Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field
aa r X i v : . [ m a t h . N T ] M a y STATISTICS OF THE ZEROS OF ZETA FUNCTIONS INFAMILIES OF HYPERELLIPTIC CURVES OVER A FINITEFIELD
DMITRY FAIFMAN AND ZE´EV RUDNICK
Abstract.
We study the fluctuations in the distribution of zeros of zetafunctions of a family of hyperelliptic curves defined over a fixed finitefield, in the limit of large genus. According to the Riemann Hypothe-sis for curves, the zeros all lie on a circle. Their angles are uniformlydistributed, so for a curve of genus g a fixed interval I will containasymptotically 2 g |I| angles as the genus grows. We show that for thevariance of number of angles in I is asymptotically π log(2 g |I| ) andprove a central limit theorem: The normalized fluctuations are Gauss-ian. These results continue to hold for shrinking intervals as long as theexpected number of angles 2 g |I| tends to infinity. Introduction
Let C be a smooth, projective, geometrically connected curve of genus g ≥ F q of cardinality q . The zeta function of thecurve is defined as(1.1) Z C ( u ) := exp ∞ X n =1 N n u n n , | u | < /q where N n is the number of points on C with coefficients in an extension F q n of F q of degree n . The zeta function is a rational function of the form Z C ( u ) = P C ( u )(1 − u )(1 − qu )where P C ( u ) ∈ Z [ u ] is a polynomial of degree 2 g , with P (0) = 1, satisfiesthe functional equation P C ( u ) = ( qu ) g P C ( 1 qu )and has all its zeros on the circle | u | = 1 / √ q (this is the Riemann Hy-pothesis for curves [19]). Moreover, there is a unitary symplectic matrixΘ C ∈ USp(2 g ), defined up to conjugacy, so that P C ( u ) = det( I − u √ q Θ C )The eigenvalues of Θ C are of the form e πiθ C,j , j = 1 , . . . , g . Date : May 4, 2008.Supported by the Israel Science Foundation (grant No. 925/06).
Our goal is to study the statistics of the set of angles { θ j,C } as we draw C at random from a family of hyperelliptic curves of genus g defined over F q where q is assumed to be odd. The family, denoted by H g +2 ,q , is thatof curves having an affine equation of the form y = Q ( x ), with Q ∈ F q [ x ] amonic, square-free polynomial of degree 2 g + 2. The corresponding functionfield is called a real quadratic function field. The measure on H g +2 ,q issimply the uniform probability measure on the set of such polynomials Q .A fundamental statistic is the counting function of the angles. Thus foran interval I = [ − β , β ] (which may vary with the genus g or with q ), let N I ( C ) = { j : θ j,C ∈ I} The angles are uniformly distributed as g → ∞ (see Proposition 5.1): Forfixed I , N I ( C ) ∼ g |I| . We wish to study the fluctuations of N I as we vary C in H g +2 ,q . This is inanalogy to the work of Selberg [15, 16, 17], who studied the fluctuations inthe number N ( t ) of zeros of the Riemann zeta function ζ ( s ) up to height t .By the Riemann-von Mangoldt formula, N ( t ) = t π log t πe + 78 + S ( t ) + O ( 1 t )with S ( t ) = π arg ζ ( + it ). Selberg showed that the variance of S ( t ),for t picked uniformly in [0 , T ], is π log log T , and that the moments of S ( t ) / q π log log t are those of a standard Gaussian.Katz and Sarnak [9] showed that for fixed genus, the conjugacy classes { Θ C : C ∈ H g +2 ,q } become uniformly distributed in USp(2 g ) in the limit q → ∞ of large constant field size. In particular the statistics of N I arethe same as those of the corresponding quantity for a random matrix inUSp(2 g ). That is, if U ∈ USp(2 g ) is a unitary symplectic matrix, witheigenvalues e πiθ j ( U ) , j = 1 , . . . , g , set b N I ( U ) = { j : θ j ( U ) ∈ I} Then the work of Katz and Sarnak [9] gives(1.2) lim q →∞ Prob H g +2 ,q ( N I ( C ) = k ) = Prob USp(2 g ) (cid:16) b N I ( U ) = k (cid:17) In the limit of large matrix size, the statistics of b N I ( U ) and related quan-tities, such as the logarithm of the characteristic polynomial of U , havebeen found to have Gaussian fluctuations in various ensembles of randommatrices [13, 3, 2, 8, 11, 18, 4, 6, 20]. In particular, when averaged overUSp(2 g ), the expected value of b N I is 2 g |I| , the variance is π log(2 g |I| ) andthe normalized random variable ( b N I − g |I| ) / q π log(2 g |I| ) has a normal Due to the functional equation, it suffices to restrict the discussion to symmetricintervals.
TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 3 distribution as g → ∞ . Moreover this holds for shrinking intervals, that is ifwe take the length of the interval |I| → g → ∞ as long as the expectednumber of angles tends to infinity , that is as long as 2 g |I| → ∞ . Thus(1.2) implies that for the iterated limit lim g →∞ (lim q →∞ ) we get a Gaussiandistribution:lim g →∞ lim q →∞ Prob H g +2 ,q a < N I ( C ) − g |I| q π log(2 g |I| ) < b = 1 √ π Z ba e − x / dx In this paper we will study these problems for a fixed constant field F q in the limit of large genus g → ∞ , that is without first taking q → ∞ ,which was crucial to the approach of Katz and Sarnak. We will show thatas g → ∞ , for both the global regime ( |I| fixed) and the mesoscopic regime( |I| → g |I| → ∞ ), the expected value of N I is 2 g |I| , the varianceis asymptotically π log(2 g |I| ) and that the fluctuations are Gaussian, thatis for fixed a < b ,(1.3) lim g →∞ Prob H g +2 ,q a < N I − g |I| q π log(2 g |I| ) < b = 1 √ π Z ba e − x / dx Our argument hinges upon the fact that P C ( u ) is the L-function attachedto a quadratic character of F q [ x ]. Thus for Q monic, square free, of degree2 g +2 the quadratic character χ Q is defined in terms of the quadratic residuesymbol as χ Q ( f ) = (cid:16) Qf (cid:17) (see § L ( u, χ Q ) = Y P (1 − χ Q ( u ) u deg P ) − the product taken over all monic irreducible polynomials P ∈ F q [ x ]. Then P C ( u ) = (1 − u ) − L ( u, χ Q )as was found in E. Artin’s thesis [1]. Thus one may tackle the problem usingSelberg’s original arguments [15] adapted to the function field setting; thiswas carried out in the M.Sc. thesis of the first-named author [5]. Insteadwe follow a quicker route, via the explicit formula, used recently by Hughes,Ng and Soundararajan [7].An important challenge is to investigate the local regime, when the lengthof the interval is of order 1 / g as g → ∞ . Due to the Central Limit Theorem This is sometime called the “mesoscopic” regime The paper [15] is under the Riemann hypothesis; [16, 17] are unconditional.
DMITRY FAIFMAN AND ZE´EV RUDNICK for random matrices, we may rewrite (1.3) as(1.4) lim g →∞ Prob H g +2 ,q a < N I − g |I| q π log(2 g |I| ) < b = lim g →∞ Prob
USp(2 g ) a < b N I − g |I| q π log(2 g |I| ) < b and ask if (1.4) remains valid also for shrinking intervals of the form I = g J where J is fixed, when the result is no longer a Gaussian. An equivalentform of (1.4) was conjectured in [10]. Acknowledgement:
We thank Chris Hughes, Jon Keating, EmmanuelKowalski and Igor Shparlinski for discussions and comments on earlier ver-sions of the paper.2.
Background on Dirichlet characters and L-functions f ∈ F q [ x ] is defined as || f || = q deg f .The zeta function of the rational function field is ζ q ( s ) := Y P (1 − || P || − s ) − , ℜ ( s ) > F q [ x ]. Interms of the more convenient variable u = q − s the zeta function becomes Z ( u ) = Y P (1 − u deg P ) − , | u | < /q . By the fundamental theorem of arithmetic in F q [ x ], Z ( u ) can be expressedas a sum over all monic polynomials: Z ( u ) = X f monic u deg f and hence Z ( u ) = 11 − qu . Given a monic polynomial Q ∈ F q [ x ], a Dirichlet character modulo Q isa homomorphism χ : ( F q [ x ] /Q F q [ x ]) × → C × A character modulo Q is primitive if there is no proper divisor ˜ Q of Q andsome character ˜ χ mod ˜ Q so that χ ( n ) = ˜ χ ( n ) whenever gcd( n, Q ) = 1. TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 5
For a Dirichlet character χ modulo Q of F q [ x ], we form the L-function(2.1) L ( u, χ ) = Y P (1 − χ ( P ) u deg P ) − (convergent for | u | < /q ), where P runs over all monic irreducible polyno-mials. It can be expressed as a series(2.2) L ( u, χ ) = X f χ ( f ) u deg f where the sum is over all monic polynomials. If χ is nontrivial, then it iseasy to show that X deg f = n χ ( f ) = 0 , n ≥ deg Q and hence the L-function is in fact a polynomial of degree at most deg Q − χ ( cH ) = χ ( H ), ∀ c ∈ F × q . The analogue for ordinary Dirichlet char-acters is χ ( −
1) = 1. For even characters, the L-function has a trivial zeroat u = 1.We assume from now on that deg Q > χ is primitive. Onethen defines a “completed” L-function L ∗ ( u, χ ) = (1 − λ ∞ ( χ ) u ) − L ( u, χ )where λ ∞ ( χ ) = 1 if χ is “even”, and is zero otherwise. The completedL-function L ∗ ( u, χ ) is then a polynomial of degree D = deg Q − − λ ∞ ( χ )and satisfies the functional equation L ∗ ( u, χ ) = ǫ ( χ )( q / u ) D L ∗ ( 1 qu , χ − )with | ǫ ( χ ) | = 1. We express L ∗ ( u, χ ) in term of its inverse zeros as(2.3) L ∗ ( u, χ ) = D Y j =1 (1 − α j,χ u ) . The Riemann Hypothesis in this setting, proved by Weil [19], is that all | α j,χ | = √ q . We may thus write(2.4) α j,χ = √ qe πiθ j,χ for suitable phases θ j,χ ∈ R / Z . As a consequence, for any nontrivial char-acter, not necessarily primitive, the inverse zeros of the L-function all haveabsolute value √ q or 1. DMITRY FAIFMAN AND ZE´EV RUDNICK
Lemma 2.1.
Let χ be a non-trivial Dirichlet character modulo f . Then for n < deg f , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X deg B = n χ ( B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) deg f − n (cid:19) q n/ (the sum over all monic polynomials of degree n ).Proof. Indeed, all we need to do is compare the series expansion (2.2) of L ( u, χ ), which is a polynomial of degree at most deg f −
1, with the expres-sion in terms of the inverse zeros: X ≤ n< deg f ( X deg B = n χ ( B )) u n = deg f − Y j =1 (1 − α j u )to get X deg B = n χ ( B ) = ( − n X S ⊂{ ,..., deg f − } S = n Y j ∈ S α j and then use | α j | ≤ √ q . (cid:3) Note that for n ≥ deg f the character sum vanishes.2.2. Quadratic characters.
We assume from now on that q is odd. Let P ( x ) ∈ F q [ x ] be monic and irreducible. The quadratic residue symbol (cid:16) fP (cid:17) ∈{± } is defined for f coprime to P by (cid:18) fP (cid:19) ≡ f || P ||− mod P .
For arbitrary monic Q , the Jacobi symbol (cid:16) fQ (cid:17) is defined for f coprime to Q by writing Q = Q P j as a product of monic irreducibles and setting (cid:18) fQ (cid:19) = Y (cid:18) fP j (cid:19) . If f, Q are not coprime we set (cid:16) fQ (cid:17) = 0. If c ∈ F ∗ q is a scalar then(2.5) (cid:18) cQ (cid:19) = c q − deg Q . The law of quadratic reciprocity asserts that if
A, B ∈ F q [ x ] are monic andcoprime then(2.6) (cid:18) AB (cid:19) = (cid:18) BA (cid:19) ( − q − deg A deg B = (cid:18) BA (cid:19) ( − || A ||− · || B ||− . This relation continues to hold if A and B are not coprime as both sidesvanish. TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 7
Given a square-free Q ∈ F q [ x ], we define the quadratic character χ Q by χ Q ( f ) = (cid:18) Qf (cid:19) If deg Q is even, this is a primitive Dirichlet character modulo Q . Note thatby virtue of (2.5), χ Q is an even character (that is trivial on scalars) if andonly if deg Q is even.It is important for us that the numerator P C ( u ) of the zeta function (1.1)of the hyperelliptic curve y = Q ( x ) coincides with the completed DirichletL-function L ∗ ( u, χ Q ) associated with the quadratic character χ Q .2.3. The Explicit Formula.Lemma 2.2.
Let h ( θ ) = P | k |≤ K b h ( k ) e ( kθ ) be a trigonometric polynomial,which we assume is real valued and even: h ( − θ ) = h ( θ ) = h ( θ ) . Then for aprimitive character χ we have (2.7) D X j =1 h ( θ j,χ ) = D Z h ( θ ) dθ + λ ∞ ( χ ) 1 πi Z h ( θ ) ddθ log(1 − e πiθ √ q ) dθ − X f b h (deg f ) Λ( f ) || f || / (cid:16) χ ( f ) + χ ( f ) (cid:17) Proof.
By computing the logarithmic derivative u L ′ L in two different ways,either using the Euler product (2.1) or the zeros (2.3) we get an identity, for n > − D X j =1 α nj,χ = X deg f = n Λ( f ) χ ( f ) + λ ∞ ( χ )where Λ( f ) = deg P if f = P k is a prime power, and Λ( f ) = 0 otherwise.Therefore we get an explicit formula in terms of the phases θ j,χ − D X j =1 e πinθ j,χ = λ ∞ ( χ ) q | n | / + X deg f = | n | Λ( f ) || f || / ( χ ( f ) n < χ ( f ) n > n both positive and negative.Now let h ( θ ) = P | k |≤ K b h ( k ) e ( kθ ) be a trigonometric polynomial, whichwe assume is real valued and even: h ( − θ ) = h ( θ ) = h ( θ ). Then the Fouriercoefficients are also real and even: b h ( − k ) = b h ( k ) = b h ( k ). Using the Fourier DMITRY FAIFMAN AND ZE´EV RUDNICK expansion of h we get D X j =1 h ( θ j ) = D b h (0) + X j K X k =1 b h ( k )( e ( kθ j ) + e ( − kθ j ))= D Z h ( θ ) dθ − K X k =1 b h ( k ) λ ∞ ( χ ) q k/ + X deg f = k Λ( f ) || f || / (cid:16) χ ( f ) + χ ( f ) (cid:17) = D Z h ( θ ) dθ − λ ∞ ( χ ) K X k =1 b h ( k ) q k/ − X f b h (deg f ) Λ( f ) || f || / (cid:16) χ ( f ) + χ ( f ) (cid:17) Note that since h is real valued, K X k =1 b h ( k ) q k/ = Z h ( θ ) q − / e πiθ − q − / e πiθ = 12 πi Z h ( θ ) ddθ log 11 − e πiθ √ q dθ which gives the claim. (cid:3) χ Q , with Q square-free of degree 2 g + 2,we get λ ∞ = 1, D = 2 g , and the explicit formula reads(2.8) g X j =1 h ( θ j,Q ) = 2 g Z h ( θ ) dθ + 1 πi Z h ( θ ) ddθ log(1 − e πiθ √ q ) dθ − X f b h (deg f ) Λ( f ) || f || / χ Q ( f )3. Averaging over H g +2 ,q Let H d,q ⊂ F q [ x ] be the set of all square-free monic polynomials of degree d . The cardinality of H d,q is H d,q = ( (1 − q ) q d , d ≥ q, d = 1as may be seen by expressing the generating function P ∞ d =0 H d,q u d in termsof the zeta function Z ( u ) of the rational function field: Z ( u ) = Z ( u ) ∞ X d =0 H d,q u d In particular we have(3.1) H g +2 ,q = (1 − q ) q g +2TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 9 We denote by h•i the mean value of any quantity defined on H g +2 ,q , thatis h F i := 1 H g +2 ,q X Q ∈H g +2 ,q F ( Q ) Lemma 3.1. If f ∈ F q [ x ] is not a square then h χ Q ( f ) i ≤ deg f − (1 − q ) q g +1 Proof.
We use the Mobius function to pick out the square free monic poly-nomials via the formula X A | Q µ ( A ) = ( , Q square-free0 , otherwisewhere we sum over all monic polynomials whose square divides Q . Thus thesum over all square-free polynomials is given by X Q ∈H g +2 ,q χ Q ( f ) = X deg Q =2 g +2 X A | Q µ ( A ) (cid:18) Qf (cid:19) = X deg A ≤ g +1 µ ( A ) (cid:18) Af (cid:19) X deg B =2 g +2 − A (cid:18) Bf (cid:19) To deal with the inner sum, note that (cid:16) • f (cid:17) is a non-trivial character since f is not a square, so we can use Lemma 2.1 to get(3.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X deg B =2 g +2 − A (cid:18) Bf (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) deg f − g + 2 − A (cid:19) q g +1 − deg A if 2 g + 2 − A < deg f , and the sum is zero otherwise. Hence we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Q ∈H g +2 ,q χ Q ( f ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X deg A ≤ g +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X deg B =2 g +2 − A (cid:18) Bf (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X g +1 − deg f < deg A ≤ g +1 (cid:18) deg f − g + 2 − A (cid:19) q g +1 − deg A = q g +1 X g +1 − deg f Let P , ..., P k be prime polynomials. Then * χ Q ( k Y j =1 P j ) + = 1 + O k X j =1 || P j || Proof. We have χ Q ( Q kj =1 P j ) = 1 if gcd( Q kj =1 P j , Q ) = 1, and χ Q ( Q kj =1 P j ) =0 otherwise. Since for primes P , ..., P k the condition gcd( Q kj =1 P j , Q ) = 1is equivalent to P j dividing Q for some j , we may write χ Q ( k Y j =1 P j ) = 1 − ( , ∃ P j | Q , otherwiseand hence * χ Q ( k Y j =1 P j ) + = 1 − H g +2 ,q { Q ∈ H g +2 ,q : ∃ P j | Q } Replacing the set of square-free Q by arbitrary monic Q of degree 2 g + 2gives { Q ∈ H g +2 ,q : ∃ P j | Q } ≤ { deg Q = 2 g + 2 : ∃ P j | Q } ≤ k X j =1 q g +2 || P j || so that recalling H g +2 ,q = (1 − q ) q g +2 , we have1 − − q ) k X j =1 || P j || ≤ * χ Q ( k Y j =1 P j ) + ≤ * χ Q ( k Y j =1 P j ) + = 1 + O k X j =1 || P j || as claimed. (cid:3) For a polynomial Q ∈ F q [ x ] of positive degree, set η ( Q ) = X P | Q || P || the sum being over all monic irreducible (prime) polynomials dividing Q . Lemma 3.3. The mean values of η and η are uniformly bounded as g → ∞ : h η i ≤ , (cid:10) η (cid:11) ≤ − q ) Proof. We consider the first moment: We have h η ( Q ) i = 1 H g +2 ,q X Q ∈H g +2 ,q X P | Q || P || = 1 H g +2 ,q X deg P ≤ g +2 || P || { Q ∈ H g +2 ,q : P | Q } TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 11 We bound the number of square-free Q divisible by P by the number of all Q of degree 2 g + 2 divisible by P , which is q g +2 / || P || , to find h η ( Q ) i ≤ − q ) q g +2 X deg P ≤ g +2 || P || { deg Q = 2 g + 2 : P | Q }≤ − q ) q g +2 X deg P ≤ g +2 q g +2 || P || ≤ − q − X f || f || = 1(the last sum is over all monic polynomials) proving that h η ( Q ) i is uniformlybounded.For the second moment of η , we have (cid:10) η (cid:11) = 1 H g +2 ,q X Q ∈H g +2 ,q X P | Q || P || = 1 H g +2 ,q X deg P , deg P ≤ g +2 || P || · || P || { Q ∈ H g +2 ,q : P | Q, P | Q } For squarefree Q , if two primes P | Q and P | Q then necessarily P = P and then Q is divisible by both iff it is divisible by their product, hence { Q ∈ H g +2 ,q : P | Q, P | Q } = { Q ∈ H g +2 ,q : P P | Q }≤ { Q : deg Q = 2 g + 2 , P P | Q } = ( q g +2 || P P || , deg( P P ) ≤ g + 20 , otherwiseand hence the contribution of such pairs is bounded by1(1 − q ) q g +2 X P X P q g +2 || P || || P || ≤ − q ) X f || f || = 1(1 − q ) Thus we see (cid:10) η (cid:11) ≤ (1 − q ) − which is again uniformly bounded. (cid:3) Beurling-Selberg functions Let I = [ − β/ , β/ 2] be an interval, symmetric about the origin, of length0 < β < 1, and K ≥ I ± K aretrigonometric polynomials approximating the indicator function I satisfy-ing (see the beautiful exposition in [12, Chapter 1.2]): • I ± K are trigonometric polynomials of degree ≤ K • Monotonicity:(4.1) I − K ≤ I ≤ I + K • The integral of I ± K is close to the length of the interval:(4.2) Z I ± K ( x ) dx = Z I ( x ) dx ± K + 1 • I ± K ( x ) are even .As a consequence of (4.2), the non-zero Fourier coefficients of I ± K satisfy(4.3) (cid:12)(cid:12)(cid:12) b I ± K ( k ) − b I ( k ) (cid:12)(cid:12)(cid:12) ≤ K + 1and in particular(4.4) | b I ± K ( k ) | ≤ K + 1 + min (cid:18) β, π | k | (cid:19) , < | k | ≤ K Proposition 4.1. Let I = [ − β/ , β/ be an interval and K ≥ an integerso that Kβ > . Then (4.5) X n ≥ b I ± K (2 n ) = O (1)(4.6) X n ≥ n b I ± K ( n ) = 12 π log Kβ + O (1) where the implied constants are independent of K and β .Proof. To bound the sum (4.5), we may use (4.3) to write b I ± K (2 n ) = sin 2 πnβ πn + O ( 1 K )and hence X n ≥ b I ± K (2 n ) = X ≤ n ≤ K/ sin 2 πnβ πn + O (1)We treat separately the range n < /β and 1 /β < n < K . To bound thesum over n < /β , use sin 2 πnβ ≪ nβ and hence X ≤ n< /β sin 2 πnβ πn ≪ X ≤ n< /β nβn = O (1)For the sum on n > /β , we apply summation by parts. The partial sumsof sin 2 πnβ are(4.7) N X n =1 sin 2 πnβ = cos πβ − cos(2 N + 1) πβ πβ = O ( 1 β )Therefore X /β 2] which is symmetric about theorigin. TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 13 To prove (4.6), we use (4.3) to write X n> n b I ± K ( n ) = 1 π X n ≤ K (sin πnβ ) n + O (1)We split the sum into two parts: The sum over 1 ≤ n ≤ /β , where we use | sin πnβ | ≪ nβ to see that it gives a bounded contribution, and the sumover 1 /β < n ≤ K , where we use sin( y ) = (1 − cos(2 y )) to get X n> n b I ± K ( n ) = 12 π X β Let χ be a primitive Dirichlet character. We denote by N I ( χ ) the num-ber of angles θ j,χ of the L-function L ∗ ( u, χ ) (see (2.4)) in the interval I =[ − β/ , β/ S I ( χ ) by N I ( χ ) = 2 g |I| + 2 π arg(1 − e iπ |I| √ q ) + S I ( χ )Set N ± K ( χ ) = D X j =1 I ± K ( θ j,χ )Here K will depend on deg Q . This will be our approximation to the countingfunction N I ( χ ). Then by virtue of (4.1),(5.1) N − K ( χ ) ≤ N I ( χ ) ≤ N + K ( χ )Using the explicit formula (2.7), we find(5.2) N ± K ( χ ) =: D ( β ± K + 1 ) + λ ∞ ( χ ) 1 πi Z I ± K ( θ ) ddθ log(1 − e πiθ √ q ) dθ + S ± K ( χ ) where S ± K ( χ ) is(5.3) S ± K ( χ ) := − X deg f ≤ K b I ± K (deg f ) Λ( f ) || f || / n χ ( f ) + χ ( f ) o the sum taken over all prime powers f ∈ F q [ x ] (of degree ≤ K ).Note that since || I − I ± K || L = K +1 , we have1 πi Z I ± K ( θ ) ddθ log(1 − e πiθ √ q ) dθ = 1 πi Z β/ − β/ ddθ log(1 − e πiθ √ q ) dθ + O ( 1 K )= 2 π arg(1 − e iπβ √ q ) + O ( 1 K )(5.4)5.1. Quadratic characters. For the case at hand, of quadratic characters,we write N I ( Q ) for N I ( χ Q ), with similar meaning for S I ( Q ), N ± K ( Q ) and S ± K ( Q ). We have(5.5) S ± K ( Q ) := S ± K ( χ Q ) = − X deg f ≤ K b I ± K (deg f ) Λ( f ) || f || / χ Q ( f )We may now deduce that the zeros are uniformly distributed: Proposition 5.1. Every fixed (symmetric) interval I = [ − β/ , β/ con-tains asymptotically g |I| angles θ j,Q , in fact N I ( Q ) = 2 g |I| + O ( g log g ) Proof. Indeed from (5.1) it suffices to show that for the smooth countingfunctions N ± K ( χ Q ) we have N ± K ( χ Q ) = 2 g |I| + O ( g log g )Now from (5.2), (5.4) it follows that N ± K ( χ Q ) = 2 g |I| + O ( gK ) + O (1) + | S ± K ( Q ) | To bound S ± K ( Q ), use (5.5) and (4.4) in the form b I ± K (deg f )Λ( f ) = O (1) todeduce that S ± K ( Q ) ≪ X deg f ≤ K p || f || ≪ q K/ and hence (cid:12)(cid:12) N ± K ( χ Q ) − g |I| (cid:12)(cid:12) ≪ gK + q K/ Taking K ≈ log q g − log q log g gives the result. (cid:3) TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 15 Expected value We first bound the expected value of S I : Proposition 6.1. Assume that either the interval I = [ − β/ , β/ is fixedor that it shrinks to zero with g → ∞ in such a way that gβ → ∞ . Then h S I i = O (1) Proof. Using (5.1), (5.2) and (5.4), we find that for any K , (cid:10) S − K (cid:11) ≤ h S i + O ( gK ) ≤ (cid:10) S + K (cid:11) Taking K ≈ g/ 100 gives the remainder term above is bounded. So it remainsto bound the expected value of S ± K for such K .Recall that S ± K is a sum over prime powers. We separate out the contri-bution of even powers, which is not oscillatory, from that of the odd powers: S ± K = even + oddWe claim that the even powers give(6.1) even = − X n ≥ b I ± K (2 n ) + O ( η ( Q ))where η ( Q ) = X P | Q || P || the sum over prime divisors of Q .To see (6.1), note that for an even power of a prime, say f = g , we have χ Q ( f ) = 1 if gcd( g, Q ) = 1 and 0 otherwise. Writing the even powers of aprime as f = g , and noting that Λ( f ) = Λ( g ), we haveeven = − X gcd( g,Q )=1 b I ± K (2 deg g )Λ( g ) || g || = − X n ≥ b I ± K (2 n ) q n X deg g = n Λ( g ) + O X P | Q || P || where the remainder term is a sum over all prime divisors of Q . By theprime number theorem, P deg g = n Λ( g ) = q n and hence(6.2) − X gcd( g,Q )=1 b I ± K (2 deg g )Λ( g ) || g || = − X n ≥ b I ± K (2 n )proving (6.1).It now follows that expected value of the even powers is bounded: In-deed, the sum P n ≥ b I ± K (2 n ) is bounded by Proposition 4.1 (note that ourchoice K ≈ g/ 100 and the condition gβ → ∞ guarantees Kβ → ∞ , hence Proposition 4.1 is applicable). As for the term η ( Q ) = P P | Q || P || , it is notbounded individually, but its mean is bounded by Lemma 3.3.The expected value of the odd powers is h odd i = − X deg f odd b I ± K (deg f )Λ( f ) p || f || h χ Q ( f ) i To estimate the expected value of the odd powers, we use Lemma 3.1 and(4.4) in the form b I ± K (deg f )Λ( f ) = O (1) to find h odd i ≪ X deg f ≤ K p || f || deg f q g +1 ≪ (2 √ q ) K q g +1 which for K ≈ g/ 100 is bounded. (cid:3) Hence we see that(6.3) * S I q π log( gβ ) + → , g → ∞ A sum over primes Consider the sum over primes T ± K ( Q ) := − X P b I ± K (deg P ) deg P p || P || χ Q ( P )This will be our approximation to S I . From now on assume that K ≈ g log log( gβ )which will guarantee log Kβ ∼ log gβ and K = o ( g ). Theorem 7.1. Assume that g → ∞ and either < β < is fixed or β → while βg → ∞ . Take K ≈ g/ log log( gβ ) . Theni) (cid:10) | T ± K | (cid:11) ∼ π log βg ii) (7.1) (cid:10) | T + K − T − K | (cid:11) = O (1) iii) (7.2) (cid:10) | S ± K − T ± K | (cid:11) = O (1)The rest of this section is devoted to the proof of Theorem 7.1. TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 17 Computing D(cid:0) T ± K (cid:1) E . We have D(cid:0) T ± K (cid:1) E = 4 X P ,P b I ± K (deg P ) b I ± K (deg P ) deg P deg P p || P |||| P || h χ Q ( P P ) i The sum is over deg P , deg P ≤ K < g . Consider the contribution of pairssuch that P P is not a perfect square (the “off-diagonal pairs”). We mayuse Lemma 3.1 to bound their contribution by ≪ q g +1 X deg P ≤ K | b I ± K (deg P ) | deg P deg P p | P | Using (4.4) in the form | b I ± K ( k ) | ≪ / | k | gives that the inner sum is boundedby ≪ X deg P ≤ K deg P p | P | deg P deg P ≪ (2 √ q ) K Hence the off-diagonal contribution is bounded by ≪ (4 q ) K q g +1 which is negligible since we take K = o ( g ).Consider the contribution of pairs such that P · P is a square. Since P and P are primes, this forces P = P . These contribute(7.3) 4 X P (deg P ) || P || b I ± K (deg P ) (cid:10) χ Q ( P ) (cid:11) = 4 X P (deg P ) || P || b I ± K (deg P ) + O X P (deg P ) || P || b I ± K (deg P ) ! by Lemma 3.2.Using the prime number theorem { P : deg P = n } = q n /n + O ( q n/ )gives4 X P (deg P ) || P || b I ± K (deg P ) = 4 X ≤ n ≤ K (cid:18) n + O ( n q n/ ) (cid:19) b I ± K ( n ) + O (1)= 4 X ≤ n ≤ K n b I ± K ( n ) + O (1)By Proposition 4.1 we find(7.4) 4 X P (deg P ) || P || b I ± K (deg P ) = 2 π log Kβ + O (1)(note that if gβ → ∞ then Kβ ≈ gβ/ log log( gβ ) → ∞ ). To bound theremainder term in (7.3) use (4.4) in the form b I ± K (deg P ) deg P = O (1) to find that the sum is at most P P / || P || = O (1). Therefore we find D(cid:0) T ± K (cid:1) E = 2 π log( Kβ ) + O (1) . Bounding (cid:10) | T + K − T − K | (cid:11) . Next we compute the variance of the differ-ence D(cid:12)(cid:12) T + K − T − K (cid:12)(cid:12) E . Arguing as above, one sees that the only terms whichmay significantly contribute to the average are again the diagonal terms D(cid:12)(cid:12) T + K − T − K (cid:12)(cid:12) E = 4 X deg P ≤ K (deg P ) || P || (cid:16)b I + K (deg P ) − b I − K (deg P ) (cid:17) (cid:10) χ Q ( P ) (cid:11) + o (1)Since by (4.3) (cid:12)(cid:12)(cid:12) b I + K ( n ) − b I − K ( n ) (cid:12)(cid:12)(cid:12) ≤ K + 1we get D(cid:12)(cid:12) T + K − T − K (cid:12)(cid:12) E ≪ K X deg P ≤ K (deg P ) || P || Using the Prime Number Theorem, this is easily seen to be O (1). Hence wefind D(cid:12)(cid:12) T + K − T − K (cid:12)(cid:12) E = O (1)7.3. Bounding (cid:10) | S ± K − T ± K | (cid:11) . Next we show that (cid:10) | S ± K − T ± K | (cid:11) = O (1).We have S ± K − T ± K = − X f = P j ,j ≥ b I ± K (deg f ) Λ( f ) || f || / χ Q ( f )= even + odd(7.5)where the term “even” is a sum over the even powers of primes, and “odd” isthe sum over odd powers of primes where the exponent is at least 3. We willshow that the second moments of both the odd and even terms are bounded.We first argue that the second moment of the even powers contribute abounded amount. As we saw in the proof of Proposition 6.1, see (6.1), wehave even ≪ X P | Q || P || the sum being over all prime divisors of Q . This is not bounded individually,but its second moment is bounded by Lemma 3.3.It remains to bound the contribution of the odd powers. We have (cid:10) | odd | (cid:11) = 4 X f ,f b I ± K (deg f ) b I ± K (deg f ) Λ( f )Λ( f ) || f f || / h χ Q ( f f ) i where the sum is over odd higher prime powers, that is over f = P j with j ≥ TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 19 The pairs where f · f is not a square contribute o (1) by the same ar-gument as above. Consider the contribution of pairs such that f · f is asquare. If f and f are odd higher prime powers but f · f is a square, thennecessarily f = P r , f = P s with P prime, r, s ≥ 2, (and r = s mod 2).Necessarily then r + s ≥ 4. The contribution of such pairs can be bounded,using (4.4) in the form b I ± K (deg f )Λ( f ) = O (1), by X P X r + s ≥ || P || ( r + s ) / ≪ X P X j ≥ j || P || j/ ≪ X P || P || = O (1)Hence (cid:10) | odd | (cid:11) = O (1) and therefore (cid:10) | S ± K − T ± K | (cid:11) = O (1)8. Higher moments of T ± K In this section we show that all moments of T ± K are Gaussian. Theorem 8.1. Assume the setting of Theorem 7.1 and let r ≥ . Then (cid:12)(cid:12)(cid:10) ( T ± K ) r − (cid:11)(cid:12)(cid:12) = o (1) and (cid:10) ( T ± K ) r (cid:11) = (2 r )! r ! π r log r ( βK ) + O (cid:0) log r − ( βK ) (cid:1) Proof. For the odd moments, we have D(cid:0) T ± K (cid:1) r − E = − r − X P ,...,P r − Q b I ± K (deg P j ) deg P j p || Q P j || D χ Q ( Y P j ) E Since Q j P j cannot be a perfect square, we may apply lemma 3.1 and obtainthe bound (cid:12)(cid:12)(cid:12)D(cid:0) T ± K (cid:1) r − E(cid:12)(cid:12)(cid:12) ≪ q g +1 X deg P ≤ K | b I ± K (deg P ) | deg P deg P p | P | r − As was already calculated in § ≪ X deg P ≤ K deg P p | P | deg P deg P ≪ (2 √ q ) K Hence (cid:12)(cid:12)(cid:12)D(cid:0) T ± K (cid:1) r − E(cid:12)(cid:12)(cid:12) ≪ (2 √ q ) (2 r − K q g +1 which vanishes assuming K ≈ g/ log log( gβ ).To compute the even moments, write D(cid:0) T ± K (cid:1) r E = 2 r ( T rsq + T rnsq ) where both T rsq and T rnsq have the form X P ,...,P r Q b I ± K (deg P j ) deg P j p || Q P j || DY χ Q ( P j ) E where T rsq is the sum over prime 2 r -tuples { P j } for which Q rj =1 P j is aperfect square, and T rnsq contains the remaining (off-diagonal) terms.The term T rnsq can be bounded as was done for the odd moments: T rnsq ≪ q g +1 X deg P ≤ K | b I ± K (deg P ) | deg P deg P p | P | r ≪ (2 √ q ) rK q g +1 Now T rsq = X P ····· P r = (cid:3) Q b I ± K (deg P j ) deg P j p || Q P j || DY χ Q ( P j ) E the sum taken over only those primes for which Q P j is a square, whichimplies all P j appear in equal pairs in each summand. Note that in particularall summands are positive. By lemma 3.2 we may replace h Q χ Q ( P j ) i with1 by introducing an error of O (cid:16)P j / || P j || (cid:17) .The total error produced by this substitution is, keeping in mind that theprimes P , . . . P r must come in identical pairs, bounded by r X j =1 X P ,...,P r Q rk =1 b I ± K (deg P k ) (deg P k ) || P j || Q k = j || P k || ≪≪ X P ,...,P r Q rk =2 b I ± K (deg P k ) (deg P k ) Q rk =2 || P k || X P b I ± K (deg P ) (deg P ) || P || The inner sum is bounded, and hence the total error introduced is ≪ X P ,...,P r Q rk =2 b I ± K (deg P k ) (deg P k ) Q rk =2 || P k || ≪ (log( βK )) r − by (7.4).So far we showed that T rsq = X P ····· P r = (cid:3) Q b I ± K (deg P j ) deg P j p || Q P j || + O (log r − ( βK ))Now we show that pairs of equal P j in X P ····· P r = (cid:3) Q b I ± K (deg P j ) deg P j p || Q P j || TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 21 can be taken all distinct, for the remaining terms are bounded by ≪ X P = P = P = P b I ± K (deg P ) deg P || P || X Q rj =5 P j = (cid:3) Q b I ± K (deg P j ) deg P j p || Q P j ||≪ ∞ X j =0 q j j j q j log r − ( βK ) ≪ log r − ( βK )Finally, the sum over distinct pairs is(2 r )! r !2 r X P ,...,P r distinct Q b I ± K (deg P j ) deg P j || Q P j || Now we remove the restriction that P , . . . P r are distinct, introducing (again)an error of O (log r − ( βK )), and obtain T rsq = (2 r )! r !2 r X P b I ± K (deg P ) deg P || P || ! r + O (log r − ( βK ))Summarizing all said above, and using (7.4) yields T rsq = (2 r )! r ! π r r log r ( βK ) + O (log r − ( βK ))and (cid:10) ( T ± K ) r (cid:11) = (2 r )! r ! π r log r ( βK ) + O (cid:0) log r − ( βK ) (cid:1) as claimed. (cid:3) Corollary 8.2. Under the assumption of Theorem 7.1, T ± K / q π log gβ hasa standard Gaussian limiting distribution. Indeed, the main-term expressions for the moments of T ± K imply all mo-ments of T ± K / q π log gβ are asymptotic to standard Gaussian moments,where the odd moments vanish and the even moments are1 √ π Z ∞−∞ x r e − x / dx = 1 · · · · · · (2 r − 1) = (2 r )!2 r r !9. Conclusion In this section we prove the claim (1.3) in our introduction. Recall thatwe wrote N I ( Q ) = 2 g |I| + 2 π arg(1 − e iπ |I| √ q ) + S I ( Q )and thus (1.3) is equivalent to: Theorem 9.1. Assume either that the interval I = [ − β/ , β/ is fixed, orthat its length β shrinks to zero while gβ → ∞ . Then (cid:10) | S I | (cid:11) ∼ π log gβ and S I / q π log βg has a standard Gaussian distribution. To prove this, it suffices to show that the second moment of the difference S I − T ± K is negligible relative to log( gβ ): Proposition 9.2. Assume that K ≈ g/ log log gβ , and that either β is fixedor β → while gβ → ∞ . Then (9.1) *(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S I − T ± K q π log gβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + → S I is close to thatof T ± K and and the distribution of S I / q π log βg coincides with that of T ± K / q π log( gβ ), that is by Corollary 8.2 we find that S I / q π log βg hasa standard Gaussian distribution. Thus we will have proved Theorem 9.1once we establish Proposition 9.2.9.1. Proof of Proposition 9.2. Assume that K ≈ g/ log log( gβ ). Then itsuffices to show(9.2) (cid:10) | S I − T ± K | (cid:11) ≪ ( gK ) . We first show(9.3) (cid:10) | S I − S ± K | (cid:11) ≪ ( gK ) . By (5.1), we have S − K ≤ S I + O ( gK ) ≤ S + K . Hence 0 ≤ S I − S − K + O ( gK ) ≤ S + K − S − K . Since we are dealing now with positive quantities, we may take absolutevalues and get | S I − S − K + O ( gK ) | ≤ | S + K − S − K | and applying the triangle inequality gives | S I − S − K | ≤ | S + K − S − K | + O ( gK ) , hence | S I − S − K | ≤ | S + K − S − K | + O (( gK ) ) . TATISTICS FOR ZEROS OF HYPERELLIPTIC ZETA FUNCTIONS 23 Taking expected values we get(9.4) (cid:10) | S I − S − K | (cid:11) ≤ (cid:10) | S + K − S − K | (cid:11) + O (cid:16) ( gK ) (cid:17) . To bound (cid:10) | S + K − S − K | (cid:11) , use the triangle inequality to get | S + K − S − K | ≤ | S + K − T + K | + | T + K − T − K | + | T − K − S − K | and hence | S + K − S − K | ≤ (cid:0) | S + K − T + K | + | T + K − T − K | + | T − K − S − K | (cid:1) . Applying (7.1) and (7.2) we find(9.5) (cid:10) | S + K − S − K | (cid:11) = O (1) . 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