methods for justifying arithmetic hypotheses and computer algebra

ISSN 0361-7688, Programming and Computer Software, 2006, Vol. 32, No. 3, pp. 128–133. © Pleiades Publishing, Inc., 2006.Original Russian Text © N.M. Glazunov, 2006, published in Programmirovanie, 2006, Vol. 32, No. 3.

128

1. INTRODUCTION

In mathematics and other scientific disciplines, theword “hypothesis” is used in various meanings. In thispaper, we will occasionally identify the word “hypoth-esis” with the words “problem” or “question” and, fol-lowing [1–5], imply that “hypothesis” is a question thatshould be answered. In the deterministic case, theanswer should be “YES” or “NO.” If a hypothesis isjustified by statistical criteria, the answer can be formu-lated probabilistically. The hypothesis is given by adescription of all its parameters and through a specifi-cation of properties that are matched by the proof of thehypothesis (if the hypothesis is true). The study of ahypothesis (or problem) is normally divided into aseries of stages (sub-problems). By the justification ofa hypothesis, we mean that:

(a) the hypothesis is checked against a finite datasample and the resulting information is compared withthe statement of the hypothesis on the basis of a statis-tical criterion (Section 2);

(b) the hypothesis is checked for a finite number ofexamples and, possibly, is refined (Section 3);

(c) a problem is stated or a hypothesis is put forwardon the basis of the resulting experimental data and the-oretical considerations (Section 3);

(d) the hypothesis is proved (Section 3).It has become increasingly important to study proper-ties and behavior of not only particular problems, butalso classes of problems joined together into familiesthrough their parameterization by points of a space,which is called the moduli space of a given family [5]. Aspecific feature of the methods proposed and developedin this paper is that they are represented as computer-oriented methods in the moduli space of objects underconsideration, which provides a uniform approach tostudying the properties and behavior of not only partic-

ular problems, but also classes of problems. The defini-tions of the theory of moduli used here can be found in [5](which refers also to some original sources), as well asin our list of references. However, unlike [5], whichconsiders mostly the zero-characteristic case, we dealwith the characteristic

p

. An algebraic manifold iscalled the moduli space of a given algebraic structure ifthe manifold points parameterize the objects or classesof objects (in terms of some equivalence relation) ofthis algebraic structure [5]. Let us recall that the prob-lem of moduli consists of two parts. First, one shouldseparate a class of objects and describe what the familyof these objects means over some scheme

S

. Second,one should choose an equivalence relation. Among dif-ferent moduli spaces, the classical algebraic geometrydistinguishes moduli and parametric (parameterizing)spaces. The points of parametric spaces are bijectivelymatched by objects of parameterizing algebraic struc-ture, while the moduli parameterize the classes, each ofwhich consists, for example, of birationally isomorphicobjects. We will use a version of this terminology, call-ing the moduli spaces with a trivial equivalence relationby parametric spaces. We consider two types of para-metric spaces: quasi-projective manifolds and arithmeticsurfaces [6, 7]. For the given problem of moduli, theobject that bijectively corresponds to a point of theparametric space is said to lie over that point. If, for aparametric space

�

, the concept of a hypersurface

P

isdefined, by the (hypersmooth) section, we call the set ofzeros

P

= 0 in

�

. The set of objects lying over the sec-tion points are said to lie over the section. Let us notethat the methods developed in this paper can also beused without recourse to the moduli space. This corre-sponds to the case of a single-point moduli space. Inmany applied and mathematical problems, it is requiredto find (Hilbert’s 10th problem) integer rational solu-tions to algebraic equations with rational coefficients

Methods for Justifying Arithmetic Hypothesesand Computer Algebra

N. M. Glazunov

Glushkov Cybernetics Institute, National Academy of Sciences of Ukraine,pr. Akademika Glushkova 40, Kiev, 03680 Ukraine

e-mail: [email protected]

Received June 29, 2005

Abstract

—Computer algebra methods for justifying hypotheses in arithmetic geometry are considered.The specific feature of these methods is that they are designed for parametric problems. The methods developedcan be used for solving computational problems and justifying hypotheses on equidistribution, as well as forthe theory of algebraic curves over finite fields.

DOI:

10.1134/S0361768806030029

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No. 3

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METHODS FOR JUSTIFYING ARITHMETIC HYPOTHESES 129

(Diophantine equations) and investigate Diophantineinequalities. Yu.V. Matiyasevich proved that Hilbert’s10th problem is algorithmically undecidable [4]. At thesame time, if the Diophantine equation is reduced to afinite field, there always can be found algorithms forchecking whether a solution of this equation in thefinite field exists. However, such algorithms are oftenclassified among the class of NP-complete algorithms.This dictates the need of developing alternative meth-ods to check whether there are solutions of Diophantineequations in finite fields. The analog of Riemann’shypothesis over finite fields and the hypotheses of A. Weylfor algebraic curves over finite fields have stimulatedthe development and application of methods for study-ing algebraic manifolds over finite fields. One of thesemethods is the estimation of the number of points ofalgebraic manifolds in prime finite fields. It is thismethod that is used in this paper and investigated on theexample of algebraic curves. In this paper, we investi-gate the class of Diophantine equations that are alge-braic curves of the form

y

2

=

f

(

x

) upon reduction in aprime finite field

F

p

, where

f

(

x

) is a polynomial of an oddorder over

F

p

with the leading coefficient equal to one.The hypothesis of J. Tate about the equidistribution ofangles of the Frobenius endomorphism of elliptic curveswithout complex multiplication and its experimentalverification by M. Sato have stimulated studies of equi-distribution and development of suitable methods andtheir application techniques. The finding that the Kloos-terman trigonometric sums can serve as an importantcomponent in stating and investigating a large numberof mathematical problems and, in particular, their rela-tionship with Artin–Schreier curves (coverings), havestimulated the study and application of these sums.Such problems and relevant hypotheses arise in the the-ory of communication and cryptography, dynamicalsystems, mathematical and theoretical physics, and inmany other fields. For example, in some problems,Lagrangian is defined in terms of its density function.As far as the author knows, the method proposed belowcan be extended to problems of checking hypothesesabout the form of the Lagrangian density function.

2. STUDY OF THE DISTRIBUTION OF ANGLES OF KLOOSTERMAN TRIGONOMETRIC SUMS

The concept of the uniform distribution modulo 1 wasintroduced by G. Weil [8] in the context of studying prob-lems of celestial mechanics and some arithmetic prob-lems. To take into account more general laws (i.e., distri-butions with rather general density functions and dis-tributions modulo), the Sobolev generalized functions(distributions in the sense of L. Schwarz) were required.G. Weil, I.M. Vinogradov, N.M. Korobov, A.G. Postni-kov, M. Sato, J. Tate, B. Birch, J.-P. Serre, P. Deligne,H. Yoshida, P. Livné, N. Katz, A. Adolphson, and otherresearchers (see [8–12] and references therein) pro-posed methods for proving the equidistribution of manyinfinite arithmetic sequences and proved a series of the-

orems on equidistribution. However, the hypothesis of theequidistribution of the angles of Kloosterman trigonomet-ric sums (see below) with the Sato–Tate density has notbeen proved or disproved yet. Not pretending to a histori-cal review of studies on the equidistribution problems, werecall briefly some facts. Let

be an elliptic curve over

Z

. Outside of a finite set ofprimes that are divisors of the discriminant, the curve

E

has good reduction to the field

F

p

. The number of points#

E

(

F

p

) on the curve

E

in the localization by (mod

p

)is expressed by the formula #

E

p

= 1 +

p

–

a

p

, where

a

p

= 2 cos

ϕ

p

and

E

itself is taken to be a projectivecurve. The Sato–Tate hypothesis [13] states that, foran elliptic curve without complex multiplications, theangles

ϕ

p

corresponding to

a

p

are equidistributed on theinterval [0,

π

] with the density

Although the Sato–Tate hypothesis has not been provedyet, its functional analogs were studied. For example,the functional analog for the case of

∞

has beenproved by Birch [12], and the case where = const withalgebraic extensions of over

F

p

as

r

∞

has been

proved by Yoshida [14]. Similar problems can be statedfor the angle distributions of the Kloosterman trigono-metric sums [15, 16]. These sums arise in many mathe-matical disciplines [17]. In recent years, there haveappeared major and interesting studies on the Klooster-man sums that further stress the importance and rele-vance of this line of scientific research [18, 19].

Below, we present results of the study of distribu-tions of angles

θ

p

of the Kloosterman trigonometricsums on the interval [0,

π

] for the following two cases:(A) the case proved by Katz [15] and Adolphson [20],

when

c

and

d

(

cd

is not divided by

p

) run independentlyover

F

p

, as

p

approaches to infinity;(B) checking the hypothesis for the sum

on a sample of 1600 sequential primes, when the sum isfixed (the fixed parameters are

c

and

d

). The provencase (A) was compared with the computational resultsfor case (B).

Computation technique and processing.

In thecourse of computations, for each prime

p

, the values

E: y2 x3 ax b, a b,+ + Z∈=

p

2π--- t.sin

2

Fp

r

T p c d,( ) e

2πicx

dx---+

p---------------

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

,x 1=

p 1–

∑=

T p c d,( ) e

2πicx

dx---+

p---------------

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

, cx 1=

p 1–

∑ d 1= = =

130

PROGRAMMING AND COMPUTER SOFTWARE

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No. 3

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GLAZUNOV

of

T

p

, cos

θ

p

, and

θ

p

were computed. We present tablesshowing examples of large and small absolute values ofthe Kloosterman sums

T

p

for different primes

p

(seeTables 1 and 2).

We divide the interval [0,

π

] into 20 subintervals

where

i

is the number of the subinterval

U

i

,

ν

(

U

i

) is thenumber of angles

θ

j

(where

j

, 1

≤

j ≤ n, is the number ofthe prime) occurring in the interval Ui, h(Ui) is thehypothetical number of angles θj in the interval Ui for a

given sample for the sin2t-distribution, pi = ,

h(Ui) = ||npi ||, where ||α|| is the integer nearest to α, andn is the number of elements in the sample.

Case (A). The theory and computations show thatthe distributions of angles θp(c, d) of the sums

over the intervals

for c = const, 1 ≤ d ≤ p – 1, is the same for different 1 ≤c ≤ p – 1. In view of this, hereafter, we present theexperimental values of distributions of angles of sumsTp(c, d) for c = const, 1 ≤ d ≤ p – 1. For example, forp = 1597, c = 890, 1 ≤ d ≤ p – 1, the computationalresults are summed in Table 3 (the notation was intro-duced above).

Now, let us compute (using Pearson’s χ2-criterion)the probability of disproving the hypothesis about thesin2t-distribution using the data given in this table.Since each interval must include at least 10 values, wecombine the intervals U1, U2, and U3 and the intervalsU19 and U20.

Proposition 1. In case (A), for Table 3 with 16 degreesof freedom, we have

χ2 = 7.52.

Uii 1–( )π

20------------------- i( )π

20----------, , i 1 2 … 20,, , ,= =

2π--- tsin

2td∫

T p c d,( ) e

2πicx

dx---+

p---------------

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

x 1=

p 1–

∑=

Uii 1–( )π

20------------------- i( )π

20----------, , i 1 2 … 20, , ,= =

Case (B). The computational results on a sample of1600 sequential primes from 2 to 13499 are presentedin Table 4.

The computational results based on Pearson’s χ2-cri-terion are similar to those obtained for case (A). Sinceeach interval must include at least 10 values, we com-bine the intervals U1, U2, and U3 and the intervals U18,U19, and U20.

Proposition 2. In case (B) with 15 degrees of free-dom, we have

χ2 = 10.1806.

Comparing the computed value of χ2 with that in thetables of the χ2-distribution for case (B), we find thatthe hypothesis about the equidistribution of angles θp

on the interval [0, π] with the density function sin2t is

true with a 5% significance level by Pearson’s χ2-criterion.

3. RATIONAL POINTS OF ALGEBRAIC CURVES OVER MODULI SPACES

OVER PRIME FINITE FIELDS

Let n ≥ 3 be an odd number, p ≥ 2 be a prime, andf(x) be a unitary polynomial of degree n, f(x) ∈ Fp[x].Let us consider the curve

(1)

The curves of form (1) are parameterized by points onthe affine space An(Fp) of dimension n over the field Fp.We call the parametric space over the field a k-factorizingspace if any curve lying over this space has a point in thefield k. Otherwise, the space is said to be non-factorizing.If we fix the degree n of the polynomial f, the knownestimates obtained by Hasse and Weil |#C(Fp) – p | ≤ 2gfor the number of solutions of (1) give the followingresult.

Statement 3. If p ≥ p0 (the boundary p0 depends on n)is a sufficiently large number, then An(Fp) is a factoriz-ing space for curves (1).

Let us consider the case of small n. For n = 3, curve (1)always have a solution as early as for p > 3; i.e., theparametric space of cubic curves of form (1) is factor-

2π---

y2 f x( ).=

Table 1. Large absolute values of cosθp

p |cosθp| p |cosθp|

2 0.35355 3041 0.75232

7 0.38721 5059 0.78164

29 0.47823 7537 0.85773

103 0.52564 10181 0.95269

1549 0.66391 13171 0.96537

Table 2. Small absolute values of cosθp

p |cosθp| p |cosθp|

2 0.35355 709 0.04543

41 0.31430 1613 0.03436

97 0.10634 2161 0.02577

383 0.08503 3719 0.01499

487 0.05637 10889 0.00492

PROGRAMMING AND COMPUTER SOFTWARE Vol. 32 No. 3 2006


izing even for p > 3. For n ≥ 5, it follows from [22] that,

for p > – 2 (Mit’kin’s estimate), An(Fp) are

factorizing spaces for curves (1). If Mit’kin’s esti-mate is not satisfied, the space cannot be factorizing.The curve y2 = x5 + x4 + 7x3 + 8x2 + 3x + 8 lies over thepoint (1, 7, 8, 3, 8) ∈ A5(F11) and has no solution in F11.At the same time, Mit’kin’s estimate (just as the Hasseand Weil estimates) is not necessary. It follows from thecomputational results [23, 24] that, although in this

case p < – 2, A5(F13) is a factorizing space for

curves (1). Let us consider a hyperelliptic curve Cg ofgenus g ≥ 2 over Fp. For the projective closure of Cg, thequasi-projective space Hg, p = {P2g + 1(Fp)\Disk(Cg) = 0},where Disc(Cg) is the discriminant of the curve Cg,parameterizes all hyperelliptic curves of genus g over Fp.According to the above estimate, for p ≥ 17, any hyper-elliptic curve of genus 2 has points in Fp for suchprimes p. Similarly, for g = 3, any hyperelliptic curve ofgenus 3 has points in Fp for p ≥ 37. For p = 2, 3, 5, 7,and 11, there are examples of curves that have no pointsin Fp. According to computations performed in [23, 24],any curve of genus 2 over F13 has points in that field.Similar, but more extensive, computations show that,for p = 2, 3, 5, 7, 11, 13, and 17, there are examples ofcurves of genus 3 that have no points in Fp. Let us for-mulate these results as a separate statement.

Statement 4. The estimates of Hasse, Weil, Mit’kin,and Serre in the affine and projective versions are suf-ficient but not necessary conditions for An(Fp) and Hg, pto be factorizing spaces.

The above considerations lead to the following prob-lem, which is referred to as the problem of A.G. Postni-kov. To formulate it, we introduce the required conceptsand definitions. Let {PSg, p} be a family of parametric

n 1+( )2

2-------------------

n 1+( )2

2-------------------

spaces with parameters g and p. Let c be some elementand #c be its numerical characteristic. Let b(g, p, #c) besome estimate defined for all elements c ∈ {PSg, p}. LetPr(c, b(g, p, #c)) be a predicate on elements of sub-spaces of the family {PSg, p}. We will say that the esti-mate b(g, p, #c) separates (precisely separates) the family{PSg, p} if, for the given g = const, there exists p = p0(g)such that, for any p ≥ p0(g) and all c ∈ PSg, p, the predi-cate satisfies the condition Pr(b(g, p, #c)) = TRUEand, for p ≤ p0(g) (for each g and for each p ≤ p0(g)),there exist c ∈ PSg, p with Pr(c, b(g, p, #c)) FALSE.A precisely separating estimate will be called a preciseestimate.

Let PSg, p := Hg, p be the parametric space of hyper-elliptic curves of genus g and Cg ∈ Hg, p, #Cg be thenumber of points of the curve Cg in Fp (i.e., in terms ofthe previous notation, c = Cg).

Postnikov’s problem (precise estimate problem).For the family of spaces Hg, p of hyperelliptic curves,does there exist a precise estimate separating this fam-ily into factorizing and non-factorizing spaces? If suchan estimate exists, what is its form?

At present, the author cannot answer these questions.

The study of arithmetic properties of objects param-eterized by subspaces of moduli spaces is currently anintensively developing field [5]. Let us pose the ques-tion of whether exist parametric spaces that includesections with curves lying over them and having norational points in Fp.

Theorem 5. Let A(p – 1)/2(Fp) be a parametric space ofform (1) and the prime be p ≡ 3 (mod 4). Then, for p ≥ 11,there exist curves over its section (a1, …, a(p – 3)/2) =(0, …, 0) with a(p – 1)/2 ≠ 0 that have no solution over Fp.

Proof. The proof is based on the method proposedby Postnikov [23]. For p = 11, the equation is

y2 x5 7.+=

Table 4. Distribution of angles for 1600 sequential primesfrom 2 to 13499

i ν(Ui) h(Ui) i ν(Ui) h(Ui)

1 0 1 11 164 159

2 7 9 12 141 151

3 25 24 13 150 136

4 41 44 14 128 116

5 66 68 15 106 92

6 98 92 16 67 68

7 99 116 17 40 44

8 132 136 18 25 24

9 152 151 19 2 9

10 156 159 20 1 1

Total 776 824

Table 3. Distribution of angles θp(c, d), p = 1597, c = 890,1 ≤ d ≤ p – 1

i ν(Ui) h(Ui) i ν(Ui) h(Ui)

1 0 1 11 154 158

2 6 9 12 158 151

3 29 24 13 141 136

4 51 44 14 109 116

5 65 67 15 99 92

6 90 92 16 70 67

7 111 116 17 44 44

8 134 136 18 18 24

9 154 151 19 7 9

10 153 158 20 3 1

Total 793 803

132


GLAZUNOV

Let p > 11. Let us denote a = a(p – 1)/2. The values of y2

are quadratic residues and 0. The monomial takes three values: –1, 0, and 1. Namely, its value is 0for x = 0, and 1 or –1 for x ≠ 0 (which follows fromthe little Fermat theorem). Let the series of numbers0, 1, 2, …, p – 1 have a combination “nonresidue, non-residue, nonresidue.” Then, taking the average of thesenumbers to be a, we obtain the required a. Indeed, in

this case, for all possible values of in the field Fp,we obtain either a itself or a – 1, or a + 1, which, by theassumption, are also nonresidues.

Let us derive the precise formula for the number Tof possible combinations “nonresidue, nonresidue,nonresidue” in the series 0, 1, 2, …, p – 1, and show thatthe value of T (which is a natural number) is greaterthan zero. The formula has the form

Then, we have

In order for T to become greater than zero, it is suf-

ficient that the inequality p – 3 – 2 – 4 > 0 hold,which takes place for p ≥ 19. The theorem is proved.

xp 1–

2------------

xp 1–

2------------

T18--- 1

x 1–p

-----------⎝ ⎠⎛ ⎞–⎝ ⎠

⎛ ⎞x 2=

p 2–

∑=

× 1xp---⎝ ⎠

⎛ ⎞–⎝ ⎠⎛ ⎞ 1

x 1+p

------------⎝ ⎠⎛ ⎞–⎝ ⎠

⎛ ⎞ 1x 1–

p-----------⎝ ⎠

⎛ ⎞–⎝ ⎠⎛ ⎞

x 0=

p 1–

∑=

× 1xp---⎝ ⎠

⎛ ⎞–⎝ ⎠⎛ ⎞ 1

x 1+p

------------⎝ ⎠⎛ ⎞–⎝ ⎠

⎛ ⎞

= 1x 1–

p-----------⎝ ⎠

⎛ ⎞–⎝ ⎠⎛ ⎞ 1

xp---⎝ ⎠

⎛ ⎞–⎝ ⎠⎛ ⎞ 1

x 1+p

------------⎝ ⎠⎛ ⎞–⎝ ⎠

⎛ ⎞x 0=

p 1–

∑ 12---Θ,+

Θ 1.≤

T18--- p

x 1–( )xp

-------------------⎝ ⎠⎛ ⎞

x 0=

p 1–

∑+=

+x 1–( ) x 1+( )

p----------------------------------⎝ ⎠

⎛ ⎞x 0=

p 1–

∑ x x 1+( )p

--------------------⎝ ⎠⎛ ⎞

x 0=

p 1–

∑+

–18--- x 1–( )x x 1+( )

p-------------------------------------⎝ ⎠

⎛ ⎞x 0=

p 1–

∑ 12---Θ+

= 18--- p 3–

x 1–( )x x 1+( )p

-------------------------------------⎝ ⎠⎛ ⎞

x 0=

p 1–

∑+

+12---Θ 1

8--- p 3– 2 p–( )> 1

2---.–

p

At present, there has been intensive research onproblems of the existence and exact values of the num-ber of rational points of curves of a genus greater thanone over fields. Theorem 5 makes it possible to specifythe class of curves of a genus greater than one that haveno solutions in Fp. Let Disc(f) denote the discriminantof the polynomial f in Theorem 5 and p = 4m + 3.

Corollary 6. Under the assumptions of Theorem 5,over the section (a1, …, a(p – 3)/2) = (0, …, 0) of theparametric space of hyperelliptic curves over Fp forp ≥ 11, there are hyperelliptic curves of genus (p – 1)/4that have no solution over Fp.

Proof. In this case, the parametric space is

The discriminant of the right-hand side of the equation

which exists for p ≥ 11 by Theorem 5, is equal to

The corollary is proved.

ACKNOWLEDGMENTS

The author is grateful to S.A. Abramov and an anon-ymous reviewer for useful comments.

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y2 x p 1–( )/2 a p 1–( )/2,+=

p 1–2

------------⎝ ⎠⎛ ⎞

p 1–( )/2

a p 1–( )/2 0.≠



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