE FROBENIUS CLASSIN THE MODULI SPACE OF HYPERELLIPTIC CURVES
IAKOVOS JAKE CHINIS
Abstract.
The Zeta function of a curve C over a finite field may be ex-pressed in terms of the characteristic polynomial of a unitary matrix Θ C .Following the work of Rudnick [1], we compute the expected value of tr(Θ nC )over the moduli space of hyperelliptic curves of genus g , over a fixed finitefield F q , in the limit of large genus. As an application, we compute theexpected value of the number of points on C in F q n as the genus tends toinfinity. We also look at biases in both expected values for small values of n . Introduction
Let C be a smooth projective curve of genus g ≥ F q of odd cardinality q . If we let C ( F q n ) denote the number of points on C in finite extensions F q n of degree n of F q , then the Zeta function associatedto the curve C is defined byZ C ( u ) := exp (cid:16) ∞ (cid:88) n =1 C ( F q n ) n u n (cid:17) , | u | < q . (1.1)It is known that Z C ( u ) is a rational function in u of the formZ C ( u ) = P C ( u )(1 − u )(1 − qu ) , where P C ( u ) ∈ Z [ u ] is a polynomial of degree 2 g , with P C (0) = 1, satisfyingthe functional equation P C ( u ) = ( qu ) g P C ( 1 qu ) . (1.2)It was proven by Weil [7] that the zeros of P C ( u ) all lie on the circle | u | = 1 /q .Hence, P C ( u ) = g (cid:89) j =1 (1 − q e iθ j ( C ) u ) , Date : September 13, 2018. a r X i v : . [ m a t h . N T ] O c t IAKOVOS JAKE CHINIS for some angles θ j ( C ), 1 ≤ j ≤ g , andZ C ( u ) = exp (cid:16) ∞ (cid:88) n =1 C ( F q n ) n u n (cid:17) = (cid:81) gj =1 (1 − q e iθ j ( C ) u )(1 − u )(1 − qu ) . (1.3)Now, we may define a unitary symplectic matrix Θ C ∈ USp(2 g ) byΘ C jk := (cid:40) e iθ j ( C ) if k = j ≤ j, k ≤ g . Then it is clear that the Zeta function associated to C canbe expressed in terms of the characteristic polynomial of Θ C Z C ( u ) = det( I − u √ q Θ C )(1 − u )(1 − qu ) , with Θ C unique up to conjugacy. We call the conjugacy class of Θ C the unitarized Frobenius class of C .Let H g be the moduli space of hyperelliptic curves of genus g over F q ; i.e.,the set of hyperelliptic curves given by the affine equation C Q : y = Q ( x ) , where Q ∈ F q [ x ] is any squarefree polynomial of degree 2 g + 1 or 2 g + 2. In thismodel, the point at infinity is not smooth, but we may consider the smoothportion of C Q and account for the point at infinity separately; in the smoothmodel, the point at infinity will be replaced by 0, 1, or 2 points and we getthat the number of points at infinity is Q ) is even and sgn( Q ) (cid:54) = (cid:3) Q ) is odd2 if deg( Q ) is even and sgn( Q ) = (cid:3) .(1.4)Note the relation between equations (1.4) and (2.3); namely, the number ofpoints at infinity is λ Q + 1, with λ Q as in (2.3). Remark.
The smooth model is the closure of C Q , denoted C Q , under the map [1 , x, x , . . . , x g − , y ] : C Q → P g +2 . One can show that this closure consists of two affine components: the first is C Q itself and the second is the curve given by y = x g +2 Q ( x ) . In fact, C Q isisomorphic to C Q (cid:84) { x (cid:54) = 0 } ; we refer the reader to Silverman [8] . For any function F on H g , we define the expected value of F over H g (cid:104) F (cid:105) H g := 1 H g · (cid:88) C Q ∈ H g F ( C Q ) . (1.5) RACES OF HIGH POWERS 3
In this paper, we study the traces of high powers of the Frobenius class of C Q over H g over a fixed finite field F q of odd cardinality q as g → ∞ . Inparticular, we concern ourselves with the expected values of (cid:104) tr(Θ nC Q ) (cid:105) H g as g → ∞ and we compare our work with the Random Matrix results of [3].From the work of Diaconis and Shahshahani [3], the expected value of thetraces of powers over the unitarized symplectic group USp(2 g ) is given by (cid:90) USp(2 g ) tr( U n ) du = (cid:40) − η n if 1 ≤ n ≤ g n > g ,(1.6)where η n = (cid:40) n is even0 if n is odd.We will prove the following theorem and the accompanying corollary: Theorem 1.1.
For n odd, (cid:104) tr(Θ nC ) (cid:105) H g = 0 , and for n even, (cid:104) tr(Θ nC ) (cid:105) H g = 1 q n · (cid:88) deg( P ) | n deg( P ) (cid:54) =1 deg( P ) | P | + 1 + O ( gq − g ) + − < n < g − − q − n = 2 gO ( nq n − g ) 2 g < n ,where the sum is over all monic irreducible polynomials P ∈ F q [ x ] and where | P | := q deg( P ) . Corollary 1.2. If n is odd, then (cid:104) tr(Θ nC ) (cid:105) H g = (cid:90) USp(2 g ) tr( U n ) dU. For n even with q ( g ) < n < g − q ( g ) and n (cid:54) = 2 g , (cid:104) tr(Θ nC ) (cid:105) H g = (cid:90) USp(2 g ) tr( U n ) dU + o ( 1 g ) . In [1], Rudnick considers the mean value of tr(Θ nC ) over a family of hyperl-liptic curves given by the affine equation C : y = Q ( x ) where Q ( x ) ∈ F g +1 ,with F g +1 := { f ∈ F q [ x ] : f monic, squarefree, and deg( f ) = 2 g + 1 } , IAKOVOS JAKE CHINIS and obtains that (cid:104) tr(Θ nC ) (cid:105) F g +1 = η n q n (cid:88) deg( P ) | n deg( P ) | P | + 1 + O ( gq − g ) + − η n < n < g − − q − n = 2 gO ( nq n − g ) 2 g < n ;in particular, if 3 log q ( g ) < n < g − q ( g ) and n (cid:54) = 2 g , then (cid:104) tr(Θ nC ) (cid:105) F g +1 = (cid:90) USp(2 g ) tr( U n ) dU + o ( 1 g ) . Rudnick then points out that there is a slight deviation in (cid:104) tr(Θ nC ) (cid:105) F g +1 fromthe Random Matrix Theory results for small values of n and for n = 2 g ;namely, (cid:104) tr(Θ C ) (cid:105) F g +1 ∼ (cid:90) USp(2 g ) tr( U ) dU + 1 q + 1and (cid:104) tr(Θ gC ) (cid:105) F g +1 ∼ (cid:90) USp(2 g ) tr( U g ) dU − q − . By considering the average value of tr(Θ nC ) over H g , we no longer get a devia-tion from the RMT results for n = 2 and the deviation at n = 2 g diminishes: (cid:104) tr(Θ C ) (cid:105) H g ∼ (cid:90) USp(2 g ) tr( U ) dU and (cid:104) tr(Θ gC ) (cid:105) H g ∼ (cid:90) USp(2 g ) tr( U g ) dU − q − . Furthermore, our results for odd n are exact and coincide with the RMT re-sults for all values of g . At first glance, this may seem counterintuitive as oneexpects to have an error term, as in the even case and as in [1]. Using anotherapproach, one can quickly verify the first result of Theorem 1.1 (this is donein section 10).Now, we may apply Theorem 1.1 to compute the average number of pointson C Q in finite extensions F q n of F q over H g , denoted (cid:104) C ( F q n ) (cid:105) H g :By taking logarithmic derivatives in (1.3), C Q ( F q n ) = q n + 1 − q n g (cid:88) j =1 e inθ j ( C Q ) = q n + 1 − q n tr(Θ nC Q ) . RACES OF HIGH POWERS 5
In fact, (cid:104) C Q ( F q n ) (cid:105) H g = q n + 1 − q n (cid:104) tr(Θ nC Q ) (cid:105) H g ∼ q n + η n q n + 1 − η n (cid:88) deg( P ) | n deg( P ) (cid:54) =1 deg( P ) | P | + 1 . More precisely,
Corollary 1.3. (i) If n is odd, then (cid:104) C Q ( F q n ) (cid:105) H g = q n + 1 . (ii) If n is even, then (cid:104) C Q ( F q n ) (cid:105) H g ∼ q n + q n + 1 − (cid:88) deg( P ) | n deg( P ) (cid:54) =1 deg( P ) | P | + 1 . Once again, our results for odd n are exact and hold for all values of g .Although we continue to get deviations from the RMT results for even n ≥ n = 2 and our deviations are different from those obtainedin [1].Another approach to computing (cid:104) C Q ( F q n ) (cid:105) H g is the work of Alzahrani [5]who uses the distribution of points on H g over F q in F q n . Using these methods,the results of Alzahrani agree with the Corollary above (albeit with a largererror term).Finally, we would like to mention that some of the computations done insections 3 through 8 were done independently by E. Lorenzo, G. Meleleo, andP. Milione in their study of statistics for biquadratic curves; their work iscollected in [9]. 2. Background
In this section, we establish some notation and we introduce the main resultsof [1]. Since the majority of what follows is based off of the work in [1], we usethe same notation and list important results for the convenience of the reader.We use [2] as a general reference.Throughout this paper, F q is a fixed finite field of odd cardinality q , P represents monic irreducible polynomials in F q [ x ], and Q will be used to denotesquarefree polynomials of degree 2 g + 1 or 2 g + 2 with g ≥
1. Unless otherwisestated, it is understood that sums and products are over all monic elementsin F q [ x ]; in the case where a sum involves elements B ∈ F q [ x ] that are notnecessarily monic, we write the sum over B n.n.m.. IAKOVOS JAKE CHINIS
Given any polynomial D ∈ F q [ x ] that is not a perfect square, we define thequadratic character χ D by the quadratic residue symbol for F q [ x ] χ D ( f ) := (cid:16) Df (cid:17) , where f is any monic polynomial in F q [ x ].The Zeta function associated to the hyperelliptic curve C Q : y = Q ( x ) isthen given by Z C Q ( u ) = L ∗ ( u, χ Q ) ζ q ( u ) , where ζ q ( u ) := 1(1 − u )(1 − qu )is the Zeta function of F q ( x ) and where L ∗ ( u, χ Q ) := (1 − λ Q · u ) − (cid:89) P ∈ F q [ x ] (1 − χ Q ( P ) · u deg( P ) ) − (2.1) = det( I − u √ q · Θ C Q ) , (2.2)with λ Q := − Q ) is even and sgn( Q ) (cid:54) = (cid:3) Q ) is odd1 if deg( Q ) is even and sgn( Q ) = (cid:3) ,(2.3)which relates to the count in equation (1.4).Taking logarithmic derivatives in equations (2.1) and (2.2), we see that g (cid:88) j =1 e inθ j ( C Q ) = tr(Θ nC Q ) = − λ nQ q n − q n (cid:88) deg( f )= n Λ( f ) χ Q ( f ) , (2.4)where Λ( f ) := (cid:40) deg( P ) if f = P k von Mangoldt function.Let F d := { f ∈ F q [ x ] : f monic, squarefree, and deg( f ) = d } and let (cid:98) F d := { f ∈ F q [ x ] : f squarefree and deg( f ) = d } . RACES OF HIGH POWERS 7
Then (cid:98) F d = ( q − F d and it is easy to see that (see Lemma 3 of [6], for example) F d = (cid:40) (1 − q ) q d , d ≥ q, d = 1 . Using these sets of polynomials, every curve in the moduli space of hyperel-liptic curves of genus g has a model C Q : y = Q ( x ), where Q ∈ (cid:98) F g +1 (cid:83) (cid:98) F g +2 .Now, let F be any family of squarefree polynomials of degree d in F q [ x ]. Forany function F on F , we define the expected value of F over F(cid:104) F (cid:105) F := 1 F · (cid:88) Q ∈F F ( Q ) . In particular, (cid:104) tr(Θ nC Q ) (cid:105) F = 1 F · (cid:88) Q ∈F (cid:16) − λ nQ q n − q n (cid:88) deg( f )= n Λ( f ) χ Q ( f ) (cid:17) . (2.5)In [1], Rudnick averages the trace over F = F g +1 . We begin by consideringthe average over F = F g +2 and then obtain the average over H g by combiningour results and also considering the contribution of the point at infinity whichdiffers on each component (cid:98) F g +1 , (cid:98) F g +2 .Let µ denote the M¨obius function. Since (cid:88) A | Q µ ( A ) = (cid:40) Q is squarefree0 otherwise,we may compute the expected value of F by summing over all elements ofdegree d in F q [ x ] and sieving out the squarefree terms; namely, (cid:104) F ( Q ) (cid:105) F = 1 F (cid:88) α + β = d (cid:88) deg( B )= βB n.n.m. (cid:88) deg( A )= α µ ( A ) F ( A B ) . (2.6)For all A, B ∈ F q [ x ], χ A B ( f ) = (cid:16) Bf (cid:17) · (cid:16) Af (cid:17) = (cid:40)(cid:16) Bf (cid:17) if ( A, f ) = 10 otherwise.With that said, taking F ( Q ) = χ Q in equation (2.6), (cid:104) χ Q ( f ) (cid:105) F = 1 F · (cid:88) α + β = dα,β ≥ σ ( f ; α ) (cid:88) deg( B )= βB n.n.m. (cid:16) Bf (cid:17) , IAKOVOS JAKE CHINIS where σ ( f ; α ) := (cid:88) deg( A )= α ( A,f )=1 µ ( A ) . We are now is a position to provide the necessary results from [1]:For any P ∈ F q [ x ] with deg( P ) = n , we define σ n ( α ) := σ ( P k ; α ) = (cid:88) deg( A )= α ( A,P k )=1 µ ( A ) = (cid:88) deg( A )= α ( A,P )=1 µ ( A ) = σ ( P ; α ) . Lemma 2.1. [1, Lemma 4] (i) For n = 1 , σ (0) = 1 , σ ( α ) = 1 − q ∀ α ≥ . (2.7) (ii) If n ≥ , then σ n ( α ) = α ≡ n − q α ≡ n otherwise. (2.8)Recall that the Dirichlet L-series associated to χ Q , denoted L ( u, χ Q ) for | u | < /q , is a polynomial in u of degree at most deg( Q ) − L ( u, χ Q ) := (cid:89) P ∈ F q [ x ] (1 − χ Q ( P ) · u deg( P ) ) − = (cid:88) β ≥ A Q ( β ) u β , where A Q ( β ) := (cid:88) deg( B )= β χ Q ( B )and A Q ( β ) = 0 for β ≥ deg( Q ).Let S ( β ; n ) := (cid:88) deg( P )= n (cid:88) deg( B )= β (cid:32) BP (cid:33) . By the
Law of Quadratic Reciprocity [2], S ( β ; n ) = ( − q − βn (cid:88) deg( P )= n A P ( β ) ⇒ S ( β ; n ) = 0 ∀ n ≤ β. (2.9) RACES OF HIGH POWERS 9
We let π q ( n ) denote the number of monic irreducible polynomials of degree n in F q [ x ]. From the Prime Polynomial Theorem [2], π q ( n ) := { P ∈ F q [ x ] : deg( P ) = n } = q n n + O (cid:16) q n n (cid:17) . Lemma 2.2. [1, Proposition 7] (i) n odd, ≤ β ≤ n − : S ( β ; n ) = q β − n − S ( n − − β ; n )(2.10) and S ( n − n ) = π q ( n ) q n − . (2.11) (ii) n even, ≤ β ≤ n − : S ( β ; n ) = q β − n (cid:16) − S ( n − − β ; n ) + ( q − n − β − (cid:88) j =0 S ( j ; n ) (cid:17) (2.12) and S ( n − n ) = − π q ( n ) q n − . (2.13) Lemma 2.3. [1, Lemma 8] If β < n , then S ( β ; n ) = η β π q ( n ) q β + O ( βn q n + β ) , (2.14) where η β = 1 for β even and η β = 0 for β odd. Improved Estimate for S ( β ; n ) when β is even Initially, we concern ourselves with (cid:104) tr(Θ nC Q ) (cid:105) F g +2 ; in doing so, we need toestimate S ( β ; n ) for when β is even (see sections 5 and 7). The following theo-rem makes use of Lemmas 2.2 and 2.3; it is the analogous result to Proposition9 of [1] (since Rudnick considers the average value over F g +1 , estimates for S ( β ; n ) in [1] involve β odd). Furthermore, this result will allow us to compute (cid:104) tr(Θ nC Q ) (cid:105) F g +2 for n near 4 g (just as Proposition 9 in [1] allows Rudnick tocompute (cid:104) tr(Θ nC Q ) (cid:105) F g +1 for n near 4 g ). Theorem 3.1. If β is even, β (cid:54) = 0 , and β < n , then S ( β ; n ) = π q ( n )( q β − η n q β − n ) + O ( q n ) , (3.1) where η n = (cid:40) n even n odd. Remarks. (i) The result above is essentially the same as Proposition 9 in [1] with one additonal term; namely, π q ( n ) q β .(ii) As Rudnick points out in [1] , the main tool in proving Theorem 3.1 isduality; it allows us to improve the error term in estimates of S ( β ; n ) and toget results holding for n < g and not only for n < g . We would like tomention that the duality present in our character sums S ( β ; n ) is based on thefunctional equation (1.2) L ∗ ( u, χ P ) = ( uq ) (cid:98) deg( P ) − (cid:99) L ∗ ( 1 qu , χ P ) , for prime characters χ P (see the proof of Proposition 7 in [1] ).Proof. (i) If n is odd, we apply (2.10) to S ( β ; n ) and then apply (2.14) to S ( n − − β ; n ): S ( β ; n ) = q β − n − S ( n − − β ; n )= q β − n − (cid:32) π q ( n ) q n − − β + O (cid:16) n − − βn q n + n − − β (cid:17)(cid:33) = π q ( n ) q β + O ( q n ) . (ii) If n is even, we apply (2.12) to S ( β ; n ) and then apply (2.14) to S ( n − − β ; n ): S ( β ; n ) = q β − n (cid:32) − S ( n − − β ; n ) + ( q − n − β − (cid:88) j =0 S ( j ; n ) (cid:33) = q β − n (cid:32) O (cid:16) n − − βn q n + n − − β (cid:17) + ( q − n − β − (cid:88) j =0 (cid:16) η j π q ( n ) q j + O ( jn q n + j ) (cid:17)(cid:33) . The two error terms are O ( q n ). Since both n and β are even, n − β − π q ( n ) q β − n ( q − n − β − (cid:88) j =0 q j . Hence, S ( β ; n ) = π q ( n ) q β − n ( q − n − β − (cid:88) j =0 q j + O ( q n )= π q ( n ) q β − n ( q n − β −
1) + O ( q n )= π q ( n )( q β − q β − n ) + O ( q n ) . RACES OF HIGH POWERS 11 (cid:3) Computing tr(Θ nC Q ) for Q ∈ F g +2 For the time being, we restrict ourselves to F g +2 . Let Q ∈ F g +2 andconsider the curve C Q : y = Q ( x ). The trace of the powers of Θ C Q is givenby equation (2.4):tr(Θ nC Q ) = − q n − q n (cid:88) deg( f )= n Λ( f ) χ Q ( f )(4.1) = − q n − q n (cid:88) P,k deg( P k )= n deg( P ) χ Q ( P k )(4.2) = − q n + P n + (cid:3) n + H n , (4.3)where P n corresponds to k = 1, (cid:3) n corresponds to the sum over all k even,and H n corresponds to the sum over all odd k ≥ P n , (cid:3) n , and H n . Not surprinsingly, our results will only slightlydiffer from Rudnick’s. The addition of − /q n from (1.1) will be the maindifference. We will also have different cut-off points for n when estimating P n (see section 5.3 of [1]).5. Contribution of the Primes: P n The contribution of the primes in (4.1) is given by: P n = − q n (cid:88) deg( P )= n nχ Q ( P ) . So, (cid:104)P n (cid:105) F g +2 = − n ( q − q g +1+ n (cid:88) deg( P )= n (cid:88) α + β =2 g +2 σ n ( α ) (cid:88) deg( B )= β (cid:32) BP (cid:33) = − n ( q − q g +1+ n (cid:88) α + β =2 g +2 σ n ( α ) S ( β ; n ) . From Lemma 2.1, if n > g + 1, then σ n ( α ) (cid:54) = 0 ⇒ ( α ≡ n ) or α ≡ n )) ⇒ ( α = 0 or α = 1) , which follows from the fact that 0 ≤ α ≤ g + 1.Since σ n (0) = 1 and σ n (1) = − q , when n > g + 1, we have (cid:104)P n (cid:105) F g +2 = − n ( q − q g +1+ n ( S (2 g + 2; n ) − qS (2 g ; n )) . We now compute (cid:104)P n (cid:105) F g +2 by considering the case n ≤ g + 1 and the case n > g + 1, which we break into four (non-distinct) ranges:(i) n ≤ g + 1 : If S ( β ; n ) (cid:54) = 0, then β < n ; since β is even, S ( β ; n ) = π q ( n ) q β + O ( βn q β + n )= q n + β n + O ( q n + β n ) + O ( βn q β + n ) . Then S ( β ; n ) (cid:28) βn q n + β , which implies that (cid:104)P n (cid:105) F g +2 (cid:28) nq g + n (cid:88) β By replacing π q (2 g + 1) and simpligying, we obtain (cid:104)P n (cid:105) F g +2 = 2 g + 1( q − q g + · (cid:32)(cid:32) q g +1 g + 1 + O (cid:16) q g g + 1 (cid:17)(cid:33) q g (cid:33) = q q − O ( q − g ) . (iv) n = 2 g + 2: Similarly, (cid:104)P n (cid:105) F g +2 = n ( q − q g +1+ n · q · S (2 g ; n )= 2 g + 2( q − q g +1 · S (2 g ; 2 g + 2) . From Theorem 3.1, (cid:104)P n (cid:105) F g +2 = 2 g + 2( q − q g +1 · (cid:32)(cid:16) q g +2 g + 2 + O ( q g g + 2 ) (cid:17) ( q g − q g − ) + O ( q g ) (cid:33) = 1 + O ( q − g ) + O ( gq − g ) . (v) n > g + 2 : We apply Theorem 3.1 to get (cid:104)P n (cid:105) F g +2 = − n ( q − q g +1+ n (cid:0) S (2 g + 2; n ) − qS (2 g ; n ) (cid:1) = − n ( q − q g +1+ n (cid:32) π q ( n )( q g +22 − η n q g +2 − n ) − q · π q ( n )( q g − η n q g − n ) + O ( q n ) (cid:33) . Upon further simplification, (cid:104)P n (cid:105) F g +2 = n ( q − q g +1+ n (cid:32) η n π q ( n )( q g +2 − n − q g +1 − n ) + O ( q n ) (cid:33) = nη n π q ( n ) q n + O ( nq n − g )= η n (1 + O ( q − n )) + O ( nq n − g ) . Note. When n = 2 g + 2 , (v) yields (iv). Contribution of the Squares: (cid:3) n For n even, we have the following contribution from the squares of primepowers: (cid:3) n = − q n (cid:88) deg( P k )= n Λ( P k ) χ Q ( P k )= − q n (cid:88) deg( P k )= n Λ( P k ) χ Q (( P k ) )= − q n (cid:88) deg( h )= n Λ( h ) χ Q ( h ) . Therefore, (cid:104) (cid:3) n (cid:105) F g +2 = − q n (cid:88) deg( P k )= n Λ( P k ) (cid:104) χ Q ( P k ) (cid:105) F g +2 = − q n (cid:88) deg( P k )= n deg( P ) 1( q − q g +1 (cid:88) ≤ α ≤ g +1 (cid:88) deg( A )= αP (cid:45) A µ ( A ) (cid:88) deg( B )=2 g +2 − αP (cid:45) B . Although section 5 shows a slight deviation in (cid:104)P n (cid:105) F g +2 from (cid:104)P n (cid:105) F g +1 , wewill see that (cid:104) (cid:3) n (cid:105) F g +2 = (cid:104) (cid:3) n (cid:105) F g +1 . Let m = deg( P ). Since { B : deg( B ) = β , P (cid:45) B } = q β · (cid:40) m > β − | P | if m ≤ β , RACES OF HIGH POWERS 15 (cid:104)P n (cid:105) F g +1 = − q n (cid:88) deg( P k )= n deg( P ) 1( q − q g +1 (cid:88) ≤ α ≤ g +1 (cid:88) deg( A )= αP (cid:45) A µ ( A ) (cid:88) deg( B )=2 g +2 − αP (cid:45) B − q ( q − q n (cid:88) deg( P k )= n deg( P ) · (cid:32) (cid:88) g +1 − m <α ≤ g +1 σ m ( α ) q α + (cid:16) − | P | (cid:17) · (cid:88) ≤ α ≤ g +1 − m σ m ( α ) q α (cid:33) = − q ( q − q n (cid:88) deg( P k )= n deg( P ) · (cid:32)(cid:16) − | P | (cid:17) · (cid:88) α ≥ σ m ( α ) q α + (cid:88) g +1 − m <α ≤ g +1 σ m ( α ) q α − (cid:16) − | P | (cid:17) · (cid:88) α>g +1 − m σ m ( α ) q α (cid:33) = − q ( q − q n (cid:88) deg( P k )= n deg( P ) · (cid:32)(cid:16) − | P | (cid:17) · (cid:88) α ≥ σ m ( α ) q α − (cid:88) α>g +1 σ m ( α ) q α + 1 | P | · (cid:88) α>g +1 − m σ m ( α ) q α (cid:33) = − q ( q − q n (cid:88) deg( P k )= n deg( P ) · (cid:32)(cid:16) − | P | (cid:17) · (cid:88) α ≥ σ m ( α ) q α + O ( q − g ) (cid:33) . Solving the recurrence relation in Lemma 2.1, (cid:104) (cid:3) n (cid:105) F g +2 = − q ( q − q n (cid:88) deg( P k )= n deg( P ) · (cid:32)(cid:16) − | P | (cid:17) · − q − | P | + O ( q − g ) (cid:33) = − q n (cid:88) deg( P ) | n deg( P ) · (cid:16) | P || P | + 1 + O ( q − g ) (cid:17) = − q n (cid:88) deg( P ) | n deg( P ) · (cid:16) − | P | + 1 + O ( q − g ) (cid:17) = − q n (cid:88) deg( P ) | n deg( P ) + 1 q n (cid:88) deg( P ) | n deg( P ) · (cid:16) | P | + 1 + O ( q − g ) (cid:17) . Since q l = (cid:88) deg( h )= l Λ( h ) = (cid:88) deg( P k )= l deg( P ) = (cid:88) deg( P ) | l deg( P ) , (cid:104) (cid:3) n (cid:105) F g +2 = − q n (cid:88) deg( P ) | n deg( P ) | P | + 1 + O ( q − g ) . Contribution of the Higher Prime Powers: H n Using trivial bounds for H n , we obtain slightly different results from Rudnickin the case where n > g . This is due to the fact that our estimates of H n involve S ( β ; n ) for β even, as opposed to β odd, and, from Lemma 2.3, S ( β ; n ) (cid:28) (cid:40) q n + β if β is even q n + β if β is odd.We shall see that our bounds for n > g are absorbed in the error term from (cid:104)P n (cid:105) F g +2 when n > g + 1.The contribution to tr(Θ nC Q ) from the higher odd prime powers in (4.1) is: H n = − q n (cid:88) d | n ≤ d : odd (cid:88) deg( P )= nd nd χ Q ( P d )= − q n (cid:88) d | n ≤ d : odd (cid:88) deg( P )= nd nd χ Q ( P ) , where the last equality follows from the fact that χ Q ( P d ) = χ Q ( P ) for odd d .This implies that (cid:104) H n (cid:105) F g +2 = − q − q g +1+ n (cid:88) d | n ≤ d : odd nd (cid:88) deg( P )= nd (cid:88) α + β =2 g +2 σ nd ( α ) (cid:88) deg( B )= β (cid:16) BP (cid:17) = − q − q g +1+ n (cid:88) d | n ≤ d : odd nd (cid:88) α + β =2 g +2 σ nd ( α ) S ( β ; nd ) . If S ( β ; nd ) (cid:54) = 0, then β < nd ; also, S ( β ; nd ) (cid:28) q nd + β . Hence, RACES OF HIGH POWERS 17 (cid:104) H n (cid:105) F g +2 (cid:28) q g + n (cid:88) d | n ≤ d : odd nd (cid:88) β ≤ min( nd , g +2) q nd + β (cid:28) nq g + n (cid:88) d | n ≤ d : odd q nd +min(2 g, nd ) . If n ≤ g , then min( nd , g ) = nd for all d ≥ 3; and so, (cid:104) H n (cid:105) F g +2 (cid:28) nq g + n (cid:88) d | n ≤ d : odd q nd (cid:28) nq g + n · q n = nq g · q n (cid:28) gq g · q g = gq − g . If n > g , then min( nd , g ) ≤ n for all d ≥ 3; therefore, (cid:104) H n (cid:105) F g +2 (cid:28) nq g + n · q n (cid:28) nq n − g . Computing (cid:104) tr(Θ nC Q ) (cid:105) F g +2 Since (cid:104) tr(Θ nC Q ) (cid:105) F g +2 = − q n + (cid:104)P n (cid:105) F g +2 + (cid:104) (cid:3) n (cid:105) F g +2 + (cid:104) H n (cid:105) F g +2 , we obtainthe following: (cid:104) tr Θ nC Q (cid:105) F g +2 = − q n + O ( gq − g ) 0 < n ≤ g + 10 g + 1 < n < g + 1 q q − + O ( q − g ) n = 2 g + 1 η n (1 + O ( q − n )) + O ( nq n − g ) 2 g + 1 < n + η n ( − q n (cid:88) deg( P ) | n deg( P ) | P | + 1 + O ( q − g ))+ (cid:40) O ( gq − g ) n ≤ gO ( nq n − g ) 6 g < n. In particular, Theorem 8.1. (cid:104) tr Θ nC Q (cid:105) F g +2 = − q n + η n q n · (cid:88) deg( P ) | n deg( P ) | P | + 1 + O ( gq − g )+ − η n < n < g + 1 q q − n = 2 g + 1 O ( nq n − g ) 2 g + 1 < n. Computing (cid:104) tr(Θ nC Q ) (cid:105) F g +1 (cid:83) F g +2 From [1], we have the following: Theorem 9.1. (cid:104) tr Θ nC Q (cid:105) F g +1 = η n q n · (cid:88) deg( P ) | n deg( P ) | P | + 1 + O ( gq − g ) + − η n < n < g − − q − n = 2 gO ( nq n − g ) 2 g < n. We would like to find the expected value of tr(Θ nC Q ) over all curves C Q : y = Q ( x ) of genus g with Q monic. To do this, we use Theorems 8.1 and9.1. By identifying the family of curves described above with F g +1 (cid:83) F g +2 ,we see that (cid:104) tr(Θ nC Q ) (cid:105) F g +1 (cid:83) F g +2 = F g +1 F g +1 (cid:83) F g +2 ) (cid:104) tr(Θ nC Q ) (cid:105) F g +1 + F g +2 F g +1 (cid:83) F g +2 ) (cid:104) tr(Θ nC Q ) (cid:105) F g +2 , where F g +1 (cid:91) F g +2 ) = F g +1 + F g +2 = ( q − q g + ( q − q g +1 = q g ( q − q + 1) . We obtain the following: Corollary 9.2. (cid:104) tr(Θ nC Q ) (cid:105) F g +1 (cid:83) F g +2 = − q n qq + 1 + η n q n · (cid:88) deg( P ) | n deg( P ) | P | + 1 + O ( gq − g )+ − η n < n < g − − q − n = 2 g q q − n = 2 g + 1 O ( nq n − g ) 2 g + 1 < n. RACES OF HIGH POWERS 19 Note. The first main term in Corollary 9.2 does not appear in Theorem 9.1,neither does the term q q − corresponding to n = 2 g + 1 . Similarly, for n = 2 g ,the constant q − in Theorem 9.1 is scaled down to q − in Corollary 9.2. Inthe next section, we shall see that these differences are diminished when weconsider the average of tr(Θ nC Q ) over H g . Computing (cid:104) tr(Θ nC Q ) (cid:105) H g As we mentioned in the introduction, averaging over monic squarefree poly-nomials of a fixed degree is not the same as averaging over the moduli space ofhyperelliptic curves of genus g : in the latter case, we consider polynomials ofdegree 2 g + 1 and 2 g + 2. Also, by restricting ourselves to monic polynomials,we introduce a bias in the average value of the trace: the contribution of thepoint at infinity is related to the leading coefficient of Q , as seen by equation(2.4).We now turn our attention to finding the average of tr(Θ nC Q ) over H g : fromequations (1.5) and (2.4), (cid:104) tr(Θ nC Q ) (cid:105) H g = 1 H g · (cid:88) Q ∈ H g (cid:16) − λ nQ q n − q n (cid:88) deg( f )= n Λ( f ) χ Q ( f ) (cid:17) , (10.1)where λ Q = − Q ) is even and sgn( Q ) (cid:54) = (cid:3) Q ) is odd1 if deg( Q ) is even and sgn( Q ) = (cid:3) .Since there are exactly ( q − / q − / F ∗ q , if n is odd, 1 H g · (cid:88) Q ∈ H g λ nQ = 1 H g · (cid:88) Q ∈ H g λ Q = 0 . On the other hand, if n is even,1 H g · (cid:88) Q ∈ H g λ nQ = 1 H g · (cid:88) Q ∈ H g | λ Q | = (cid:98) F g +2 H g = qq + 1 . Also, given D ∈ F q [ x ] with deg( D ) = d , we may write D = A B , where A, B ∈ F q [ x ] with A monic, B not necessarily monic, and deg( A ) = α , deg( B ) = β ,so that d = 2 α + β . From here, we can take the character sum above over all elements in F q [ x ] of genus g by sieving out the squarefree terms (as we didearlier): (cid:88) Q ∈ H g χ Q ( f ) = (cid:88) α + β = dd =2 g +1 , g +2 (cid:88) deg( B )= βB n.n.m. (cid:88) deg( A )= α µ ( A ) (cid:16) Af (cid:17) (cid:16) Bf (cid:17) = (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) deg( B )= βB n.n.m. (cid:16) Bf (cid:17) = (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) a ∈ F ∗ q (cid:88) deg( B )= β (cid:16) aBf (cid:17) = (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) a ∈ F ∗ q (cid:88) deg( B )= β (cid:16) af (cid:17) · (cid:16) Bf (cid:17) = (cid:88) a ∈ F ∗ q (cid:16) af (cid:17) (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) deg( B )= β (cid:16) Bf (cid:17) . Hence, (cid:104) tr(Θ nC Q ) (cid:105) H g = − q n · H g (cid:32) (cid:88) Q ∈ H g λ nQ + (cid:88) deg( f )= n Λ( f ) (cid:88) a ∈ F ∗ q (cid:16) af (cid:17) (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) deg( B )= β (cid:16) Bf (cid:17)(cid:33) . If f is a power of some prime in F q [ x ], say f = P k , then for all a ∈ F ∗ q (seeProposition 3.2 of [2]), (cid:16) af (cid:17) = (cid:16) aP (cid:17) k = (cid:16) a q − deg( P ) (cid:17) k = a q − deg( f ) . If deg( f ) = deg( P k ) = n is even, then (cid:16) af (cid:17) = 1 because | F ∗ q | = q − 1. Thistells us that (cid:80) a ∈ F ∗ q (cid:16) af (cid:17) = q − 1. If deg( f ) = deg( P k ) = n is odd, then wehave that (cid:80) a ∈ F ∗ q (cid:16) af (cid:17) = 0 because there are exactly ( q − / F ∗ q andexactly ( q − / F ∗ q .So, for n odd, (cid:104) tr(Θ nC Q ) (cid:105) H g = 0 , RACES OF HIGH POWERS 21 and for n even, (cid:104) tr(Θ nC Q ) (cid:105) H g = − q n qq + 1 − H g · q n (cid:88) deg( f )= n Λ( f ) (cid:88) a ∈ F ∗ q (cid:16) af (cid:17) (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) deg( B )= β (cid:16) Bf (cid:17) = − q n qq + 1 − F g +1 (cid:83) F g +2 ) · q n (cid:88) deg( f )= n Λ( f ) (cid:88) α + β = dd =2 g +1 , g +2 σ ( f ; α ) (cid:88) deg( B )= β (cid:16) Bf (cid:17) = − q n qq + 1 − q n (cid:88) deg( f )= n Λ( f ) (cid:104) χ Q ( f ) (cid:105) F g +1 (cid:83) F g +2 . In other words, Theorem 10.1. For n odd, (cid:104) tr(Θ nC Q ) (cid:105) H g = 0 , and for n even, (cid:104) tr(Θ nC Q ) (cid:105) H g = (cid:104) tr(Θ nC Q ) (cid:105) F g +1 (cid:83) F g +2 = 1 q n · (cid:88) deg( P ) | n deg( P ) (cid:54) =1 deg( P ) | P | + 1 + O ( gq − g ) + − < n < g − − q − n = 2 gO ( nq n − g ) 2 g < n. In particular, Corollary 10.2. If n is odd, then (cid:104) tr(Θ nC Q ) (cid:105) H g = (cid:90) USp(2 g ) tr( U n ) dU. For n even with q ( g ) < n < g − q ( g ) and n (cid:54) = 2 g , (cid:104) tr(Θ nC Q ) (cid:105) H g = (cid:90) USp(2 g ) tr( U n ) dU + o ( 1 g ) . Proof. The first part is clear. To prove the second part, we treat each non-mainterm in Theorem 10.1 separately and show that each of them contributesan error term of o ( g ) in the desired region.Fix (cid:15) > 0. If n < g − (4 + (cid:15) ) log q ( g ), thenlim g →∞ g · nq n − g ≤ lim g →∞ g g − − (cid:15) = lim g →∞ g − (cid:15) = 0; i.e., O ( nq n − g ) = o ( 1 g ) . Note that 1 q n (cid:88) deg( P ) | n deg( P ) (cid:54) =1 deg( P ) | P | + 1 = O ( nq n ) . If n = (2 + (cid:15) ) log q ( g ), thenlim g →∞ g nq n (cid:28) (cid:15) lim g →∞ g log q ( g ) g (cid:15) = 0 . So, for n > (2 + (cid:15) ) log q ( g ),1 q n (cid:88) deg( P ) | n deg( P ) (cid:54) =1 deg( P ) | P | + 1 = O ( nq n ) = o ( 1 g ) . We have actually shown a stronger version of our statement; namely, for anyfixed (cid:15) > n with (2 + (cid:15) ) log q ( g ) < n < g − (4 + (cid:15) ) log q ( g )and n (cid:54) = 2 g , (cid:104) tr(Θ nC Q ) (cid:105) H g = − o ( 1 g ) . (cid:3) We now look at another approach which quickly verifies the first result ofTheorem 10.1. This argument was provided by Dr. Ze´ev Rudnick: fix a finitefield F q of odd cardinality q , let Q be any monic, squarefree polynomial in F q [ x ] of degree 2 g + 1 or 2 g + 2, and let a ∈ F ∗ q . Then C ( F q n ) = (cid:88) x ∈ P ( F qn ) (cid:16) χ n ( a ( Q ( x )) + 1 (cid:17) = q n + 1 + (cid:88) x ∈ P ( F qn ) χ n ( a ( Q ( x )) , where χ n is a multiplicative character on F q n defined by χ n ( α ) := α is a square in F ∗ q n α = 0 − α is not a square in F ∗ q n . RACES OF HIGH POWERS 23 When x is the point at infinity, Q ( x ) is defined by the evaluation of x g +2 Q ( x )at x = 0; i.e., χ n ( Q ( ∞ )) yields λ Q according to the count of (1.4). Moreover, − q n tr(Θ nC ) = (cid:88) x ∈ P ( F qn ) χ n ( aQ ( x ))Therefore, (cid:104) tr(Θ nC ) (cid:105) H g = − q n H g (cid:88) a ∈ F ∗ q (cid:88) Q ∈ F q [ x ] (cid:48) (cid:88) x ∈ P ( F qn ) χ n ( aQ ( x ))= − q n H g (cid:88) a ∈ F ∗ q χ n ( a ) (cid:88) Q ∈ F q [ x ] (cid:48) (cid:88) x ∈ P ( F qn ) χ n ( Q ( x )) , where (cid:80) Q ∈ F q [ x ] (cid:48) indicates that the sum is over all monic, squarefree polynomi-als Q ∈ F q [ x ] such that deg( Q ) = 2 g + 1 or deg( Q ) = 2 g + 2. When n is odd,there are exactly q − squares and q − non-squares in F ∗ q ⊂ F q n , which tells usthat (cid:88) a ∈ F ∗ q χ n ( a ) = 0 . So, for odd n , (cid:104) tr(Θ nC ) (cid:105) H g = 0 . On the other hand, when n is even, computing the average over the entiremoduli space reduces to computing the average over the moduli space withthe restriction that a = 1: for even n , every element of F ∗ q is a square in F ∗ q n so that (cid:88) a ∈ F q χ n ( a ) = q − (cid:104) tr(Θ nC ) (cid:105) H g = − ( q − q n H g (cid:88) Q ∈ F q [ x ] (cid:48) (cid:88) x ∈ P ( F qn ) χ n ( Q ( x ))= (cid:104) tr(Θ nC ) (cid:105) (cid:99) H g , where (cid:99) H g := { C Q ∈ H g : Q monic } . Evidently, H g = ( q − (cid:99) H g . Acknowledgements We thank Dr. Chantal David for many valuable suggestions which havevastly improved this paper and for her continuous support throughout thisresearch. We also thank Dr. Ze´ev Rudnick and Manal Alzahrani for theirvaluable input and insight. References [1] Rudnick, Z., Traces of High Powers of the Frobenius Class in the Hyperelliptic Ensemble, Acta Arith., 143(1):81-99, 2010.[2] Rosen, M., Number Theory in Function Fields , Springer-Verlag, New York, 2002.[3] Diaconis, P. and Shahshahani, M., On the Eigenvalues of Random Matrices , Studies inApplied Probabilities, J. Appl. Probab. 31A: 49-62, 1994.[4] Bucur, A., David, C., Feigon, B., Kaplan, N., Lalin, M., Ozman, E., and MatchettWood, M., The Distribution of F q -Points on Cyclic l -Covers of Genus g , 2014.[5] Alzahrani, M., The Distribution of Points on Hyperelliptic Cuvres Over F q of Genus g in Finite Extensions of F q (Master’s Thesis), Concordia University, Canada, 2015.[6] Kurlberg, P. and Rudnick, Z., The Fluctuations in the Number of Points on a Hyperel-liptic curve over a Finite Field, Journal of Number Theory 129, no. 3 (2009):580-87.[7] Weil, A., Sur les courbes alg ´ e briques et les vari ´ e t ´ e s qui s’en d ´ e duisent , Publications del’Insitut de Math´ematique de l’Universit´e de Strasbourg 7, Paris: Hermann et Cie., 1948.[8] Silverman, J., The Arithmetic of Elliptic Curves , Springer, Dordrecht, 2009.[9] Meleleo, G., Questions Related to Primitive Points on Elliptic Curvesand Statistics for Biquadratic Curves Over Finite Fields (Ph. D. Thesis),Universit`a degli Studi Roma Tre, Italy, 2015. Retrieved from