Traces, high powers and one level density for families of curves over finite fields
Alina Bucur, Edgar Costa, Chantal David, João Guerreiro, David Lowry-Duda
aa r X i v : . [ m a t h . N T ] O c t TRACES, HIGH POWERS AND ONE LEVEL DENSITY FORFAMILIES OF CURVES OVER FINITE FIELDS
ALINA BUCUR, EDGAR COSTA, CHANTAL DAVID, JO ˜AO GUERREIRO,AND DAVID LOWRY-DUDA
Abstract.
The zeta function of a curve C over a finite field may be expressedin terms of the characteristic polynomial of a unitary matrix Θ C . We developand present a new technique to compute the expected value of tr(Θ nC ) forvarious moduli spaces of curves of genus g over a fixed finite field in the limitas g is large, generalizing and extending the work of Rudnick [Rud10] andChinis [Chi15]. This is achieved by using function field zeta functions, explicitformulae, and the densities of prime polynomials with prescribed ramificationtypes at certain places as given in [BDF +
16] and [Zha]. We extend [BDF + L -functions L (1 / it, χ ). Asapplications, we compute the one-level density for hyperelliptic curves, cyclic ℓ -covers, and cubic non-Galois covers. Introduction and statement of results
Let F q be a finite field of odd cardinality, and let C be a smooth curve over F q .The Weil conjectures tell us that the Hasse-Weil zeta function has the form Z C ( u ) := exp ∞ X n =1 C ( F q n ) u n n ! = P C ( u )(1 − u )(1 − qu ) , where P C ( u ) := det (cid:0) − u Frob | H (cid:0) C (cid:0) F q (cid:1) , Q ℓ (cid:1)(cid:1) ∈ Z [ u ]is the characteristic polynomial of the Frobenius automorphism, whose roots haveabsolute value q − / and are stable (as a multiset) under complex conjugation.Furthermore, P C ( u ) corresponds to a unique conjugacy class of a unitary symplec-tic matrix Θ C ∈ USp(2 g ) such that the eigenvalues e iθ j correspond to the zeros q − / e iθ j of P C ( u ). This conjugacy class Θ C is called Frobenius class of C .For many different families of curves C , Katz and Sarnak [KS99] showed that as q → ∞ , the Frobenius classes Θ C become equidistributed in certain subgroups ofunitary matrices, where the group depends on the monodromy group of the familyof curves. Stated more precisely, suppose F ( g, q ) is a natural family of curves ofgenus g over F q with symmetry type M(2 g ), equipped with the Haar measure. The The first author was partially supported by Simons Foundation grant expected value of a function F evaluated on the eigenangles of curves in F ( g, q ) isdefined as h F (Θ C ) i F ( q,g ) := 1 F ( q, g ) X C ∈F ( q,g ) F ( C ) . Katz and Sarnak predicted thatlim q →∞ h F (Θ C ) i F ( q,g ) = Z M(2 g ) F ( U ) d U, where the integral is taken with the respect to the Haar measure. This means thatmany statistics of the eigenvalues can be computed, in the limit, as integrals overthe corresponding unitary monodromy groups.One particularly important and well-studied statistic is the one-level density,which concerns low-lying zeroes. The definition of the one-level density W f ( U ) ofa N × N unitary matrix U and with test function f in the function field setting isgiven by (2.4) in Section 2.The work of Katz and Sarnak concerns the q -limit. Recently, there has beenwork exploring another type of limit, examined by fixing a constant finite field F q and looking at statistics of families of curves as their genus g → ∞ , such as thework of Kurlberg and Rudnick [KR09] who first investigated that type of limit forthe distribution of tr (cid:0) Θ C (cid:1) for the family of hyperelliptic curves. The statisticsare then given by a sum of q + 1 independent and identically distributed randomvariables, and not as distributions in groups of random matrices. In a subsequentwork, Rudnick [Rud10] investigated the distribution of tr (cid:0) Θ nC (cid:1) for the same familyof hyperelliptic curves. Denote by F g +1 the family of hyperelliptic curves of genus g given in affine form by C : Y = Q ( X )where Q ( X ) is a square-free, monic polynomial of degree 2 g + 1. Rudnick showedthat the g -limit statistics for trace of high powers tr (cid:0) Θ nC (cid:1) over the family F g +1 agrees (for n in a certain range) with the corresponding statistics over USp(2 g )given by:(1.1) Z USp(2 g ) tr U n d U = g n = 0 , − η n < | n | < g, | n | > g. where(1.2) η n = ( n even , n odd . More precisely:
Theorem 1.1. [Rud10]
For all n > h tr Θ nC i F g +1 = η n q − n X deg v | n deg vq deg v + 1 + O (cid:0) gq − g (cid:1) + − η n < n < g, − − q − n = 2 g,O (cid:0) nq n − g (cid:1) n > g, where the sum is over all finite places of F q [ X ] . RACES, HIGH POWERS AND ONE LEVEL DENSITY 3
Furthermore, if q g < n < g − q g and n = 2 g , then h tr Θ nC i = Z USp(2 g ) tr U n d U + o (cid:18) g (cid:19) . Moreover, if f is an even test function in the Schwartz space S ( R ) with Fouriertransform ˆ f supported in ( − , , then h W f i F g +1 = Z USp(2 g ) W f ( U ) d U + dev( f ) g + o (cid:18) g (cid:19) , where dev( f ) = ˆ f (0) X v deg vq v − − ˆ f (1) 1 q − and the sum is over all finite places of F q [ X ] . We remark that the bias towards having more points over F q n whenever n iseven (and in a certain range with respect to g ), which follows from the symplecticsymmetry, was first pointed out by Brock and Granville [BG01].The results of Rudnick [Rud10] hold for statistics over the space F g +1 , whichis only a subset of the moduli space of hyperelliptic curves of genus g , H g (cf.Section 2). The statistics for the whole moduli space of hyperelliptic curves ofgenus g , H g , were obtained by Chinis [Chi15], and they differ slightly from thestatistics for F g +1 . Theorem 1.2. [Chi15]
For n odd, (cid:10) tr (cid:0) Θ nC (cid:1)(cid:11) H g = 0 , and for n even, (cid:10) tr (cid:0) Θ nC (cid:1)(cid:11) H g = q − n X deg v | n deg v =1 deg vq deg v + 1 + O (cid:0) gq − g (cid:1) + − < n < g, − − q − n = 2 g,O (cid:0) nq n − g (cid:1) g < n, where the sum is over all finite places of F q [ X ] . It is interesting that when studying the distribution of zeta zeros for hyperellipticcurves Faifman and Rudnick [FR10] can restrict to half of the moduli space (inthis case, polynomials of even degree) without it affecting the result; but whenone restricts to F g +1 the one-level density is not quite the same as the one-leveldensity on the whole moduli space H g . The difference is explained by the fact thatthe infinite place behaves differently in F g +1 and F g +2 . work of The results ofRudnick were vastly generalized in a recent paper of Bui and Florea [BF14], whichgive formulas for the one-level density which are uniform in q and d , and theycan then identify lower order terms when the support of the test function holds invarious ranges. For the one-level density of classical Dirichlet L-functions associatedto quadratic characters, some recent work of Fiorilli, Parks and Sodergren [FPS16]exhibits all the lower order terms which are descending powers of log X .We are interested in a different generalization, extending the statistics of Rudnickand Chinis to statistics of families of curves for fixed q and as g varies. In Section 2,we first present a new proof for Theorem 1.2 using function field zeta functions andexplicit formulae, specifically relying on densities of prime polynomials of differentramification types, as described in [BDF + A. BUCUR, E. COSTA, C. DAVID, J. GUERREIRO, AND D. LOWRY-DUDA what is used in [Rud10] and [BF14], and the result presented in Section 2 is weakerthan the results of Rudnick and Chinis (as our result holds for a more limitedrange of n ), but it has the benefit of having clear generalization to many familiesof curves. We present two such generalizations here. In Section 3 we generalize theresult from Section 2 for cyclic ℓ -covers curve, and in Section 4, we do the samefor cubic curves corresponding to non-Galois extensions. We summarize our mainresults in the following theorems, whose details can be found in Section 3 and 4.Throughout this paper, all explicit constants in the error terms can depend on ℓ and q . Theorem 1.3.
Let ℓ be an odd prime and let H g,ℓ be the moduli space of ℓ coversof genus g . For any ε > and n such that q g < n < (1 − ε ) (cid:16) gℓ − + 2 (cid:17) , as g → ∞ we have h tr Θ nC i H g,ℓ = Z U(2 g ) tr U n d U + O (cid:18) g (cid:19) . Let f be an even test function in the Schwartz space S ( R ) with supp ˆ f ⊂ (cid:16) − ℓ − , ℓ − (cid:17) ,then h W f (Θ C ) i H g,ℓ = Z U(2 g ) W f ( U ) d U − ˆ f (0) ℓ − g X v deg v (1 + ( ℓ − q − deg v )( q ℓ deg v/ −
1) + O (cid:18) g − ε (cid:19) , where the sum is over all places v of F q ( X ) . Theorem 1.4.
Let E ( g ) be the space of cubic non-Galois extensions of F q ( X ) withdiscriminant of degree g + 4 , and let δ, B > be fixed constants as in Theorem 4.1.For q g < n < δgB +1 / , and as g → ∞ , h tr Θ nC i E ( g ) = Z USp(2 g ) tr U n d U + O (cid:18) g (cid:19) . Let f be an even test function in the Schwartz space S ( R ) with supp ˆ f ⊂ ( − δ B +1 , δ B +1 ) ,then for any ε > , h W f (Θ C ) i E ( g ) = Z USp(2 g ) W f ( U ) d U − ˆ f (0) g κ + O (cid:18) g − ε (cid:19) , where κ is defined by (4.2) . We remark that the one-level densities exhibit the predicted symmetries: unitaryfor cyclic covers of order ℓ (for ℓ an odd prime), and symplectic for cubic non-Galoisextensions.The main theorems of Sections 2 and 3 rely on results concerning the densi-ties of prime polynomials with prescribed ramification types at particular placesfrom [BDF + L ( s, χ ). RACES, HIGH POWERS AND ONE LEVEL DENSITY 5
Acknowledgements
The proof of the Lindel¨of Hypothesis presented in Section 5 was suggested tous by Soundararajan, following his work with Chandee for a similar bound forthe Riemann zeta function [CS11], and we are very grateful for his suggestion andhelp. We would also like to thank Daniel Fiorilli and Lior Bary-Soroker for helpfuldiscussions, and Zeev Rudnick for helpful discussions related to this work, andcomments on previous versions of this paper. We would like to thank the ArizonaWinter School for creating opportunities for research and providing an excellentplatform for starting the collaboration that lead to this paper. We also wish tothank ICERM (Providence, RI) for its hospitality during Fall 2015 when this paperwas finalized. 2.
Hyperelliptic covers
In this section we present a weaker version of Theorem 1.2, using a differenttechnique, namely using the results of [BDF +
16] to count the function field ex-tensions corresponding to the hyperelliptic curves in H g with prescribed ramifica-tion/splitting conditions. Let H g be the moduli space of hyperelliptic curves ofgenus g . Every such curve has an affine model C : Y = Q ( X ) , with Q ( X ) is a square-free polynomial of degree 2 g + 1 or 2 g + 2. Theorem 2.1.
Let E ( Z / Z , g ) be the set of quadratic extensions of genus g of F q [ X ] , let v be a place, and let E ( Z / Z , g, v , ω ) be the subset of E ( Z / Z , g ) withprescribed behavior ω ∈ { ramified , split , inert } at the place v . Then for any ε > , E ( Z / Z , g, v , ramified ) E ( Z / Z , g ) = q − deg v q − deg v + O (cid:0) q − g (cid:1) E ( Z / Z , g, v , split ) E ( Z / Z , g ) = E ( Z / Z , g, v , inert ) E ( Z / Z , g )= 12 (cid:0) q − deg v (cid:1) + O (cid:0) q ( ε − g +1)+ ε deg v (cid:1) . Proof.
It is shown in [BDF +
16] that E ( Z / Z , g ) = 2 q g +2 (cid:0) − q − (cid:1) E ( Z / Z , g, v , ramified) = q − deg v q − deg v q g +2 (cid:0) − q − (cid:1) + O (1) E ( Z / Z , g, v , split) = E ( Z / Z , g, v , inert)= 12 (cid:0) q − deg v (cid:1) q g +2 (cid:0) − q (cid:1) + O v (cid:0) q ( g +1)(1+ ε ) (cid:1) , where O v o indicates that the implicit constant may depend on v . We prove inSection 5 that keeping track of the dependence on v gives E ( Z / Z , g, v , ω ) = 12(1 + q − deg v ) 2 q g +2 (1 − q ) + O (cid:0) q ( g +1)(1+ ε )+ ε deg v (cid:1) , for ω ∈ { split , inert } which proves the theorem. (cid:3) A. BUCUR, E. COSTA, C. DAVID, J. GUERREIRO, AND D. LOWRY-DUDA
Lemma 2.2.
Let C be a fixed F q -point in the moduli space H g , F q ( C ) its functionfield and tr Θ nC be the n -th power of the trace of C . Then (2.1) − q n tr Θ nC = X deg v | nv split in F q ( C ) deg v + η n X deg v | n v inert in F q ( C ) v − X deg v | nv inert in F q ( C ) deg v, where the sums are over all places v of F q ( X ) (including the place at infinity) withthe prescribed behavior.Proof. For any function field K , over F q ( X ), we denote its zeta function by ζ K ( s ).The lemma follows by taking the logarithmic derivative on both sides of g Y j =1 (cid:0) − q / q − s e iθ j (cid:1) = P C (cid:0) q − s (cid:1) = ζ F q ( C ) ( s ) ζ F q ( X ) ( s )with respect to q − s after expressing ζ F q ( C ) ( s ) /ζ F q ( X ) ( s ) as an Euler product. (cid:3) Theorem 2.3.
The average n -th moment of the trace over hyperelliptic curves ofgenus g is given by (cid:10) − q n tr Θ nC (cid:11) H g = η n q n − η n X deg v | n deg v =1 deg v q deg v + O (cid:0) q ( ε − g +1)+ n (1+ ε ) (cid:1) for all ε > , and where the sum is over all finite places v of F q ( X ) .Proof. We start out by averaging equation (2.1) over hyperelliptic curves of genus g , hence (cid:10) − q n tr Θ nC (cid:11) H g equals1 E ( k, Z / Z , g ) X C ∈H g X deg v | nv split in F q ( C ) deg v + η n X deg v | n v inert in F q ( C ) v − X deg v | nv inert in F q ( C ) deg v . Swapping the order of summation gives us (cid:10) − q n tr Θ nC (cid:11) H g = X deg v | n deg v E ( Z / Z , g, v, split) E ( k, Z / Z , g )+ η n X deg v | n v E ( Z / Z , g, v, inert) E ( Z / Z , g ) − X deg v | n deg v E ( Z / Z , g, v, inert) E ( Z / Z , g ) . RACES, HIGH POWERS AND ONE LEVEL DENSITY 7
Applying Theorem 2.1 we get (cid:10) − q n tr Θ nC (cid:11) H g = X deg v | n deg v (cid:18) q − deg v ) + O (cid:0) q ( ε − g +1)+ ε deg v (cid:1)(cid:19) + η n X deg v | n deg v (cid:18)
11 + q − deg v + O (cid:0) q ( ε − g +1)+ ε deg v (cid:1)(cid:19) − X deg v | n deg v (cid:18) q − deg v ) + O (cid:0) q ( ε − g +1)+ ε deg v (cid:1)(cid:19) . The main terms of the first and the third sums cancel, but their error terms do not.Therefore (cid:10) − q n tr Θ nC (cid:11) H g = η n X deg v | n deg v q − deg v + O q ( ε − g +1) X deg v | n deg v q ε deg v = η n X deg v | n deg v q − deg v + O (cid:0) q ( ε − g +1)+ n (1+ ε ) (cid:1) , where the last equality follows from the prime number theorem for F q [ X ] (as provedin [Ros02] for instance). Using the following identity(2.2) q n = X d | n d π ( d ) , where π ( d ) is the number of irreducible polynomials of degree d defined over F q ,we have for n even that X deg v | n deg v q − deg v = X deg v | n deg v − X deg v | n deg v q deg v = X d | n d π ( d ) + 1 − X deg v | n deg v q deg v = q n − X deg v | n deg v =1 deg v q deg v . We remark that in the second equality above, the extra 1 arises from the place atinfinity. (cid:3)
As expected from [Rud10] and [Chi15], the previous theorem agrees with corre-sponding statistics over USp(2 g ). Recall from [DS94] that(2.3) Z USp(2 g ) tr U n d U = g n = 0 − η n < n < g n > g. Corollary 2.4.
For any ε > , and as g → ∞ , h tr Θ nC i H g = − η n (cid:18) −
11 + q n (cid:19) + O (cid:16) q − n + q ( ε − g + n ( ε + ) (cid:17) . A. BUCUR, E. COSTA, C. DAVID, J. GUERREIRO, AND D. LOWRY-DUDA
Moreover, for any ε ′ > and n such that q g < n < g (cid:0) − ε ′ (cid:1) , we have as g → ∞ h tr Θ nC i H g = Z USp(2 g ) tr U n d U + O (cid:18) g (cid:19) . Proof.
Applying the prime number theorem to Theorem 2.1, we have η n X deg v | n deg v =1 deg v q deg v = η n q n q n + η n O (cid:0) q n (cid:1) . To prove the second statement, we apply the first statement choosing ε small enoughsuch that ( ε −
1) + 2 (cid:0) − ε ′ (cid:1) (cid:0) ε + (cid:1) < (cid:3) We can apply the last result to determine the one-level density of hyperellipticcurves, as done in [Rud10], and we recall the definition of the one-level density inthe function field setting with the relevant properties below for completeness. Wewill also apply this to other families of curves in the following sections.Let f be an even test function in the Schwartz space S ( R ), and for any integer N ≥
1, we define F ( θ ) := X k ∈ Z f (cid:18) N (cid:18) θ π − k (cid:19)(cid:19) , which has period 2 π and is localized in an interval of size approximatively 1 /N in R / π Z . Then, for a unitary matrix N × N matrix U with eigenvalues e iθ j , j =1 , . . . , N , we define the one-level density(2.4) W f ( U ) := N X j =1 F ( θ j ) , counting the number of angles θ j in an interval of length approximatively 1 /N around 0 (weighted with the function f ). Using the Fourier expansion, we havethat W f ( U ) = Z ∞−∞ f ( x ) dx + 1 N X n =0 ˆ f (cid:16) nN (cid:17) tr U n . Katz and Sarnak conjectured that for any fixed q , the expected value of W f (Θ c )over H g will converge to R USp(2 g ) W f ( U ) d U as g → ∞ for any test function, andwe show in the next theorem that this holds for test functions on a limited support(which is more restrictive than the support obtained in [Rud10, Corollary 3]). Theorem 2.5.
Let f be an even test function in the Schwartz space S ( R ) with supp ˆ f ⊂ ( − , . Then for any ε > , h W f (Θ C ) i H g = Z USp(2 g ) W f ( U ) d U + ˆ f (0) g X deg v =1 deg vq v − O (cid:18) g − ε (cid:19) , where the sum is over all finite places v of F q ( X ) . Moreover, lim g →∞ h W f (Θ C ) i H g = lim g →∞ Z USp(2 g ) W f ( U ) d U = Z R f ( x ) (cid:18) − sin(2 πx )2 πx (cid:19) d x. RACES, HIGH POWERS AND ONE LEVEL DENSITY 9
As we mentioned in the introduction, a vast generalization of the formula abovewas obtained by Bui and Florea [BF14] in some recent work.
Proof.
As ˆ f is continuous, its support is contained in [ − α, α ] for some 0 < α < W f (Θ C ) = g X j =1 X k ∈ Z f (cid:18) g (cid:18) θ j π − k (cid:19)(cid:19) = Z R f ( x ) d x + 12 g X n =0 ˆ f (cid:18) n g (cid:19) tr Θ nC = ˆ f (0) + 1 g αg X n =1 ˆ f (cid:18) n g (cid:19) tr Θ nC , where the last equality follows from f being even and the condition on the supportof ˆ f . Averaging W f (Θ C ) over our family of curves and applying Theorem 2.3 with0 < ε < − α α , we get h W f (Θ C ) i H g = ˆ f (0) − g αg X n =1 ˆ f (cid:18) ng (cid:19) + 1 g αg X n =1 ˆ f (cid:18) ng (cid:19) q n X deg v | n deg v =1 deg v q deg v + O (cid:0) q − εg (cid:1) = Z USp(2 g ) W f ( U ) d U + 1 g αg X n =1 ˆ f (cid:18) ng (cid:19) q n X deg v | n deg v =1 deg v q deg v + O (cid:0) q − εg (cid:1) , where we note that by (2.3) and recalling that f is even and supp ˆ f ⊂ ( − , Z USp(2 g ) W f ( U ) d U = ˆ f (0) − g αg X n =1 ˆ f (cid:18) ng (cid:19) . We now compute(2.6) αg X n =1 ˆ f (cid:18) ng (cid:19) q n X deg v | n deg v =1 deg v q deg v = X deg v ≤ αg deg v =1 deg v q deg v X k deg v ≤ αg ˆ f (cid:18) k deg vg (cid:19) q k deg v Suppose φ ( g ) is a function tending to 0 as g tends to infinity, to be specified later.We break the range of the inside sum of the right hand side at gφ ( g ). For the first range, we use the Taylor expansion for ˆ f to write ˆ f ( x ) = ˆ f (0) + O ( x ) = ˆ f (0) + o (1),explicitly, ˆ f (cid:18) k deg vg (cid:19) = ˆ f (0) + O (cid:18) k deg vg (cid:19) . Thus, (2.6) can be rewritten as (cid:16) ˆ f (0) + O ( φ ( g )) (cid:17) X deg v ≤ αg deg v =1 deg v q deg v (cid:18) q deg v − O (cid:16) q − gφ ( g ) (cid:17)(cid:19) = ˆ f (0) X deg v ≤ αg deg v =1 deg vq v − O (cid:16) φ ( g ) + q − gφ ( g ) (cid:17) = ˆ f (0) X deg v =1 deg vq v − O (cid:16) φ ( g ) + q − gφ ( g ) + q − αg (cid:17) . For the remaining range, X deg v ≤ αg deg v =1 deg vq deg v + 1 X gφ ( g ) ≤ k deg v ≤ αg ˆ f (cid:18) k deg vg (cid:19) q k deg v ≪ X deg v ≤ αg deg v =1 deg vq deg v + 1 X gφ ( g ) ≤ k deg v ≤ αg q − gφ ( g ) ≪ αgq − gφ ( g ) . Thus, by choosing φ ( g ) = g − ε , we get that1 g αg X n =1 ˆ f (cid:18) ng (cid:19) q n X deg v | n deg v =1 deg v q deg v = ˆ f (0) g X deg v =1 deg vq v − O (cid:18) g − ε (cid:19) , which proves the first statement. Taking the limit g → ∞ we get the second partof the theorem. (cid:3) General cyclic ℓ -covers Let ℓ be an odd prime and assume that q ≡ ℓ . Let H g,ℓ be the modulispace of general ℓ -covers of genus g . Every such cover has an affine model C : Y ℓ = Q ( X ) , where Q ( X ) is an ℓ -powerfree polynomial in F q [ X ].We first state an explicit form of Corollary 1.2 of [BDF +
16] for the number ofcyclic extensions with prescribed behavior at a given place v , keeping the depen-dence on the place v . All implied constants in the error term of this section candepend on q and ℓ . Theorem 3.1.
Let E ( Z /ℓ Z , d ) be the set of cyclic extensions of degree ℓ of F q [ X ] with conductor of degree d , let v be a place, ω ∈ { ramified, split, inert } , and E ( Z /ℓ Z , d, v , ω ) be the subset of E ( Z /ℓ Z , d ) with prescribed behavior ω at the place v . Then for any ε > , we have E ( Z /ℓ Z , d, v , ω ) E ( Z /ℓ Z , d ) = c v ,ω P v ,ω ( d ) P ( d ) + O (cid:16) q ( ε − ) d + ε deg v (cid:17) , RACES, HIGH POWERS AND ONE LEVEL DENSITY 11 where c v ,ω = ( ℓ − q − deg v ℓ − q − deg v if ω = ramified , ℓ (1 + ( ℓ − q − deg v ) if ω = split or inert , and where P ( x ) , P v , split ( x ) , P v , ramified ( x ) ∈ R [ x ] are monic polynomials of degree ℓ − and (3.1) P v , inert ( x ) = ( ℓ − P v , split ( x ) . Furthermore, (3.2) P v , inert ( d ) P ( d ) = ( ℓ −
1) + O deg v d + · · · + (cid:18) deg v d (cid:19) ℓ − ! . Finally, if ω = ramified, the error term can be written as O (cid:16) q ( ε − ) d (cid:17) , i.e., thereis not dependence on the place v in that case.Proof. This follows from Corollary 1.2 of [BDF + v as done in Section 5. This gives E ( Z /ℓ Z , d ) = C ℓ q d P ( d ) + O (cid:16) q ( + ε ) d (cid:17) E ( Z /ℓ Z , d, v , ramified) = c v ,ω C ℓ q d P v ,ω ( d ) + O (cid:16) q ( + ε ) d (cid:17) E ( Z /ℓ Z , d, v , split) = E ( Z /ℓ Z , d, v , inert)= c v ,ω C ℓ q d P v ,ω ( d ) + O (cid:16) q ( + ε ) d + ε deg v (cid:17) To bound the quotient P v , inert ( d ) P ( d ) , we also need the dependence on the coefficientsof P v , inert ( x ) = ( ℓ − x ℓ − + a v ,ℓ − x ℓ − + · · · + a v , for the place v . It follows from the computations of [BDF +
16] on page 4327 that a v ,i ≪ (deg v ) ℓ − − i for 0 ≤ i ≤ ℓ − . (This comes from the residue computation at u = q − ). The bound (3.2) thenfollows. (cid:3) Recall that for a function field extension
L/K cyclic of order ℓ , the discriminantand conductor of L/K are related bydeg Disc(
L/K ) = ( ℓ −
1) deg Cond(
L/K ) , (as given in Theorem 7.16 of [Ros02]) and from the Riemann–Hurwitz formula, wehave 2 g + 2( ℓ −
1) = deg Disc(
L/K ) . Thus we can interpret Theorem 3.1 in terms of the genus g by taking(3.3) d = 2 gℓ − . Further, put λ n := ( ℓ | n, . Lemma 3.2.
Let C be a given curve in H g,ℓ , F q ( C ) its function field and tr Θ nC bethe n -th power of the trace of C . Then (3.4) − q n tr Θ nC = ( ℓ − X deg v | nv split in F q ( C ) deg v + λ n ℓ X deg v | nℓ v inert in F q ( C ) deg v − X deg v | nv inert in F q ( C ) deg v, where the sums are over all places v of F q ( X ) (including infinity) with the prescribedbehavior.Proof. Mutatis mutandis Lemma 2.2. (cid:3)
Theorem 3.3.
For any ǫ > , we have (cid:10) − q n tr Θ nC (cid:11) H g,ℓ = λ n X deg v | nℓ ( ℓ −
1) deg v ℓ − q − deg v + O (cid:18) q n/ℓ n ℓ − d + q ( ε − ) d + n (1+ ε ) (cid:19) , where d is defined by (3.3) .Proof. We average (3.4) over E ( Z /ℓ Z , d ) with d = 2 g/ ( ℓ −
1) + 2 using Theorem 3.1to obtain (cid:10) − q n tr Θ nC (cid:11) H g,ℓ =( ℓ − X deg v | n deg v (cid:18) c v, split P v, split ( d ) P ( d ) + O (cid:16) q ( ε − ) d + ε deg v (cid:17)(cid:19) + λ n ℓ X deg v | nℓ deg v (cid:18) c v, inert P v, inert ( d ) P ( d ) + O (cid:16) q ( ε − ) d + ε deg v (cid:17)(cid:19) − X deg v | n deg v (cid:18) c v, inert P v, inert ( d ) P ( d ) + O (cid:16) q ( ε − ) d + ε deg v (cid:17)(cid:19) . Since c v, inert = c v, split and P v, inert ( x ) = ( ℓ − P v, split ( x ), the main term in the firstand the third sum cancel. Thus, using (3.2) (cid:10) − q n tr Θ nC (cid:11) H g,ℓ = λ n ℓ X deg v | nℓ P v, inert ( d ) P ( d ) c v, inert deg v + O q ( ε − ) d X deg v | n deg vq ε deg v = λ n X deg v | nℓ ( ℓ −
1) deg v ℓ − q − deg v + O d X deg v | nℓ deg v ℓ − + O (cid:16) q ( ε − ) d + n (1+ ε ) (cid:17) . = λ n X deg v | nℓ ( ℓ −
1) deg v ℓ − q − deg v + O (cid:18) q n/ℓ n ℓ − d + q ( ε − ) d + n (1+ ε ) (cid:19) . (cid:3) RACES, HIGH POWERS AND ONE LEVEL DENSITY 13
The previous theorem agrees with the corresponding statistics over the unitarygroup U(2 g ), as we have, by [DS94], Z U(2 g ) tr U n d U = ( g n = 0 , n = 0 . Corollary 3.4.
For any ε > and n such that q g < n < (1 − ε ) (cid:16) gℓ − + 2 (cid:17) ,as g → ∞ we have h tr Θ nC i H g,ℓ = Z U(2 g ) tr U n d U + O (cid:18) g (cid:19) . Proof.
Using Theorem 3.3, we have that h tr Θ nC i H g,ℓ = O (cid:16) q n/ℓ − n/ n ℓ − + q (2 ε − g/ ( ℓ − n (1 / ε ) (cid:17) , and we proceed as in the proof of Corollary 2.4. (cid:3) Theorem 3.5.
Let f be an even test function in the Schwartz space S ( R ) with supp ˆ f ⊂ (cid:16) − ℓ − , ℓ − (cid:17) , then h W f (Θ C ) i H g,ℓ = Z U(2 g ) W f ( U ) d U − ˆ f (0) ℓ − g X v deg v (1 + ( ℓ − q − deg v )( q ℓ deg v/ −
1) + O (cid:18) g − ε (cid:19) , where the sum is over all places v of F q ( X ) . Moreover, lim g →∞ h W f (Θ C ) i H g,ℓ = lim g →∞ Z U(2 g ) W f ( U ) d U = Z R f ( x ) d x = ˆ f (0) . Proof.
Pick α ∈ (cid:16) , ℓ − (cid:17) , such that the support of ˆ f is contained in [ − α, α ]. Bywriting out the definition of the one-level density and obtaining the Fourier expan-sion for each variable θ j , we get W f (Θ C ) = g X j =1 X k ∈ Z f (cid:18) g (cid:18) θ j π − k (cid:19)(cid:19) = Z R f ( x ) d x + 12 g X n =0 ˆ f (cid:18) n g (cid:19) tr Θ nC = ˆ f (0) + 1 g αg X n =1 ˆ f (cid:18) n g (cid:19) tr Θ nC , where the last equality follows from f being even and the condition on the supportof ˆ f .Averaging W f (Θ C ) over our family of curves and applying Theorem 3.3 with0 < ε < − α ( ℓ − ℓ + 1 + 2 α ( ℓ − , we get h W f (Θ C ) i H g,ℓ = ˆ f (0) − ℓ − g αg/ℓ X n =1 ˆ f (cid:18) ℓn g (cid:19) q ℓn/ X deg v | n deg v ℓ − q − deg v + O g αg X n =1 ˆ f (cid:18) n g (cid:19) q n/ℓ − n/ n ℓ − ! + O ( q − εg )= ˆ f (0) − ℓ − g αg/ℓ X n =1 ˆ f (cid:18) ℓn g (cid:19) q ℓn/ X deg v | n deg v ℓ − q − deg v + O (cid:18) g (cid:19) . We now compute αg/ℓ X n =1 ˆ f (cid:18) ℓn g (cid:19) q ℓn/ X deg v | n deg v ℓ − q − deg v = X deg v ≤ αg/ℓ deg v ℓ − q − deg v X k deg v ≤ αg/ℓ ˆ f (cid:18) kℓ deg v g (cid:19) q ℓk deg v/ . As in the proof of Theorem 2.5, let φ ( g ) be a function which tends to 0 as g tendsto ∞ , and we split the range of the inner sum at gφ ( g ). We start by addressing thefirst range, k deg v ≤ gφ ( g ). From the Taylor expansion of ˆ f ( x ) at 0, we haveˆ f (cid:18) kℓ deg v g (cid:19) = ˆ f (0) + O (cid:18) k deg vg (cid:19) , thus X deg v ≤ αg/ℓ deg v ℓ − q − deg v X k deg v ≤ gφ ( g ) ˆ f (cid:18) kℓ deg v g (cid:19) q ℓk deg v/ = (cid:16) ˆ f (0) + O ( φ ( g )) (cid:17) X deg v ≤ αg/ℓ deg v ℓ − q − deg v (cid:18) q ℓ deg v/ − O (cid:16) q − ℓgφ ( g ) / (cid:17)(cid:19) = ˆ f (0) X deg v ≤ αg/ℓ deg v (1 + ( ℓ − q − deg v )( q ℓ deg v/ −
1) + O (cid:16) φ ( g ) + q − gφ ( g ) (cid:17) = ˆ f (0) X v deg v (1 + ( ℓ − q − deg v )( q ℓ deg v/ −
1) + O (cid:16) φ ( g ) + q − gφ ( g ) + q − (2+ ℓ ) αgℓ (cid:17) . For the remaining range, X deg v ≤ αg/ℓ deg v ℓ − q − deg v X gφ ( g ) ≤ k deg v ≤ αg/ℓ ˆ f (cid:18) kℓ deg v g (cid:19) q ℓk deg v/ ≪ X deg v ≤ αg/ℓ deg v ℓ − q − deg v X gφ ( g ) ≤ k deg v ≤ αg/ℓ q − k deg v ≪ αgq − gφ ( g ) . Using φ ( g ) = g − ǫ , this completes the proof of the first statement of the theorem.As lim g →∞ Z U(2 g ) W f ( U ) d U = Z R f ( x ) d x = ˆ f (0) , RACES, HIGH POWERS AND ONE LEVEL DENSITY 15 we get the second part of the theorem by taking the limit g → ∞ . (cid:3) Cubic non-Galois Covers
In this section, we consider the family of cubic non-Galois curves. As a first stepwe need to count the number of cubic non-Galois extensions of genus g of F q ( X )with prescribed splitting at given places v with an explicit error term (in the genus g and in the place v ). The following result was recently obtained by Zhao [Zha]. Thecount was previously established by Datskovsky and Wright [DW88], but withoutan error term which is needed for the present application. As the final version ofthe preprint [Zha] is not available, we write the explicit constants appearing in theerror term as general constants, δ for the power saving in the count, and B forthe dependence on the place v . This allows to get a general result that could beapplied to different versions of Theorem 4.1. The same convention was adopted byYang [Yan09] who considered the one level-density for cubic non-Galois extensionsof Q , and this also allows us to compare our results with his. Theorem 4.1. [Zha]
Let E ( g ) be the set of cubic non-Galois extensions of F q ( X ) with discriminant of degree g + 4 . For any finite set of primes S , and any set Ω ofsplitting conditions for the primes contained in S , define E ( g, S , Ω) to be the subsetof E ( g ) consisting of the cubic extensions satisfying those splitting conditions.Then, as g → ∞ , E ( g, S , Ω) E ( g ) = Y v ∈S c v + O q − δg Y v ∈S q B deg v ! , where δ, B > are fixed constants, and c v = q v q deg v + q v / v totally split , / v partially split , / v inert ,q − deg v v partially ramified ,q − v v totally ramified . We also need the explicit formulas for the curves C associated to the cubic non-Galois extensions in E ( g ). This is proven following exactly the same lines as theproofs of the explicit formulas for the families of elliptic curves and cyclic covers oforder ℓ in lemmas 2.2 and 3.2. The result can also be found in a paper of Thorneand Xiong [TX14, Proposition 3] who computed other statistics for the same family. Proposition 4.2.
Let C be a given curve with function field F q ( C ) ∈ E ( g ) , and tr Θ nC be the n -th power of the trace of C . Then − q n tr Θ nC = X deg v | nv totally split in F q ( C ) v + X deg v | n v partially split in F q ( C ) v + X deg v | nv partially ramified in F q ( C ) deg v + X deg v | n v inert in F q ( C ) v − X deg v | nv inert in F q ( C ) deg v, (4.1) where the sums are over all places v of F q ( X ) (including the place at infinity) withthe prescribed behavior. For convenience, write τ n := ( | n, , and we recall that η n is given by (1.2). Theorem 4.3.
Let δ, B > be as in Theorem 4.1. The average n -th moment ofthe trace over cubic non-Galois curves in E ( g ) is given by (cid:10) − q n tr Θ nC (cid:11) E ( g ) = η n q n/ + η n − η n X deg v | n ( q deg v + 1) deg v q deg v + q v + X deg v | n q deg v deg v q deg v + q v + τ n X deg v | n q v deg v q deg v + q v + O (cid:16) q − δg q ( B +1) n (cid:17) , where the sums are over all places v of F q ( X ) .Proof. We rewrite equation (4.1) as − q n tr Θ nC = X α X v ∈V α ( C )deg v | ndα δ α deg v, where α = 1 , , , , α to refer to the type of ramification associated to the curve C in each term, moreprecisely as v ∈ V α ( C ). Note that δ = δ = 2, δ = 1, δ = 3, δ = −
1, and d = d = d = 1, d = 2 and d = 3.We now average over our family of curves with genus g to obtain (cid:10) − q n tr Θ nC (cid:11) E ( g ) = 1 E ( g ) X F q ( C ) ∈ E ( g ) X α X v ∈V α ( C )deg v | ndα δ α deg v = X α X deg v | ndα δ α deg v E ( g, v, α ) E ( g )= X α X deg v | ndα (cid:0) δ α deg v c v,α + O (cid:0) deg v q − δg q B deg v (cid:1)(cid:1) , where the second equality is obtained by swapping the order of the sums and thethird equality follows from Theorem 4 .
1. Note that the sum of the error terms is O (cid:16) q − δg q ( B +1) n (cid:17) . Writing A ( v ) = q v deg v q deg v + q v , we have X α X deg v | ndα δ α deg v c v,α = η n X deg v | n A ( v ) + X deg v | n A ( v ) q − deg v + τ n X deg v | n A ( v ) , RACES, HIGH POWERS AND ONE LEVEL DENSITY 17 and η n X deg v | n A ( v ) = η n X deg v | n deg v − η n X deg v | n (cid:0) q deg v + 1 (cid:1) deg v q deg v + q v = η n (cid:16) q n/ + 1 (cid:17) − η n X deg v | n (cid:0) q deg v + 1 (cid:1) deg v q deg v + q v using (2.2). (cid:3) Corollary 4.4.
For any ε > , and as g → ∞ , h tr Θ nC i E ( g ) = − η n + O (cid:16) q − n + q − δg + ( B + ) n (cid:17) . Further, for q g < n < δgB +1 / , and as g → ∞ , h tr Θ nC i E ( g ) = Z USp(2 g ) tr U n d U + O (cid:18) g (cid:19) . Proof. Mutatis mutandis
Corollaries 2.4 and 3.4. (cid:3)
Theorem 4.5.
Let δ, B > be fixed constants as in Theorem 4.1. Let f be aneven test function in the Schwartz space S ( R ) with supp ˆ f ⊂ ( − δ B +1 , δ B +1 ) , thenfor any ε > , h W f (Θ C ) i E ( g ) = Z USp(2 g ) W f ( U ) d U − ˆ f (0) g κ + O (cid:18) g − ε (cid:19) , where κ = 1 q − − X v (1 + q deg v ) deg v ( q deg v −
1) (1 + q deg v + q v )+ X v q deg v deg v (cid:0) q deg v/ − (cid:1) (1 + q deg v + q v )+ X v q v deg v (cid:0) q v/ − (cid:1) (1 + q deg v + q v ) , (4.2) where the sums are over all places of F q ( X ) . Moreover, lim g →∞ h W f (Θ C ) i E ( g ) = lim g →∞ Z USp(2 g ) W f ( U ) d U = Z R f ( x ) (cid:18) − sin(2 πx )2 πx (cid:19) d x. Proof.
Since the function ˆ f is continuous, its support is contained in [ − α, α ] forsome 0 < α < δ B +1 . Averaging W f (Θ C ) over our family of curves using (2.5) andTheorem 4.3 we get for 0 < ε < δ − α ( B + 1) that h W f (Θ C ) i E ( g ) = ˆ f (0) − g αg X n =1 ˆ f (cid:18) ng (cid:19) − g αg X n =1 ˆ f (cid:18) n g (cid:19) q − n/ F ( n ) + O (cid:0) q − εg (cid:1) where F ( n ) := η n − η n X deg v | n ( q deg v + 1) deg v q deg v + q v + X deg v | n q deg v deg v q deg v + q v + τ n X deg v | n q v deg v q deg v + q v . Moreover, the two first terms can be rewritten as Z USp(2 g ) W f ( U ) d U = ˆ f (0) − g X ≤ n ≤ αg ˆ f (cid:18) ng (cid:19) , using (2.3). Therefore, for 0 < ε < δ − α (2 B + 2) we have h W f (Θ C ) i E ( g ) = Z USp(2 g ) W f ( U ) d U − g αg X n =1 ˆ f (cid:18) n g (cid:19) q − n/ F ( n ) + O (cid:0) q − εg (cid:1) . We now compute the lower order terms for each of the sums of F ( n ) as definedabove. We have αg X n =1 ˆ f (cid:18) n g (cid:19) η n q n/ X deg v | n (1 + q deg v ) deg v q deg v + q v = X deg v ≤ αg (1 + q deg v ) deg v q deg v + q v X k deg v ≤ αg ˆ f (cid:18) k deg vg (cid:19) q k deg v ; αg X n =1 ˆ f (cid:18) n g (cid:19) q n/ X deg v | n q deg v deg v q deg v + q v = X deg v ≤ αg q deg v deg v q deg v + q v X k deg v ≤ αg ˆ f (cid:18) k deg v g (cid:19) q k deg v/ ; αg X n =1 ˆ f (cid:18) n g (cid:19) τ n q n/ X deg v | n q v deg v q deg v + q v = X deg v ≤ αg q v deg v q deg v + q v X k deg v ≤ αg ˆ f (cid:18) k deg v g (cid:19) q k deg v/ . As before, we break the range of the inside sum at gφ ( g ) where φ ( g ) is a functionwhich tends to 0 as g tends to infinity, and we use the Taylor expansion for ˆ f ( x )in the first range to get that the first, second and third sum above are respectivelyˆ f (0) X v (1 + q deg v ) deg v ( q deg v −
1) (1 + q deg v + q v ) + O (cid:16) φ ( g ) + q − gφ ( g ) + q − αg (cid:17) ˆ f (0) X v q deg v deg v (cid:0) q deg v/ − (cid:1) (1 + q deg v + q v ) + O (cid:16) φ ( g ) + q − gφ ( g ) + q − αg (cid:17) ˆ f (0) X v q v deg v (cid:0) q v/ − (cid:1) (1 + q deg v + q v ) + O (cid:16) φ ( g ) + q − gφ ( g ) + q − αg (cid:17) , RACES, HIGH POWERS AND ONE LEVEL DENSITY 19 and similarly gφ ( g ) X n =1 ˆ f (cid:18) n g (cid:19) η n q n/ = ˆ f (0) q − O (cid:16) φ ( g ) + q − gφ ( g ) (cid:17) . For the remaining range from gφ ( g ) to 2 αg , working as in the proofs of Theorems 2.5and 3.5, we have that each of the four sums is O (cid:0) αgq − gφ ( g ) (cid:1) . By choosing φ ( g ) = g − ε , we get that − g αg X n =1 ˆ f (cid:18) n g (cid:19) q − n/ F ( n ) = − g ˆ f (0) κ + O (cid:18) g − ε (cid:19) , which proves the first statement. Taking the limit g → ∞ we get the second partof the theorem. (cid:3) We now compare the results of the above theorem with the results obtainedby Yang for the one-level density of cubic non-Galois extensions over numberfields [Yan09]. Yang’s results hold for supp ˆ f ⊂ ( − c, c ), where c = 2(1 − A )2 B + 1 , and the parameters 0 < A < B > N p ( X, T ) = c P,T X + O (cid:0) X A p B (cid:1) , where N p ( X, T ) is the number of cubic non-Galois extensions of Q with discriminantbetween 0 and X and such that the splitting behavior at the prime p is of type T ,see [Yan09, Proposition 2.2.4]. In order to compare it with Theorem 4.5, we need tofind the correspondence between the A of (4.3) and the δ of Theorem 4.1 (the B ’sare the same). We rewrite (4.3) by dividing by the main term given by N ( X ), thenumber of non-Galois cubic fields of discriminant up to X which is CX for someabsolute constant C , and we rewrite (4.3) as N p ( X, T ) N ( X ) = c ′ P,T + O (cid:0) X A − p B (cid:1) . (4.4)In the situation of Theorem 4.1, since E ( g ) ∼ q g +4 , we have for one place v that E ( g, { v } , Ω) E ( g ) = c v + O (cid:16)(cid:0) q g (cid:1) − δ/ q B deg v (cid:17) . (4.5)Then, to compare (4.4) and (4.5), we set A − − δ ⇐⇒ δ = 2 − A. Then, we have that the support of the Fourier transform in Theorem 4.5 is ( − c, c )where c = δ B + 1 = 2 − A B + 1 , which agrees with the support of the Fourier transform in [Yan09, Proposition2.2.4]. Explicit error terms and the Lindel¨of bound
In this section we explain our approach to make the dependence on the place v explicit in Theorems 2.1 and 3.1. We start by reviewing how the counting offunction field extensions ramifying (or splitting or inert) at a given finite place v is obtained in [BDF + v reduces toobtaining the Lindel¨of bound for the Dirichlet L-functions L (1 / it, χ ), where χ is a Dirichlet character of modulus v and order ℓ . We conclude this section byproving this bound.The counting of function fields extensions in [BDF +
16] is done by writing ex-plicitly the generating series for the extensions, and applying the Tauberian Theo-rem [BDF +
16, Theorem 2.5] to the generating series. As usual, this involves movingthe line of integration and applying Cauchy’s residue theorem to the relevant re-gion. The main term will be given by the sum of the residues at the poles in theregion, and this is where the main terms of Theorems 2.1 and 3.1 come from. Theerror term comes from evaluating the integral at the limit of the region of analyticcontinuation of the generating series, which involves bounding the generating serieson some half line.We start by looking at the counting for cyclic extensions of degree ℓ with conduc-tor of degree d which ramify (or not ramify) at a given place v . In this case, the gen-erating series F R ( s ) and F U ( s ), respectively, converge absolutely for Re( s ) > ℓ − with a pole of order ℓ − s = ℓ − , which gives the main term. Each generatingseries has analytic continuation to Re( s ) = ℓ − + ε for any ε >
0, and the errorterm is then bounded by O (cid:16) q ( + ε ) d M (cid:17) , where M is the maximum value taken by F R ( s ) (or F U ( s )) on the line Re( s ) = ℓ − + ε . It is important to note that the generating series are absolutely boundedon this line, i.e., the bound does not depend on v , but might depend on q and ℓ ,and the results of Theorems 2.1 and 3.1 follow. There is a difference between thecase ℓ = 2 and ℓ ≥
3, as the generating series is written as the sum of two functions,one with a pole of order ℓ −
1, and one with poles of order 1. If ℓ ≥
3, the mainterm comes from the pole of order ℓ − ℓ = 2, the two poles are simple andthere is some cancellation between contributions of the residues at the two poles.It remains to deal with the error terms for the two unramified cases, namelycounting extensions split at v and inert at v . In this case the argument works thesame way for all ℓ ≥
2. Let ξ ℓ be a primitive ℓ -th root of unity, and let χ v,ℓ ( v ) = (cid:16) v v (cid:17) ℓ , be the ℓ -th power residue symbol, which is a Dirichlet character of order ℓ andmodulus v over F q ( X ).The generating series for E ( Z /ℓ Z , ℓ, v , split) is(5.1) F S ( s ) = 1 ℓ F U ( s ) + 1 ℓ ℓ − X j =0 ℓ − X k =1 ℓ − X r =0 ξ − rk deg v ℓ ! M j,k ( s, v , split) , where M j,k ( s, v , split) is given by Y v = v (cid:16) (cid:16) ξ j deg vℓ χ v,ℓ ( v ) k + · · · + ξ ( ℓ − j deg vℓ χ v,ℓ ( v ) ( ℓ − k (cid:17) N v − ( ℓ − s (cid:17) . RACES, HIGH POWERS AND ONE LEVEL DENSITY 21
As before, the count is then obtained by applying the Tauberian theorem to thegenerating series F S ( s ). This series converges absolutely for Re( s ) > ℓ − with apole of order ℓ − s = ℓ − , which gives the main term. The function F S ( s )has analytic continuation to Re( s ) = ℓ − + ε for any ε >
0. The error term isbounded by O (cid:16) q + ε M (cid:17) , where M is the maximal value of F S ( s ) for Re( s ) = ℓ − + ε. As we mentionedabove, F U ( s ) is absolutely bounded on this line, thus we have to bound F S ( s ) − ℓ F U ( s ) on the aforementioned line. We can rewrite M j,k ( s, v , split) as G ( s ) ℓ − Y r =1 Y v = v (cid:16) − ξ rj deg vℓ χ v,ℓ ( v ) rk N ( v ) − ( ℓ − s (cid:17) − , where the function G ( s ) converges absolutely for Re( s ) > ℓ − + ε , and it isuniformly bounded in that region. Hence our task is reduced to bounding theL-functions L j,k ( s ) = Y v = v − ξ j deg vℓ χ v,ℓ ( v ) k N ( v ) ( ℓ − s ! − , on the line Re( s ) = ℓ − + ε . The L j,k ( s ) (0 ≤ j, k ≤ ℓ −
1) are Dirichlet L-functions associated to some character χ of modulus v and order ℓ and we need toevaluate them at s = 1 / ε + it . Indeed, if ξ be any root of unity and we write ξ = q − iθ , then we have Y v (cid:18) − χ ( v ) ξ deg v N ( v ) s (cid:19) − = Y v (cid:16) − χ ( v ) q − ( s + iθ ) deg v (cid:17) − = L ( s + iθ, χ ) . In the following theorem we prove that the Lindel¨of Hypothesis is true for theL-functions L ( s, χ ) associated with non-trivial Dirichlet characters of F q [ X ]. Thereare two main ingredients in our proof, the Riemann Hypothesis and [CV10, Theo-rem 8.1], an Erd¨os–Tur´an-type inequality, proved by Carneiro and Vaaler, bound-ing the size of polynomials inside the unit circle. This approach was suggestedto us by Soundararajan who used the same approach in a paper in collaborationwith Chandee [CS11] to get similar bounds for ζ ( + it ). We are very thank-ful for his suggestion and his help. There are other bounds in the literature forlog | L (1 / it, χ v ) | , for example the bound proved by Altung and Tsimerman in[AT14, p.45] log | L (1 / , χ v ) | ≤ g log q ( g ) + 4 q / g / , where q is prime. Then, the bound below improves the constant from 1 to theoptimal constant log 22 (we recall that d = 2 g + 2 for hyperelliptic curves). Veryrecently, similar bounds with the constant log 22 were obtained by Florea [Flo16,Corollary 8.2] using a different proof based in similar ideas, inspired by the work ofCarneiro and Chandee [CC11]. Her proof also allows her to get better bounds forlog | L ( α + it, χ v ) | for α ≥ / Theorem 5.1.
Let v be a finite place of F q ( X ) and denote by χ v be the ℓ -th powerresidue symbol, which is a Dirichlet character of modulus v . Let d be the degree ofthe conductor of the character, and let L ( s, χ v ) be the L-function attached to χ v .For any s = σ + it with σ ≥ / , we have as d → ∞ , (5.2) log | L ( s, χ v ) | ≤ (cid:18) log 22 + o (1) (cid:19) d log q d . Hence, for any ε > , we have (5.3) L ( s, χ v ) ≪ q,ε (cid:0) q d (cid:1) ε . Proof.
Let v ( X ) be the polynomial of F q [ X ] corresponding to the place v , andconsider the curve C v : Y ℓ = v ( X ) . Let g be the genus of the curve. Then, d − g/ ( ℓ − C v writes as Z C v ( u ) = Q gj =1 (cid:0) − ue iθ j (cid:1) (1 − u )(1 − qu ) = Q ℓ − k =1 L ( u, χ kv )(1 − u )(1 − qu ) , where L ( u, χ kv ) = g/ ( ℓ − Y j =1 (cid:0) − √ qe iθ k,j u (cid:1) , renaming the roots of Z C v ( u ).Without loss of generality, take k = 1 and rewrite L ( u, χ v ) = d − Y j =1 (cid:0) − √ qe iθ j u (cid:1) . Evaluating at u = q − s for s = σ + it with σ ≥ /
2, we have(5.4) L ( s, χ v ) = d − Y j =1 (cid:16) − e i ( θ j − t log q ) q / − σ (cid:17) . We consider the polynomial F ( z ) = d − Y j =1 (cid:16) z − e i ( θ j − t log q ) q / − σ (cid:17) , and we notice that all α j = q / − σ e i ( θ j − t log q ) are such that | α j | ≤ σ ≥ / F M ( z ) = M Y m =1 ( z − α m )where | α m | ≤ ≤ m ≤ M , we have for any positive integer N thatsup | z |≤ log | F M ( z ) | ≤ log 2 MN + 1 + N X n =1 n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X m =1 α nm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (5.5) RACES, HIGH POWERS AND ONE LEVEL DENSITY 23
We then have to evaluate the sums of powers d − X j =1 α nj ≪ d − X j =1 e inθ j . Taking the logarithm derivative on both sides of (5.4) (similarly to the proofs ofLemma 2.2 and 3.2), we derive the following identity for n ≥ d − X j =1 e inθ j = − q − n X deg v | n deg v ( χ v ( v )) n deg v , where the sum is over all places v of F q ( X ) (including the place at infinity). There-fore, using(2.2), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d − X j =1 e inθ k,j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = q − n (1 + q n ) ≤ q n Replacing the bound above in (5.5) with M = 2 g/ ( ℓ −
1) = d −
2, we getsup | z |≤ log | F ( z ) | ≤ log 2 d − N + 1 + N X n =1 q n n ≤ log 2 d − N + 1 + log 2 N + 2 q N N The theorem follows by taking N = ⌊ (2 − f ( d )) log q d ⌋ , where f ( d ) is any positivefunction f ( d ) such that f ( d ) = o (1) and e − f ( d ) log q d = o (1), for example f ( d ) = log q log q d log q d . Without loss of generality assume N >
0, and we havesup | z |≤ log | F ( z ) | ≤ log 2 d (2 − f ( d )) log q d + o (cid:18) d log q d (cid:19) ≤ (cid:18) log 22 + o (1) (cid:19) d log q d , which shows (5.2), and (5.3) follows. (cid:3) References [AT14] Salim Ali Altug and Jacob Tsimerman. Metaplectic ramanujan conjecture over func-tion fields with applications to quadratic forms.
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