Moments of zeta functions associated to hyperelliptic curves over finite fields

Abstract

Let q be an odd prime power, and denote the set of square-free monic polynomials D(x)∈F_q[x] of degree d. Katz and Sarnak showed that the moments, over , of the zeta functions associated to the curves y²=D(x), evaluated at the central point, tend, as , to the moments of characteristic polynomials, evaluated at the central point, of matrices in USp(2⌊(d−1)/2⌋). Using techniques that were originally developed for studying moments of L-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and . We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.

1. Introduction

In this paper, we provide theoretical and numerical evidence in support of a conjecture of Andrade and Keating regarding the moments, at the central point, of zeta functions associated to hyperelliptic curves over finite fields of odd characteristic.

Relevant background on these zeta functions is provided in this section. Section 2 describes the Andrade–Keating conjecture. We present numerical support for the conjecture in §3 and describe the algorithms used in §9.

In §§5 and 6, we apply old work of Birch [1] to obtain formulae for all the positive integer moments when d=3 or 4. We are also able, for 5≤d≤9, to use our data to guess formulae for a few specific moments (for example, the first three moments when d=7). These are presented in §7.

We then derive, in §8, series expansions for Andrade and Keating's conjectured formula. By comparing with the actual moments, derived or guessed, we can precisely determine in certain cases the remainder term in Andrade and Keating's formula for the moments.

(a). Zeta functions of quadratic function fields according to Artin

Let q be an odd prime power, and be a square-free monic polynomial of positive degree d. Artin [2] developed the theory of quadratic function fields in analogy to that of Dedekind for quadratic number fields. Let R be the ring

1.1

Inspired by Dedekind's work on algebraic number fields, Artin [2] established that all non-zero proper ideals of R can be uniquely factored into prime ideals [3]. He further proved that every prime ideal of R divides some unique ideal 〈P〉 of R, where P is an irreducible polynomial in , and furthermore obtained the decomposition law

1.2

In the first case, explicitly: , , where B(x)²=D(x) mod  P(x); in the second case and in the third case, . Artin thus defined, for , and P irreducible, the ‘Legendre symbol’

1.3

One can extend it, multiplicatively, to non-irreducible polynomials, in analogy with the Jacobi symbol. Let , b(x)≠0, monic and b(x)=Q₁(x)^α₁…Q_r(x)^α_r be the unique factorization in , of b(x) into monic irreducible polynomials. Then define

1.4

Artin proved the law of quadratic reciprocity for , relatively prime, non-zero and monic

where .

For ease of notation, we define, for ,

1.5

for n≠0, and 0 for n=0. Artin defined the zeta function associated to R to be

1.6

where the sum is over non-zero ideals of R, and , the absolute norm of the ideal , denotes the number of residue classes . Artin also obtained the meromorphic continuation of ζ_R(s) to (see (1.8)–(1.12)) and its functional equation (see §1b).

Note, that the absolute norm is completely multiplicative, , for any ideals of R. Unique factorization of into prime ideals gives the Euler product

1.7

We can account for each ideal of R by considering the irreducible polynomial that sits below it. Now, in R, we have because there are choices for a(x) and b(x) modulo P(x) in (1.1). Thus, the decomposition of 〈P〉 into prime ideals given in (1.2) yields if or , and if .

We can use the Legendre symbol to correctly account for each local factor

1.8

Here, runs over all monic irreducible polynomials and . When χ_D(P)=1 this accounts for the two prime ideals that P sits below, each of norm . When χ_D(P)=0 there is just a single of norm . And when χ_D(P)=−1 the two factors involving P combine to give the correct norm of .

Artin proved that ζ_R(s) is a rational function of q^−s. We denote the first Euler product by . It can be expressed in closed form by unique factorization in

1.9

The second equality follows from unique factorization, the third equality gathers n's according to their degree (there are q^r monic polynomials of degree r in ), and the last equality is the sum of the stated geometric series.

Letting

1.10

one can collect together the terms n of given degree and get

1.11

Artin used quadratic reciprocity to show that

1.12

so that L(s,χ_D) is a polynomial in q^−s of degree ≤d−1, and in fact of degree d−1 by means of the functional equation, also proved by Artin, described below.

To prove (1.12), one can use the fact that the sum of (n|D) over a complete set of residue classes n modulo D is 0. Note that on applying quadratic reciprocity, each χ_D(n)=±(n|D). For fixed D and fixed , with n monic, each application of quadratic reciprocity has, by (1.5), the same ±1 factor. And, when r≥d, n runs over q^r−d copies of a complete set of residue classes modulo D, which can be seen by writing n=g(x)D(x)+h(x), with or h=0, and , g monic.

(b). Functional and ‘approximate’ functional equations

Artin also derived the functional equation for L(s,χ_D). It plays an important role in Andrade and Keating's heuristics leading to their moment conjecture, and also in allowing us to reduce the complexity of determining the zeta function associated to quadratic function fields.

In order to describe it, we let X_D(s)=|D|^1/2−sX(s), where

1.13

Then

1.14

The function X(s) plays the same role as the ratio of Gamma factors, , where , that appears in the functional equation of Dirichlet L-functions.

Note that in the case d is even, L(s,χ_D) has a trivial zero at s=0. If one defines the ‘completed’ L-function, L*(s,χ_D) by

1.15

then L* is a polynomial in q^−s of even degree

1.16

and satisfies the functional equation

1.17

Because L and L* are polynomials in u=q^−s, it is convenient to define

1.18

so that the above functional equation reads

1.19

Notice that this gives a relationship between the coefficients of (and hence of L*):

1.20

Comparing coefficients yields

1.21

thus

1.22

When d is odd, so that L=L*, then, returning to (1.10), we have

1.23

in analogy to the approximate functional equation of Dirichlet L-functions, though, here, the approximate functional equation is an identity with no correction terms. The advantage of the approximate functional equation is that it only involves terms with . This alone represents a large savings, since the number of monic polynomials n of degree r equals q^r, so that the total number of n involved is , rather than (q^2g+1−1)/(q−1) in (1.23), i.e. roughly |D|^1/2 terms compared with approximately |D| terms in (1.11).

The approximate functional equation in the case that d is even involves extra corrections terms. We define so that, when d is even, . Letting

1.24

we have

1.25

Hence, a(0)=b(0), a(1)=b(1)−b(0), a(2)=b(2)−b(0),…,a(2g)=b(2g)−b(2g−1), a(2g+1)=−b(2g). Summing, gives

1.26

The extra factor of (1−u) complicates, slightly, the approximate functional equation. Substituting (1.26) into (1.22), rearranging the resulting double sum and summing the geometric series, we obtain

1.27

Thus, multiplying by 1−u,

1.28

so that, for d even,

1.29

Hence in the d even case, the approximate functional equation has a remainder term, expressed in the second line above. Note that one can also express the remainder term using the coefficients and .

(c). Hyperelliptic curves according to Schmidt and Weil

Another point of view is obtained by considering the related hyperelliptic curve C:y²=D(x) over . One can define the zeta function associated to C as the function

1.30

where N_r(C) counts the number of points, including points at infinity, on the curve C over the field . When d is odd there is one point at infinity on the curve and when d is even there are two

1.31

We can express N_r(C) in terms of the Legendre symbol on : For , let

1.32

Then

1.33

since there are two solutions in to y²=D(x) when D(x) is a square (and non- zero), one solution if D(x)=0, and none otherwise.

For given D, we define a_q^r=a_q^r(D) to be

1.34

One can show that Z_C and ζ_R are related

1.35

so that

1.36

Weil [4] proved the Riemann Hypothesis for Z_C: that its zeros lie on the circle |u|=q^−1/2 (equivalently, that the zeros of L*(s,χ_D) lie on ). Thus, we may write

1.37

with |α_j|=q^1/2. Taking the logarithm of (1.30) and (1.36), using (1.37), and equating coefficients of their Maclaurin series gives

1.38

In more generality, Schmidt [5] obtained the rationality and functional equation of the zeta function associated to any non-singular curve over , and Weil established its Riemann hypothesis.

One can express the coefficients of L or L* in terms of the a_q^r's. Substituting (1.34) into (1.36), we get

1.39

On Taylor expanding the series on the r.h.s. above, and also using relationship (1.21), we get table 1 for the polynomials , for d≤7:

Table 1.

, for d≤7.

d
1	1
2	1−u
3	1−aqu+qu2
4	(1−u)(1−(aq−1)u+qu2)
5
6
7

(d). The hyperelliptic ensemble

We define to be the set of square-free monic polynomials of degree d in . The number of elements of is given by

1.40

This can be proven by considering the coefficient of q^−ds for .

We will also need the following formula for the number, i_n(q), of monic irreducible polynomials in of degree n≥1:

1.41

where μ is the traditional Möbius function. This can be obtained by grouping together, in (1.9), polynomials P according to their degree, so that . Taking the logarithmic derivative with respect to s, expanding the geometric series on both sides, and comparing coefficients of q^−ns, gives

1.42

Möbius inversion then yields (1.41).

2. Moments of zeta functions over the hyperelliptic ensemble

Let k be a positive integer. Katz & Sarnak [6,7] proved that

2.1

where 2g=d−1 or d−2 depending on whether d is odd or even, and dA is Haar measure on USp(2g) normalized so that , see eqn (40) and the discussion above eqn (41) in [7]. The statement of their result is given for a general class function on USp(2g), and their interest was in the statistics of zeros of zeta functions. However, one can take, in their eqn (40), for the class function, a power of the characteristic polynomial.

One can give precise formulae for the integral on the r.h.s. above. Keating & Snaith [8] used the Selberg integral to derive

2.2

This formula has the advantage of being expressed very concisely and explicitly.

Conrey et al. gave, in [9], another formula, as a k-fold contour integral

2.3

where the contours of integration enclose the origin,

2.4

is a Vandermonde determinant, and

2.5

While much more complicated than (2.2), this form is the one for which analogous formulae for the moments of have been developed, for number fields [9,10,11] and in the function field setting [12,13].

(a). Andrade–Keating conjectures

Andrade and Keating have given a conjecture for the asymptotic behaviour of the moments of , averaged over . While they restricted their discussion to the case that d is odd, it is straight-forward to adapt their analysis to include even d. For the reader's convenience, we repeat below the definition of X(s) given earlier in (1.13).

Conjecture 2.1 (Andrade–Keating) —

Let q be an odd prime power, and d a positive integer. Define

2.6

Andrade & Keating [12] conjectured the following asymptotic expansion. For q fixed, and ,

2.7

where Q_k(q;d) is the polynomial of degree k(k+1)/2 in d, with coefficients that depend on k and q, given by the k–fold residue

2.8

where

2.9

and is the Euler product, absolutely convergent for , defined by

2.10

Remarks —

(1) The above conjecture is the function field analogue of conjecture 1.5.3 in [9] for the moments of quadratic Dirichlet L-functions in the number field setting.

(2) If we substitute , then for d=2g+1 or d=2g+2,

2.11

where

2.12

Letting , we have that, Q_k(q;d) tends to as expressed on the r.h.s. of (2.3), consistent with the theorem of Katz and Sarnak.

(3) When d is odd, k=1, and q≡1 mod  4, Andrade & Keating [14] proved

2.13

where

2.14

This is consistent with the conjecture since, when d is odd,

2.15

The above formula is analogous to the formula obtained by Jutila [15] for the first moment, in the number field setting, of .

We would like to point out that Hoffstein & Rosen [16] have obtained formulae for the first moment, as , averaging over all , and also for square-free D(x), not necessarily monic. In the latter case, they did not explicitly determine a certain coefficient in their formula. In principle, their method should produce a sharper remainder term than (2.13). The second to fourth moments, as , again averaged over all , have been considered, by Chinta & Gunnels [17,18] and Bucur & Diaconu [19]. Square-free averages seem harder to get a handle on, and, for the purpose of testing Andrade and Keating's conjecture we require square-free averages.

3. Numerical data

We first present numerical evidence in support of the Andrade–Keating conjecture. We have numerically computed the moments M_k(q,d), and compared them with Andrade and Keating's Q_k(q,d) for k≤10, d≤18, and for odd prime powers q specified below:

In addition to these values, we also computed moments for a few large values of q, when d=3, such as q=10009. Later, we discovered formulae for the moments when d=3, and d=4, so that one can directly evaluate the moments in those cases quite easily using theorems 5.1 and 6.1. Our data will be made available on lmfdb.org [20].

We display a selection of data, in tables 2–20 for the pairs q,d: 10009,3; 729,3; 491,4; 343,4; 81,5; 73,5; 49,6; 23,7; 17,8; 9,9; 9,10; 5,11; 5,12; 5,13; 3,14; 3,15; 3,16; 3,17; 3,18.

Table 3.

versus Q_k(729,3).

k	Mk(729,3)	Qk(729,3)	difference	ratio
1	2	2.000000005141182844814	−5.14118×10−09	0.9999999974294085842009
2	4.999998118323576841079	4.999998118342878449719	−1.93016×10−11	0.9999999999961396768193
3	13.99998870994146104648	13.99998871043573721017	−4.94276×10−10	0.9999999999646945312655
4	41.99994919215539991743	41.99994919090068601026	1.25471×10−09	1.000000000029874176787
5	131.9997929897817046016	131.9997929893691064232	4.12598×10−10	1.000000000003125748678
6	428.9991925930345626555	428.9991925915335626766	1.50100×10−09	1.000000000003498841035
7	1429.996916910558118702	1429.996916903926491314	6.63163×10−09	1.000000000004637511668
8	4861.988351797874997796	4861.98835177088281103	2.69922×10−08	1.00000000000555167656
9	16795.95621972101045984	16795.95621961290008345	1.08110×10−07	1.000000000006436690771
10	58785.81724292694956271	58785.83581101586665318	−0.0185681	0.9999996841400881534971

Table 4.

versus Q_k(491,4).

k	Mk(491,4)	Qk(491,4)	difference	ratio
1	1.952833793133705729622	1.95283379342706162633	−2.93356×10−10	0.9999999998497793833277
2	4.72347699885310851273	4.723476998737103048879	1.16005×10−10	1.00000000002455933709
3	12.73886907319525470025	12.7388690698370324848	3.35822×10−09	1.000000000263620121774
4	36.7169899417769629311	36.71698994054792792551	1.22904×10−09	1.00000000003347319613
5	110.6950691954947392148	110.6950691966720004329	−1.17726×10−09	0.9999999999893648269375
6	344.7459728846995577523	344.7459728837781730119	9.21385×10−10	1.000000000002672648306
7	1100.405995216241690213	1100.405995213205079234	3.03661×10−09	1.000000000002759536928
8	3580.803938022174127785	3580.80393800301250494	1.91616×10−08	1.000000000005351206929
9	11834.53044485529539674	11834.52941628875665659	0.00102857	1.000000086912331074563
10	39615.88015863915407142	39615.83875603152753383	0.0414026	1.000001045102386485103

Table 5.

versus Q_k(343,4).

k	Mk(343,4)	Qk(343,4)	difference	ratio
1	1.943089189220997181719	1.943089190188867075	−9.67870×10−10	0.9999999995018911647659
2	4.667904524799604996632	4.667904524297283786135	5.02321×10−10	1.000000000107611714825
3	12.49238185342463481838	12.49238184309220107738	1.03324×10−08	1.000000000827098776741
4	35.71296913719100356228	35.71296913156088528584	5.63012×10−09	1.000000000157649123367
5	106.7587272293154491271	106.758727235013386794	−5.69794×10−09	0.9999999999466278981169
6	329.6145322715815812529	329.6145322661610210221	5.42056×10−09	1.00000000001644514941
7	1042.878983141951628492	1042.878983130103334014	1.18483×10−08	1.000000000011361140333
8	3363.515181241613078536	3363.515181166675890101	7.49372×10−08	1.000000000022279426255
9	11017.02775430122683174	11017.02965791539142027	−0.00190361	0.9999998272116692341986
10	36547.55945032561986556	36547.62883036162722131	−0.06938	0.9999981016542460418421

Table 6.

versus Q_k(81,5).

k	Mk(81,5)	Qk(81,5)	difference	ratio
1	2.987806713547357068149	2.987806692825562058199	2.07218×10−08	1.00000000693545370914
2	13.86573074840367797091	13.86573073409551151745	1.43082×10−08	1.000000001031908575743
3	82.64367981408428192661	82.64368117790658728224	−1.36382×10−06	0.9999999834975610244207
4	580.146307177966733273	580.1463667277005413773	−5.95497×10−05	0.9999998973539485492377
5	4573.824668082202791908	4573.826022502549800431	−0.00135442	0.9999997038758491588937
6	39335.1550736043829847	39335.17940837786345422	−0.0243348	0.9999993813483541583506
7	361979.8712634998365703	361980.2776302882857858	−0.406367	0.9999988773786486117298
8	3516936.691114034122135	3516935.189217477924701	1.5019	1.000000427046981360952
9	35726613.38116429736676	35726128.68407336596104	484.697	1.000013567019679562501
10	376702516.8245619561432	376679451.0864266274913	23065.7	1.000061234394572897393

Table 7.

versus Q_k(73,5).

k	Mk(73,5)	Qk(73,5)	difference	ratio
1	2.986488987117195040355	2.986488956110408111587	3.10068×10−08	1.000000010382354458511
2	13.85120391739408683316	13.85120389332632073605	2.40678×10−08	1.000000001737593806464
3	82.4967741946123427344	82.49677598164466815201	−1.78703×10−06	0.9999999783381555927079
4	578.6447454493516547044	578.6448254181129270916	−7.99688×10−05	0.9999998617999198133259
5	4558.084908449866951901	4558.086742384675503866	−0.00183393	0.9999995976524993483348
6	39165.71395519698225425	39165.74675342505698226	−0.0327982	0.9999991625787634992611
7	360109.386585466970311	360109.9246416242541557	−0.538056	0.9999985058557943958092
8	3495803.870606360195483	3495808.763850148092748	−4.89324	0.9999986002541562061777
9	35482616.7531615019917	35482487.21304819659354	129.54	1.000003650818290375134
10	373825112.8977981121039	373816499.0489997443828	8613.85	1.000023042987188317283

Table 8.

versus Q_k(49,6).

k	Mk(49,6)	Qk(49,6)	difference	ratio
1	2.816676047960577246894	2.816676013338886305786	3.46217×10−08	1.000000012291683806426
2	11.94445177181344907967	11.94445177333470101717	−1.52125×10−09	0.9999999998726394508203
3	63.85807086800793596929	63.85808011755308483439	−9.24955×10−06	0.9999998551546627797441
4	397.3481793964877073688	397.3484554832939249546	−0.000276087	0.9999993051770998284564
5	2754.623288588277155958	2754.628161624466408897	−0.00487304	0.9999982309640708896263
6	20714.1727032707348348	20714.24331890170583675	−0.0706156	0.9999965909625621436374
7	165996.9411444855213461	165997.8721434242824435	−0.930999	0.9999943915007660055879
8	1400184.057794141070937	1400195.334950112458361	−11.2772	0.9999919460123242095944
9	12319825.39035079187353	12319948.18218488950777	−122.792	0.9999900330884284727859
10	112306968.1406010439838	112308155.0209042696054	−1186.88	0.9999894319312519673762

Table 9.

versus Q_k(23,7).

k	Mk(23,7)	Qk(23,7)	difference	ratio
1	3.916667261680037602233	3.916667215072046984931	4.66080×10−08	1.000000011899910831828
2	28.36895318290689557286	28.36895361224982933044	−4.29343×10−07	0.9999999848657465613331
3	296.8271210147472574343	296.8271319379614607806	−1.09232×10−05	0.9999999632000817040224
4	3978.400255691440915331	3978.400675986332260201	−0.000420295	0.9999998943558164259889
5	63802.68692372865989707	63802.67647434303583351	0.0104494	1.000000163776603137732
6	1173290.508072928492608	1173288.388689279659737	2.11938	1.000001806362075397768
7	24046416.78084689795807	24046272.80809433006013	143.973	1.000005987320933971713
8	538361067.7094472855076	538352287.1146935750589	8780.59	1.000016310128077601687
9	12974750743.4272898467	12974141403.6447755601	609340	1.00004696571153009841
10	332891976281.3758666031	332847688903.5173317907	4.42874×107	1.000133055987272822582

Table 10.

versus Q_k(17,8).

k	Mk(17,8)	Qk(17,8)	difference	ratio
1	3.586540611827683173548	3.586540636892566051014	−2.50649×10−08	0.9999999930114041871885
2	22.548947403531964213	22.54894776512864642973	−3.61597×10−07	0.9999999839639221313939
3	197.6802683820100941163	197.6802898650704364376	−2.14831×10−05	0.9999998913242167087836
4	2166.015292026007802413	2166.014628189217440864	0.000663837	1.000000306478442814818
5	27892.99630055103627191	27892.89108878033170407	0.105212	1.000003771992310502669
6	406297.4340536546537236	406291.5092110336903502	5.92484	1.000014582737976652927
7	6525359.938112686293172	6525112.516320263663534	247.422	1.000037918394786877841
8	113486818.2305410890984	113477804.8603318981476	9013.37	1.000079428485775562689
9	2109141498.958278796091	2108834825.379838974051	306674	1.000145423233128078027
10	41466902858.6631825799	41456762858.08206808141	1.01400×107	1.000244592193940142331

Table 11.

versus Q_k(9,9).

k	Mk(9,9)	Qk(9,9)	difference	ratio
1	4.699049316413412507979	4.699049891407480099095	−5.74994×10−07	0.9999998776361007269725
2	46.24725707056031614576	46.24726745110153152897	−1.03805×10−05	0.9999997755426041904241
3	706.9332948602088630742	706.9332532286168577971	4.16316×10−05	1.000000058890414073949
4	14388.19341678737191699	14388.17906849176482149	0.0143483	1.000000997228039684076
5	356658.7872479684052459	356657.018621382592894	1.76863	1.000004958900269644989
6	10183031.33432607207208	10182911.73737028773408	119.597	1.00001174486815450132
7	322685130.7849396712488	322680691.2234089091978	4439.56	1.000013758373684926526
8	11060883575.07667143044	11060965709.40727992865	−82134.3	0.9999925743978630506471
9	402640355635.9474171249	402672245068.807641106	−3.18894×107	0.9999208054857250596119
10	15357415165127.97732483	15360969485690.78586929	−3.55432×109	0.9997686135262413285134

Table 12.

versus Q_k(9,10).

k	Mk(9,10)	Qk(9,10)	difference	ratio
1	4.249549776125279466818	4.249550011750719262062	−2.35625×10−07	0.9999999445528493267051
2	35.47122535458782164262	35.47122542617537260736	−7.15876×10−08	0.9999999979818134246947
3	442.286953846524408696	442.2870463596704975278	−9.25131×10−05	0.9999997908300800344951
4	7174.125718182284449517	7174.134434494580275984	−0.00871631	0.9999987850363865615889
5	139775.8683089006307473	139776.6034509980038277	−0.735142	0.9999947405926369444087
6	3110983.61263697065039	3111036.983659021477957	−53.371	0.9999828446198707494429
7	76480294.89533696000008	76483336.31197182891681	−3041.42	0.9999602342577935783894
8	2028259368.757841547712	2028400599.367150037059	−141231	0.9999303734137366393356
9	57039223496.5499399637	57044641791.88289192065	−5.41830×106	0.999905016577144622413
10	1679490328130.420640044	1679652778279.477297485	−1.62450×108	0.9999032834933758985671

Table 13.

versus Q_k(5,11).

k	Mk(5,11)	Qk(5,11)	difference	ratio
1	5.32482940928	5.324828316051856638322	1.09323×10−06	1.000000205307679135139
2	64.88099399827456	64.88091655935417199203	7.74389×10−05	1.000001193554661287276
3	1274.6768000899874816	1274.6704998246032173	0.00630027	1.000004942661954702197
4	33521.58695492143399567	33521.27305651990807685	0.313898	1.000009364155144008333
5	1062440.450217281671513	1062426.889916814194921	13.5603	1.000012763513984984261
6	38147507.21495457787241	38147338.69609051874127	168.519	1.000004417578521051629
7	1494075723.893608277159	1494132326.153963719466	−56602.3	0.9999621169695851894083
8	62322834399.64654047306	62331619932.572067288	−8.78553×106	0.9998590517471705265441
9	2726087327379.298589965	2726989575639.266669371	−9.02248×108	0.9996691412875105793323
10	123744101491973.6125044	123822245466828.0429362	−7.81440×1010	0.9993689019726639896186

Table 14.

versus Q_k(5,12).

k	Mk(5,12)	Qk(5,12)	difference	ratio
1	4.654401394029119045648	4.654400599565288165843	7.94464×10−07	1.00000017069090076905
2	45.47033591867196354032	45.4703054260267446037	3.04926×10−05	1.000000670605682834999
3	681.930213578023967161	681.9301580154612560079	5.55626×10−05	1.000000081478377306043
4	13331.77182957562186018	13331.78726162052641967	−0.015432	0.999998842462409448644
5	309607.9020328393226788	309608.1472707221788929	−0.245238	0.9999992079088195254194
6	8077636.649190943197238	8077624.699048311052369	11.9501	1.000001479412955835001
7	228659527.1493859795208	228658788.966048358131	738.183	1.000003228318233291249
8	6867842716.914419001117	6867841634.620738593006	1082.29	1.00000015758861924717
9	215668267720.2325918011	215671097846.3146046479	−2.83013×106	0.9999868775829943167874
10	7010909280434.801886765	7011206914849.719740156	−2.97634×108	0.9999575487617848698201

Table 15.

versus Q_k(3,13).

k	Mk(3,13)	Qk(3,13)	difference	ratio
1	5.710384491306550387427	5.710336021545693923735	4.84698×10−05	1.000008488075075368984
2	79.01975914340451932061	79.01896720095370412587	0.000791942	1.000010022181747847959
3	1770.144898438187087668	1770.108824445967349489	0.036074	1.000020379533575303827
4	51913.19970116326269693	51911.40410226095204163	1.7956	1.000034589680887334151
5	1785178.554900046977396	1785085.94328058320004	92.6116	1.000051880762274996239
6	67873237.3947317133838	67870093.08716805240916	3144.31	1.000046328322544402952
7	2760851654.820987619395	2760898873.542778898848	−47218.7	0.9999828973374418859096
8	117829045375.9911859183	117848552675.9647081734	−1.95073×107	0.9998344714505984698264
9	5212177572584.563015279	5214335433244.846855522	−2.15786×109	0.9995861676549371857998
10	237048460599876.5060545	237230552226057.5905753	−1.82092×1011	0.999232427592178055752

Table 16.

versus Q_k(3,14).

k	Mk(3,14)	Qk(3,14)	difference	ratio
1	4.707406146004197020658	4.707399252588057470547	6.89342×10−06	1.00000146437890003917
2	47.62537772288575735518	47.62540907500632349621	−3.13521×10−05	0.9999993416934116667874
3	734.5698773629301869476	734.5805919276818747064	−0.0107146	0.9999854140377932247518
4	14428.2643076236746704	14428.74431535543910537	−0.480008	0.9999667325360216135754
5	327860.1672995015230248	327878.7206791626643333	−18.5534	0.9999434138951661451567
6	8176125.594815910182649	8176771.02985183297252	−645.435	0.9999210648025282315549
7	217115876.0852813531656	217133701.6376359054959	−17825.6	0.9999179051790665801582
8	6029316864.523584287498	6029554103.41727163357	−237239	0.9999606539903916128607
9	173111253375.948678331	173097704368.0710223596	1.35490×107	1.00007827375832117143
10	5100152365967.425716091	5098632913159.453941573	1.51945×109	1.00029801180705716269

Table 17.

versus Q_k(3,15).

k	Mk(3,15)	Qk(3,15)	difference	ratio
1	6.444523381931924617441	6.444536693201652192808	−1.33113×10−05	0.9999979344877124188869
2	109.7499547245450694558	109.7507187605598090365	−0.000764036	0.9999930384418127916091
3	3183.809844081673755951	3183.853347913461213922	−0.0435038	0.9999863361069014161053
4	124342.7729226484856941	124346.6094296010463177	−3.83651	0.9999691466701813637648
5	5787791.045784771300337	5788224.648068712848916	−433.602	0.9999250888985301868983
6	301059018.8940758921497	301101235.2359253549406	−42216.3	0.9998597935281919898824
7	16884124578.35074000199	16887585330.58731680708	−3.46075×106	0.9997950712213244256056
8	999516139114.1778849258	999765221174.3504308231	−2.49082×108	0.9997508594469009758296
9	61630814297036.52818885	61647035026636.81900097	−1.62207×1010	0.9997368773762877312361
10	3923376265666177.708666	3924344564045026.44537	−9.68298×1011	0.9997532585727256678234

Table 18.

versus Q_k(3,16).

k	Mk(3,16)	Qk(3,16)	difference	ratio
1	5.441593663908911049183	5.44159992424401573962	−6.26034×10−06	0.9999988495414598933115
2	70.04046859007929975499	70.04057073846899985698	−0.000102148	0.9999985415825624619284
3	1448.922020668379138097	1448.930247315431886658	−0.00822665	0.999994322261497411048
4	39229.51451253235302518	39230.00535754890487004	−0.490845	0.9999874880206597424508
5	1247448.818507931641297	1247476.280762623550041	−27.4623	0.9999779857500175312687
6	43941730.00487174101086	43943354.00708841730286	−1624	0.9999630432803009454547
7	1658947112.231571005672	1659057502.345166009773	−110390	0.9999334621533979613097
8	65816178711.03131525193	65824479170.36143594316	−8.30046×106	0.9998739001138370098827
9	2710058461030.138083664	2710694875002.238384995	−6.36414×108	0.9997652210959008155407
10	114863654355609.5971023	114911144985484.4439671	−4.74906×1010	0.9995867186783244926063

Table 19.

vs Q_k(3,17).

k	Mk(3,17)	Qk(3,17)	difference	ratio
1	7.178737839030501043015	7.17873736485761046188	4.74173×10−07	1.000000066052408171718
2	147.3726497579550442855	147.3725161321440454976	0.000133626	1.000000906721378625103
3	5404.506101895536199984	5404.49242409700269409	0.0136778	1.000002530820188131028
4	274060.1660103541922629	274058.9817947832475316	1.18422	1.000004321024486004542
5	16832953.27879710470395	16832847.16481320120331	106.114	1.000006303983091186001
6	1167928626.57377059563	1167920813.9644438571	7812.61	1.000006689331359905344
7	88062690804.08967547081	88062582866.71670697102	107937	1.000001225689384240111
8	7052055863098.318652111	7052168134243.813217808	−1.12271×108	0.9999840799108362999548
9	591144818225498.1663163	591174439968232.2018155	−2.96217×1010	0.9999498933973944690917
10	51372433793444437.15117	51377776383644053.11002	−5.34259×1012	0.9998960135962342512948

Table 2.

versus Q_k(10009,3).

k	Mk(10009,3)	Qk(10009,3)	difference	ratio
1	2	2.00000000000199401202	−1.99401×10−12	0.9999999999990029939901
2	4.999999990017975729127	4.999999990017976230662	−5.01535×10−16	0.9999999999999998996931
3	13.99999994010785437476	13.9999999401078685067	−1.41319×10−14	0.9999999999999989905756
4	41.99999973048434738158	41.99999973048431072166	3.66599×10−14	1.000000000000000872855
5	131.9999989019673571555	131.9999989019673481792	8.97627×10−15	1.000000000000000068002
6	428.9999957176467628805	428.99999571764672006	4.28205×10−14	1.000000000000000099815
7	1429.999983649095186217	1429.999983649095000872	1.85345×10−13	1.000000000000000129612
8	4861.999938229538381732	4861.999938229537621148	7.60584×10−13	1.000000000000000156434
9	16795.99976785031236985	16795.99976785030932926	3.04059×10−12	1.000000000000000181031
10	58785.99694768653745618	58785.99912943382729291	−0.00218175	0.9999999628866171852757

Table 20.

versus Q_k(3,18).

k	Mk(3,18)	Qk(3,18)	difference	ratio
1	6.175801337371783637064	6.175800595899974008692	7.41472×10−07	1.000000120060840390576
2	98.4198929258615830515	98.41984756709154860716	4.53588×10−05	1.000000460870151252001
3	2648.54819782900739719	2648.548299692500437867	−0.000101863	0.9999999615398771272156
4	95776.99330883472578033	95777.07102537293038863	−0.0777165	0.9999991885684394752487
5	4129734.976650697670257	4129735.205366196747353	−0.228715	0.9999999446173936818269
6	199191998.5992826340305	199190826.0798038029441	1172.52	1.000005886413053788573
7	10369942932.1902759808	10369724943.99307832651	217988	1.000021021598776728345
8	570422300453.0205939942	570394265112.5694039534	2.80353×107	1.000049150810528673817
9	32711546699641.20745935	32708464677244.22990219	3.08202×109	1.00009422705796159752
10	1938245416991953.993278	1937933951306313.043464	3.11466×1011	1.00016072048556195558

For k≤10, and the above pairs of q,d, we list the difference and ratio between the actual moments M_k(q,d), and the Andrade–Keating value Q_k(q,d). The conjectured value Q_k(q,d) nicely fits the actual data M_k(q,d), spectacularly well in some cases.

The sheer number, q^d−q^d−1, of polynomials makes it prohibitive to compute the moments M_k(q,d) for d large, at least if we do so one D at a time. One can slightly reduce the amount of computation for the moments by taking advantage of the fact that many D have the same zeta functions, see §4. The largest value of d for which we determined moments was d=18 and q=3.

Our data support Andrade and Keating's conjecture in the sense that, for given q (size of field), and k, the ratio between the actual moment M_k(q,d) and their prediction Q_k(q,d) does appear to tend to 1 as d grows.

It seems quite difficult to determine, theoretically, the rate at which it approaches 1 as . However, while Andrade and Keating made their prediction for given k and q, and , we have had some success in determining the size of the remainder term for given k and d, letting q grow. We describe our findings below.

A natural quantity with which to measure the remainder term in the Andrade–Keating prediction is

3.1

It is roughly the number of terms, , being summed in the moment M_k(q,d).

For any given value of d and k, our data suggest that, as (i.e. as since, now, d is fixed), there is a constant μ(=μ(k,d)), depending on d and k, such that

3.2

with the implied constants in the Θ depending on k and d. As remarked earlier, Q_k(q,d) converges, as , to (2.3). Thus, for given k and d, Q_k(q,d) is bounded as ; hence, the above can be written as

3.3

In §§5–8, we are able to determine (conditionally, for d>4) the values of μ displayed in table 21, for a selection of d≤9 and k=1,2,3.

Table 21.

Values of μ, giving the size of the remainder term Θ(X^−μ), in the Andrade–Keating conjecture, for k=1,2,3 and the first few values of d.

d	k	μ	k	μ	k	μ
1	1	1	2	1	3	1
2	1	1	2		3	1
3	1	1	2		3
4	1		2		3
5	1		2	1	3
6	1		2		3	not determined
7	1		2		3
8	1
9	1

Interestingly, when d=3, the k=2,3 predictions fit better ( in both cases) than the k=1 prediction (μ=1), with a similar feature for d=5, and k=2 (μ=1) in comparison with k=1 ().

The d=6 entry for k=3 is missing because we did not have enough data to determine it. The formulae for even values of d seem to involve powers of 1/q^1/2, as compared with 1/q for odd values of d, and hence more terms.

One might ask about the behaviour of μ if we fix k and allow d to grow. For example, if we fix k=1 and let d grow, is it true that . This would be in analogy with the conjectured remainder term in the first moment (k=1) of quadratic Dirichlet L-functions [10]. Is there a term of size X^−1/4 that eventually (for d sufficiently large) enters when k=3, as predicted in the number field setting by Diaconu and colleagues [21,10]?

If we fix d and allow k to grow, it appears that μ is not as impressive. For example, we show in §5, for d=3 and any k≥10, that (we restrict in that section to q prime). In §6 we prove, for d=4 and any k≥9, that (again with q restricted to being prime).

4. Isomorphic hyperelliptic curves

We took advantage, in tabulating zeta functions, and also in deriving the formulae described below in §§5 and 6, of the fact that the same zeta functions in arise repeatedly.

For , let us denote its coefficients as c_n=c_n(D):

4.1

If is non-zero, i.e. if p, the characteristic of F_q does not divide d, then, on binomial expanding and rearranging the resulting double sum

4.2

We can choose u=−d⁻¹c_d−1 so as to make the coefficient of x^d−1 equal to zero. Furthermore, D(x) is square-free if and only if D(x+u) is square-free.

Thus, for , let denote the set

4.3

Thus, in the case that , the set can be partitioned into q subsets of equal size, each one obtained from by a change of variable x→x−u, .

For example, in the case that d=3 and is not of characteristic 3, each is expressed as x³+Ax+B, with . When d=3, the square-free condition is equivalent D(x) not having a repeated root in .

If we let , and , then their associated zeta functions are equal, because both have the same point counts over any as we may pair up points on y²=D(x) with points (x+u,y) on y²=D₂(x).

Therefore, for , we can write

4.4

There are yet additional isomorphisms, though we did not exploit these in our work. Given (or ), consider, for , the polynomial (resp. ). If a^d is a square (and non-zero) in (if d is even, or if a is itself a square), then the hyperelliptic curves y²=D(x) and y²=a^dD(a⁻¹x)=x^d+ac_d−1x^d−1+a²c_d−2x^d−2+⋯a^d have the same number of solutions over any . This can be seen by pairing up (x,y) on the first curve with , where denotes either square root of a^d in , on the second curve.

5. Moment formulae when d=3

In this section we assume that d=3, and the characteristic of F_q is not 3, so that each is of the form D(x)=x³+Ax+B, we have that

5.1

where

5.2

Thus,

5.3

Now the odd moments of a_q are all equal to 0

5.4

That is because may can pair up each D(x) that produces a given value of a_q=a_q(D(x)), with another curve that produces . This can be achieved as follows. Let a be a non-square in . Let . Then

5.5

the last equality holding because , and because a⁻¹x runs over all of as x does.

Birch [1] used the Selberg trace formula to determine the even moments of a_q(D(x)) for the set of all D(x)=x³+Ax+B, with , i.e. without the square-free condition. He restricted to q=p, i.e. prime fields, with p>3. Thus, for the remainder of this section, we restrict to q=p>3, as well.

For j even, Birch defines

5.6

and obtains a formula for S_j/2(p):

5.7

where tr_2l(T_n) is the trace of the Hecke operator T_n acting on the space of cusp forms of weight 2l for the full modular group, i.e. acting on :

5.8

where f runs over the eigenfunctions of the all the Hecke operators, and where λ_f(n) are their Fourier coefficients, normalized so that λ(1)=1.

The term tr_2l+2(T_p) first contributes to S_j/2(p) when j=10, because for 2l+2=2,4,6,8,10, whereas tr₁₂(T_p)=τ(p), the Ramanujan τ function.

Thus S₁(p),…,S₄(p) are polynomials in p, but the higher moments S₅(p),S₆(p),… can be expressed as polynomials in p and the coefficients of Hecke eigenforms.

We note that there is a typo in the example formulae of Birch's Theorem 2. His stated formulae for S₁(p),…,S₅(p) are all missing the factor of p−1, and should read: S₁(p)=(p−1)p², S₂(p)=(p−1)(2p³−3p), S₃(p)=(p−1)(5p⁴−9p²−5p), S₄(p)=(p−1)(14p⁵−28p³−20p²−7p), S₅(p)=(p−1)(42p⁶−90p⁴−75p³−35p²−9p−τ(p)),….

Now, Birch sums over all A,B∈F_p, whereas we are summing over square-free . If x³+Ax+B is not square-free, we can write it as

5.9

for some . Comparing coefficients of x² gives t=−2s mod  p, hence x³+Ax+B=(x+s)²(x−2s), so that

5.10

For given , (x+s|p)²=1, unless x=−s, in which case (x+s|p)²=0. Thus,

5.11

the latter equality because the full sum of (x−2s|p) over all x mod  p is 0. Thus, when j is even, a_p((x+s)²(x−2s))^j=1, when s≠0 mod  p, and equals 0 if s=0 mod   p.

Therefore, we have shown that

5.12

Combining the above with (5.7) and (5.3) gives

5.13

Simplifying, and using

5.14

(this identity is derived in greater generality below) we have

5.15

Rearranging the sum over j and l, the right side above equals

5.16

Now, the inner sum over j equals

5.17

where ₂F₁(a,b;c;z) is the Gauss hypergeometric function

5.18

One easily checks this by comparing, with a=l−k/2, , c=2l+2, each term in the above sum with the terms in the sum over j in (5.16). Note that, with this choice of a and b, the terms in the above series vanish if 2n>k−2l, and the hypergeometric series terminates.

Using Gauss' identity

5.19

we thus have, on simplifying,

5.20

Applying the Legendre duplication formula, we can simplify both the numerator (with z=k+1) and denominator (with ) to get

5.21

Returning to (5.15), we thus have the following theorem:

Theorem 5.1 —

Let p>3 be prime. Then

5.22

The fact that our final formula for the moments (in the d=3 case) can be expressed so cleanly and succinctly suggests that an alternate point of view should exist that produces the same formula more directly. Indeed, A Diaconu & V Pasol (2014, personal communication) have derived an equivalent formula using multiple Dirichlet series over finite fields, though perhaps a simpler approach can be found.

We list the first 10 moments in table 22.

Table 22.

Moment formulae for d=3, k≤10.

k
1	2
2	5−p−2
3	14−6 p−2
4	42−27 p−2−p−3
5	132−110p−2−10p−3
6	429−429p−2−65p−3−p−4
7	1430−1638p−2−350p−3−7p−4
8	4862−6188p−2−1700p−3−119p−4−p−5
9	16796−23256p−2−7752p−3−798p−4−18p−5
10	58786−87210p−2−33915p−3−4655p−4−189p−5−(τ(p)+1)p−6

It appears, from our numerical data, that (5.22) also holds for , if k≤9, i.e. if we replace p with any odd prime power q, whether divisible by 3 or not. For k≥10, one would need to adjust the terms tr_2l+2(T_p). For example, for k=10, and q=p², it appears from our tables that one should replace τ(p) by 2τ(p²)−τ(p)². We do not attempt to address the general formula here since the above suffices for the purpose of testing the Andrade–Keating conjecture, which does not see the arithmetic terms tr_2l+2(T_p).

Note, for instance, that the Fourier coefficients λ(p) of a weight 2l+2 modular form satisfies the Ramanujan bound

5.23

Thus, for given k≥10, the terms tr_2l+2(T_p) contribute, overall, an amount to (5.22) that is O(p^−1/2). Furthermore, it is known that

5.24

Therefore, in the case k≥10, d=3 and q prime, we have , since, here, X=p³, and X^−1/6=p^−1/2.

6. Moment formulae when d=4

Birch's formula can be applied to the case of d=4 as well, because there is a relationship between elliptic curves of degrees 3 and 4.

According to the table in 1c, the zeta function associated to y²=D(x) over , for , equals (1−u)(1−(a_q−1)u+qu²). Here a_q(D(x)) is defined by (1.34). Substituting u=q^−1/2, binomial expanding, and rearranging the resulting double sum, we have

6.1

where

6.2

The connection with the moments for d=3 is through the following relationship. Let

6.3

We prove, in theorem 6.2, that, for q an odd prime power, not divisible by 3, and for j≥0

6.4

Now, equation (5.12) gives, for prime q=p>3,

6.5

The extra factor of p compared with (5.12) is to account for the fact that here our sum is over rather than .

Thus, breaking the sum on the right side of (6.1) into even and odd terms j, we have, for q=p>3,

6.6

The first sum is precisely the sum that appears in (5.3), and (5.22) gives

6.7

Furthermore, substituting j=ν−1, the second sum equals

6.8

Using (6.5), as well as lemma 6.3 (below), and simplifying, the second sum becomes

6.9

Putting together (6.9) and (6.7), we arrive at the following theorem:

Theorem 6.1 —

Let p>3 be prime. Then,

6.10

The above formula seems to hold (based on our tables), for k≤8, if we replace p by any odd prime power q. The Hecke eigenvalues enter starting with k=9.

Therefore, in the case k≥9, d=4 and q prime, we have , since, here, X=p⁴ and X^−1/8=p^−1/2. Expanding this formula, for k=1,2,3,4,5, and collecting powers of p, gives table 23.

Table 23.

Moment formulae for d=4, k≤5.

k
1	2−p−1/2−p−1−p−5/2+p−3
2	5−6p−1/2−3p−1+4p−3/2−p−2−2p−5/2+7p−3−4p−7/2
3	14−28p−1/2+28p−3/2−20p−2+3p−5/2+27p−3−40p−7/2+18p−4−3p−9/2+p−5
4	42−120p−1/2+60p−1+120p−3/2−177p−2+100p−5/2+61p−3−232p−7/2
	+223p−4−100p−9/2+31p−5−8p−11/2
5	132−495p−1/2+495p−1+330p−3/2−1100p−2+1034p−5/2−230p−3−985p−7/2
	+1665p−4−1286p−9/2+614p−5−225p−11/2+55p−6−5p−13/2+p−7

Theorem 6.2 —

With m₃(q;j) and m₄(q;j) defined by (6.3) and (6.2), the relationship (6.4) holds for any odd prime power q not divisible by 3, and any j≥0.

Proof. —

While the relationship in (6.4) involves sums over and , we establish the same relationship over the simpler , . One can then recover the original sums (6.3) and (6.2) by scaling both by a factor of q.

Thus, let . To the hyperelliptic curve specified by a quadratic equation of the form E₄:y²=x⁴+Ax²+Bx+C, we can associate a cubic equation E₃:Y ²=X³+αX+β, where the two equations are related by the rational change of variables,

6.11

so that on substituting and simplifying

6.12

These can be verified by hand or, more easily, with the aid of a symbolic math package such as Maple. See Mordell [22, p. 77], where this change of variables is described, though with a slightly different normalization. We will use this association to establish the relationship specified in the statement of this lemma.

Note that, since we are in characteristic greater than 3, all coefficients appearing in the above two displays (e.g. ) are defined in . Also, the change of variable (6.11) can be inverted:

6.13

The points , satisfying y²=x⁴+Ax²+Bx+c are in one-to-one correspondence with the points satisfying Y ²=X³+αX+β, with the exception of one point.

This exception arises from the denominator, X+A/6, in (6.11). If X=−A/6, then substituting into Y ²=X³+αX+β, with α,β given by (6.12), gives Y ²=B²/64, i.e. Y =±B/8. Thus, when B≠0, there are two points on Y ²=X³+αX+β with X=−A/6, namely (−A/6,±B/8). When B=0 there is just one point, (−A/6,0).

In the former case, the point (X,Y)=(−A/6,−B/8) does not have a corresponding point , but the point (−A/6,B/8) does, namely (x,y)=((A²−4C)/(4B),(16C²+8AB²−8A²C−A⁴)/(16B²)), obtained by substituting X=−A/6 into y=−x²+2X−A/6, then substituting for y into y²=x⁴+Ax²+Bx+C to get x, and finally back-substituting into y=−x²+2X−A/6.

In the latter case, i.e. B=0, there is no point corresponding to (X,Y)=(−A/6,0). For, if there was, we would have, on substituting y=−x²+2X−A/6=−x²−A/2 into y²=x⁴+Ax²+C, that A²/4=C, so that y²=x⁴+Ax+A²/4=(x²+A/2)², violating the assumption that x⁴+Ax+Bx+C is square-free.

Thus, we have shown that −a_q(X³+αX+β)=1−a_q(x⁴+Ax²+Bx+C) (in terms of the point counting function, recalling (1.34), this gives N₁(E₄)=N₁(E₃), though, below, we work just with a_q). This allows us to relate m₄(q;j) as expressed in (6.2) with m₃(q;j) as expressed in (6.3).

By carefully examining our tables of zeta functions, we also determined that it is important to pair curves according to their value of ±a_q(X³+αX+β). Thus, fix a to be any non-square in . Given , we define its quadratic twist (depending on a), to be X³+a²αX+a³β. As explained in §5, we have a_q(X³+a²αX+a³β)=−a_q(X³+αX+β).

Now, we can count the number of curves y²=x⁴+Ax²+Bx+C that are associated to a given y²=X³+αX+β as follows. For any choice of , there is exactly one choice of such that −C/4−A²/48=α.

For given A and C, there are either 0, 1 or 2 choices of such that β=A³/864+B²/64−AC/24, i.e. such that (B/8)²=β−A³/864+AC/24. More precisely, the number of such B is given by

6.14

Thus, the total number of of curves y²=x⁴+Ax²+Bx+C that are associated under the above change of variable to a given Y ²=X³+αX+β is equal to

6.15

As already remarked, the above sum involves q pairs , since any choice of A determines C.

We will also need the number of y²=x⁴+Ax²+Bx+C that are associated to the twisted curve y²=X³+a²αX+a³β:

6.16

As A,C run over the elements of , so do a²C and aA. Thus, we can replace the condition in the last summand by −a²C/4−a²A²/48=a²α, i.e. by the same condition as in (6.15), −C/4−A²/48=α. The above sum therefore equals

6.17

the latter equality because since we have chosen a to be a non-square in .

Summing (6.15) and (6.17), the number of curves y²=x⁴+Ax²+Bx+C associated to either Y ²=X³+αX+β or to y²=X³+a²αX+a³β is given by

6.18

Thus 2q curves in are associated to each pair of curves Y ²=X³+αX+β, y²=X³+a²αX+a³β in , and all such curves have the same value of |1−a_q(x⁴+Ax²+Bx+C)|.

Special care is needed in the event that Y ²=X³+αX+β twists to itself, i.e. a²α=α and a³β=β. But, in that case, a_q(X³+αX+β)=−a_q(X³+αX+β), and thus equals 0, hence such polynomials contribute 0 to m₃(q;j), and their associated curves y²=x⁴+Ax²+Bx+C contribute 0 to m₄(q;j), so we may ignore these.

Thus, the number of curves from with given ±a_q are in 1:q proportion with the number of curves from with the same L-functions. When j is even, each term in m₃ and m₄ appear with an even exponent, and all terms summed are positive. Hence

6.19

When j is odd, then

6.20

Here, we are considering the contribution to m₄ from each particular value of a_q(X³+αX+β). The factor of q outside the sums is to account for the fact that m₄ is a sum over rather than . We run over all square-free , and also their twists X³+a²αX+a³β (where, as before, a is any fixed non-square in ), that give rise to that particular value of ±a_q. For any such pair of curves in , we count how many y²=X⁴+Ax²+Bx+C are associated to them using (6.15) and (6.17). Because j is odd, , thus resulting in (6.20) when the two are combined. The impact of running over curves and their twists (with a_q≠0) is to count each twice, hence the extra factors of 2 in front of both sides of (6.20).

Now, the inner sum equals −a_q(X³+αX+β), as one can check by substituting t=−A/6, which runs over as A does, and −C/4=α+A²/48=α+3t²/4 into the summand. Thus, the inner sum in (6.20) equals

6.21

Simplifying thus gives, when j is odd,

6.22

which, by definition, equals m₃(q;j+1). ▪

Lemma 6.3 —

6.23

If l=0, we take the v=0 term to equal 0.

Proof. —

The sum in the lemma can be expressed as

6.24

evaluated at z=1. Here (a)_n=Γ(a+n)/Γ(a)=a(a+1)…(a+n−1) (taken to be 1 if n=0). Other than the factor l+n, the sum over n is . The sum can be obtained by multiplying ₂F₁ by z^l, differentiating with respect to z, and then multiplying by z. Using

6.25

we can thus express the sum over n in (6.24) as

6.26

with a=l−k/2, and c=2l+2. Substituting z=1, and applying (5.19), we get

6.27

(we also used in simplifying). Substituting the right side into (6.24), simplifying, and using the Legendre duplication formula gives (6.23). ▪

7. Formulae suggested by our data, d≥5

We list here the formulae that one gets, experimentally, from interpolating (or guessing!) when possible, from our data.

When we did not have enough data to interpolate, we combined leading terms as derived from the Andrade–Keating conjecture with interpolation for the lower coefficients (also exploiting, via the Chinese remainder theorem, the observation that the coefficients seem to be integers). We left ourselves some leeway so that we could check our guess against at least one additional data point. We give the resulting formulae, for d=5, in table 24.

Table 24.

Moment formulae for d=5, k≤5.

k
1	3−q−1+q−2−q−3
2	14−11q−1+10q−2+5q−3−15q−4−q−5
3	84−111q−1+91q−2+98q−3−174q−4−51q−5−q−6
4	594−1133q−1+861q−2+1476q−3−1959q−4−1192q−5 −90q−6−q−7
5	4719−11869q−1+8645q−2+20416q−3−22055q−4 −21516q−5−3398q−6−145q−7−q−8

These formulae appear to hold for all prime powers q. For k>5 and d=5, presumably some extra arithmetic quantities enter, as they do for k>9 when d=3. In the case of d=5, the approach of A Diaconu & V Pasol (2014, personal communication) does appear to produce, with proof, a somewhat complicated formula for the moments involving traces of Hecke operators acting on Siegel cusp forms for certain congruence subgroups of . We have not attempted to put their formula in more concrete form. It would be a worthwhile project to do so, to provably produce and extend the above table of moment polynomials for d=5, and to better understand the contribution from the Hecke terms, presumably starting, when d=5, at k=6. We believe the Hecke terms enter at k=6 (when d=5) because we were not able to interpolate any polynomials in 1/q for k=6 in spite of having the moments for all q≤53 (19 data points).

The leading coefficients, 3,14,84,594,4719,…, are given by the Keating Snaith formula, with g=2 (so that d=2g+1=5). Interestingly, these leading coefficients also appear in the work of Kedlaya & Sutherland [23, table 4] as moments of traces in USp(2g), for g=2, and similarly for g=1 and the leading coefficients of table 22. This does not persist for g>2.

We display in tables 25–28 moment formulas guessed at from our data, for 6≤d≤9.

Table 26.

Moment formulae for d=7, k≤3.

k
1	4−2q−1+2q−2−2q−3+2q−4+2q−5−2q−6
2	30−40q−1+60q−2−66q−3+20q−4+101q−5−85q−6−36q−7−2q−8
3	330−832q−1+1674q−2−1986q−3−240q−4+4348q−5−2330q−6−3222q−7−626q−8−12q−9

Table 27.

Moment formulae for d=8, k=1.

k
1	4−q−1/2−3q−1+2q−2−q−5/2−3q−3+q−7/2+3q−4−3q−9/2−q−5+3q−11/2−3q−6−q−13/2
	+5q−7−2q−15/2

Table 25.

Moment formulae for d=6, k≤2.

k
1	3−q−1/2−2q−1+q−2−q−5/2−q−3+q−7/2−q−4−q−9/2+2q−5
2	14−12q−1/2−19q−1+14q−3/2+17q−2−24q−5/2+24q−7/2−33q−4+14q−9/2+30q−5−34q−11/2
	+14q−6−6q−13/2+q−7

Table 28.

Moment formula for d=9, k=1.

k
1	5−3q−1+3q−2−4q−3+6q−4−5q−5+q−6+5q−7−7q−8−q−9

8. Series expansions for Q_k(q;d)

When d is odd,

8.1

When d is even,

8.2

Grouping P's together according to their degree, and using formula (1.41) for the number of irreducible polynomials of given degree, we have, on expanding the above formulae in powers of 1/q or 1/q^1/2, that, for d=2g+1 odd, k=1:

8.3

and for d=2g+2 even, k=1:

8.4

Substituting d=1,2,3,…,9 into the above formulae yields table 29:

Table 29.

Expansion of Q₁(q;d) in the q-aspect, for d≤9.

d	Q1(q;d)
1	1+O(q−1)
2	1−1/q1/2+O(q−2)
3	2+O(q−3)
4	2−1/q1/2−1/q−1/q5/2+1/q3+O(q−7/2)
5	3−1/q+1/q2+O(q−4)
6	3−1/q1/2−2/q+1/q2−1/q5/2−1/q3+1/q7/2−1/q4+O(q−9/2)
7	4−2/q+2/q2−2/q3+2/q4+2/q5+O(q−6)
8	4−1/q1/2−3/q+2/q2−1/q5/2−3/q3+1/q7/2+3/q4−3/q9/2−1/q5+O(q−11/2)
9	5−3/q+3/q2−4/q3+6/q4−5/q5+1/q6+O(q−7)

In table 29, we are displaying the terms that match with the actual moments from the previous sections.

(a). Series expansions for Q_k(q;d) when k=2,3

We can work out expansions, analogous to (8.3) and (8.4) for additional values of k, and do so here for k=2,3.

We make use of the methods of Goulden et al. [11] to express the coefficients of the polynomials Q_k(q;d) more explicitly. To apply the formulae of [11], we first write Q_k(q;d) as a polynomial in 2g rather than in d. For given q and k let

8.5

Note that this actually defines two different polynomials, depending on whether d=2g+1 or d=2g+2, so that c_r(q;k) also depends (for r>0) on the parity of d. To avoid clutter, we suppress this dependence in our notation.

Define

8.6

In the last equality, we are simply grouping together factors according to the value of |P|=qⁿ.

The length of the partition λ is defined to be the number of non zero λ_is. We denote it by l(λ). Given α=(α₁,…,α_n), we write u^α to denote . Let λ be a partition of length less than or equal to n. If n≥l(λ), then

8.7

where the α ranges over distinct permutations of (λ₁,…,λ_n). If l(λ)>n, then m_λ(u₁,…,u_n)=0. For the only partition of 0, the empty partition, we define m₀=1. Thus, for example,

Let

8.8

be the power series expansion of

8.9

where H is defined in (2.12). The double product above plays the role of cancelling the poles of in (2.12).

In (8.8), the sum is over all partitions λ₁+⋯+λ_k=i, with λ₁≥λ₂≥⋯λ_k≥0. We divide the expression by a_k to ensure that the constant term in the power series is 1.

Then

8.10

and, for r≥1,

8.11

where N_λ(k) is defined by

8.12

The above is obtained by substituting (8.8) into (2.11), changing variables, u_j=(2g)z_j/2, and taking care to borrow from , thus producing the factor displayed, .

In Goulden et al. [11], we obtained several formulae for N_λ(k) and also proved that it is a polynomial in k of degree at most 2|λ| (which is the reason why we pull out the factor To exploit the formulae obtained in that paper, we also regard Q_k as a polynomial in 2g rather than in g. A list of the polynomials N_λ(k), quoted from [12], is given in table 30.

Table 30.

We display the polynomials, from [11], N_λ(k), for all |λ|≤6. N_λ(k) has, as a factor, the polynomial: , where (k)_m=k(k−1)…(k−m+1). The polynomial r_λ(k) counts the number of monomials in m_λ(z). Therefore, we separate this factor out, and list N_λ(k)/r_λ(k).

λ	Nλ(k)/rλ(k)	rλ(k)
[1]	k+1	(k)1
[1,1]	(k+2)(k+1)	(k)2/2
[2]	0	(k)1
[1,1,1]	(k+3)(k+2)(k+1)	(k)3/6
[2,1]	(k+2)(k+1)	(k)2
[3]	−(k−1)(k+2)(k+1)	(k)1
[1,1,1,1]	(k+4)(k+3)(k+2)(k+1)	(k)4/24
[2,1,1]	2(k+3)(k+2)(k+1)	(k)3/2
[2,2]	0	(k)2/2
[3,1]	−(k−2)(k+3)(k+2)(k+1)	(k)2
[4]	0	(k)1
[1,1,1,1,1]	(k+5)(k+4)(k+3)(k+2)(k+1)	(k)5/120
[2,1,1,1]	3(k+4)(k+3)(k+2)(k+1)	(k)4/6
[2,2,1]	4(k+3)(k+2)(k+1)	(k)3/2
[3,1,1]	−(k−3)(k+4)(k+3)(k+2)(k+1)	(k)3/2
[3,2]	−2(k−2)(k+3)(k+2)(k+1)	(k)2
[4,1]	−2(k−2)(k+3)(k+2)(k+1)	(k)2
[5]	2(k−1)(k−2)(k+3)(k+2)(k+1)	(k)1
[1,1,1,1,1,1]	(k+6)(k+5)(k+4)(k+3)(k+2)(k+1)	(k)6/720
[2,1,1,1,1]	4(k+5)(k+4)(k+3)(k+2)(k+1)	(k)5/24
[2,2,1,1]	10(k+4)(k+3)(k+2)(k+1)	(k)4/4
[2,2,2]	0	(k)3/6
[3,1,1,1]	−(k−4)(k+5)(k+4)(k+3)(k+2)(k+1)	(k)4/6
[3,2,1]	−(k+3)(k+2)(k+1)(3k2+3k−40)	(k)3
[3,3]	(k−2)(k−4)(k+5)(k+3)(k+2)(k+1)	(k)2/2
[4,1,1]	−4(k+3)(k+2)(k+1)(k2+k−10)	(k)3/2
[4,2]	0	(k)2
[5,1]	2(k−2)(k+3)(k+2)(k+1)(k2+k−10)	(k)2
[6]	0	(k)1

In order to compute the multivariate Taylor expansion of (8.9), i.e. the coefficients b_λ(k), we consider the series expansion of its logarithm, since it is easier to deal with a sum than a product. Let

8.13

be the power series expansion of the logarithm of (8.9). We start the sum at r=1 because the division by a_k makes the constant term 0. Now, the l.h.s. is symmetric in the u_i's, and we can find B_λ(k) by applying

8.14

where l=l(λ), and setting u₁=⋯=u_k=0. Since the partial derivatives do not involve u_l+1,…,u_k we can set these to 0 before the differentiation.

Thus, B_λ(k) is equal to (8.14) applied to

8.15

evaluated at u₁=⋯=u_l=0.

Next, by composing the series expansions (8.13) with the series for the exponential function, we can derive formulae for the coefficients b_λ(k).

In this way, we computed the following series expansions, if d=2g+1 is odd:

8.16

and

8.17

Note that the terms that are independent of q (e.g. in Q₂(q;d)), match the right side of (2.2). This is explained by the fact that, as ,

8.18

and we recover the moments of unitary sympletic matrices as given in (2.3).

If d=2g+2 is even we have

8.19

and

8.20

Substituting d=1,…,7 into the above formulae for Q₂(q;d) gives table 31. For Q₃(q;d), this yields table 32.

Table 31.

Expansion of Q₂(q;d) in the q-aspect, for d≤7.

d	Q2(q;d)
1	1+O(q−1)
2	1−2q−1/2+q−1+O(q−3)
3	5−q−2+O(q−4)
4	5−6q−1/2−3q−1+4q−3/2−q−2−2q−5/2+7q−3−4q−7/2+O(q−5)
5	14−11q−1+10q−2+5q−3−15q−4+O(q−5)
6	14−12q−1/2−19q−1+14q−3/2+17q−2−24q−5/2+24q−7/2−33q−4+14q−9/2+O(q−5)
7	30−40q−1+60q−2−66q−3+20q−4+101q−5+O(q−6)

Table 32.

Expansion of Q₃(q;d) in the q-aspect, d≤7, except d=6 where we did not have enough data to guess the moment formula for k=3.

d	Q3(q;d)
1	1+O(q−1)
2	1−3q−1/2+3q−1−q−3/2+O(q−2)
3	14−6q−2+O(q−4)
4	14−28q−1/2+28q−3/2−20q−2+3q−5/2+27q−3+O(q−7/2)
5	84−111q−1+91q−2+O(q−3)
7	330−832q−1+1674q−2−1986q−3−240q−4+O(q−5)

Again, we are displaying the terms that match with the actual moments from §7. Letting X=q^d, the above expansions yield the values of μ presented in §3.

9. Algorithms used

To tabulate zeta functions, we first looped though all monic polynomials D(x) of given degree d in or , and, for each D, checked whether to determine if D is square-free. We then used the approximate functional equation described in §1b and quadratic reciprocity to determine each zeta functions for all (when ), or (when p|d), and the values of d,q listed in §3. We implemented our code in C++ using the flint package [24] for finite field arithmetic. However, this became prohibitive as , and each application of the approximate functional equation requiring roughly q^g evaluations of χ_D(n) via quadratic reciprocity.

After gathering some data in this fashion, we switched to using Magma's built-in routine for computing the zeta function of a hyperelliptic curve. It uses a combination of exponential point counting methods and Kedlaya's algorithm [25]. Let q=pⁿ. The latter algorithm runs in time O(p^1+ϵg^4+ϵn^3+ϵ), for any ϵ>0, with the implied constant depending on ϵ, see [26, Theorem 3.1].

The other computational aspect of testing the Andrade–Keating conjecture involved numerically evaluating the coefficients of the polynomials Q_k(q;d). While formula (2.11) can be used to evaluate a few coefficients c_r(q;k) of the polynomials Q_k(q;d), it is not well suited for computing all k(k+1)/2 coefficients, except when k is small. For example, we took (2.11) as our starting point in the previous section to work out, via (8.11), formulae for Q_k(q;d) for k=1,2,3. However, it is not feasible, to compute, in this manner, all 55 coefficients, c_r(q;k), when say, k=10, as this would involve expanding the integrand in a series of 10 variables using monomials of degree less than or equal to 55. Instead, we used a technique that was developed in the number field setting, see [27, §3] and [28, §4.2]. We summarize the method, as applied in our setting, below.

In [9, Lemma 2.5.2] plays a key role, and we first paraphrase the part we need.

Lemma 9.1 (from [9]) —

Suppose F is a symmetric function of k variables, regular near (0,…,0), and f(s) has a simple pole of residue 1 at s=0 and is otherwise analytic in a neighbourhood of s=0, and let

9.1

Assume |α_i|≠|α_j| if i≠j. Then, for sufficiently small |α_j|,

9.2

and where the path of integration encloses the ±α_j's.

Note that the poles of K from the product of f's are cancelled by a portion of the factor . The condition that |α_j| be sufficiently small is needed to ensure that the numerator of the integrand in (9.2) is analytic in and on the contours.

To compute c_r(q;k) we do two things. First, we expand the exponential in (2.11) to get

9.3

Next, we view the above as the limiting case of (9.2), α_j→0, with

and evaluate it by summing the 2^k terms on the left side of (9.2). In practice, we took a_j=j10⁻⁶⁵. Now the terms being summed have poles of order k(k+1)/2 that cancel as we sum all the terms. One can see that they must cancel since the expression on the right side of (9.2) is analytic in a neighbourhood of α=0. Thus, to see our way through the enormous cancellation that takes place, we used, for example when k=10, thousands of digits of working precision.

One advantage here, over the number field setting, is that the arithmetic product A, defined in (2.10), as expressed in (2.12) (i.e. grouping together irreducible polynomials according to their degree n), converges very quickly. The relative remainder term in truncating the product over n in (2.12) at n≤N, is, for sufficiently small u_j, O(q^−N−1+ϵ), with the implied constant depending on ϵ. Thus, only a few hundred (for q=3) or handful (for q=10 009) of n were needed to achieve at least 30 digits precision for all c_r(q;k) that we computed.

Funding statement

The first author was supported in part by an NSERC Discovery grant and EPSRC grant no. EPK0343831. The second author was supported by an NSERC USRA.