Ideal arithmetic and infrastructure in purely cubic function fields

R

ENATE

S

CHEIDLER

Ideal arithmetic and infrastructure in purely

cubic function fields

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{2 (2001),}

p. 609-631

<http://www.numdam.org/item?id=JTNB_2001__13_2_609_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Ideal arithmetic and infrastructure in

_purely

cubic function fields

par RENATE SCHEIDLER

RÉSUMÉ. Dans cet

_article,

nous étudions

_{l’arithmétique}

des

idéaux fractionnnaires dans les _corps de fonctions

_cubiques

_purs,

ainsi que l’infrastructure de la classe des idéaux

_{principaux lorsque}

le _groupe des unités du corps est de _{rang 1.} Nous décrivons d’abord la

_{décomposition}

des

_polynômes

irréductibles dans l’ordre maximal du corps. Nous construisons ensuite des bases

d’idéaux,

dites _canoniques, bien

_adaptées

pour les calcul. Nous

énonçons

des

_algorithmes

_permettant de

_multiplier

les

_idéaux,

et même de les réduire

_lorsque

le groupe des unités est de _{rang 1} et la

ca-ractéristique

au moins _5, L’article se termine avec une

analyse

de l’infrastructure de l’ensemble des idéaux fractionnaires réduits

principaux

dans le cas des corps

cubiques

purs de groupe des unités

de rang 1 et de

_{caractéristique}

au moins 5.

ABSTRACT. This paper

investigates

the arithmetic of fractional ideals of a

_purely

cubic function field and the infrastructure of the

principal

ideal class when the field has unit rank one.

First,

we

describe how irreducible

_{polynomials decompose}

into

_prime

ideals

in the maximal order of the field. _{We go} on to _compute so-called canonical bases of

ideals;

such bases are very suitable for

compu-tation. We state

_algorithms

for ideal

_{multiplication and,}

in the

case of unit rank one and characteristic at least

_five,

ideal

reduc-tion. The _paper concludes with an

_analysis

of the infrastructure

in the set of reduced fractional

_principal

ideals of a

_purely

cubic

function field of unit rank one and characteristic at least five.

1. Introduction

The infrastructure of a number field of unit rank one refers to the struc-ture of the set of reduced

_{representatives}

in an

equivalence

class of ideals in

the maximal

_{(or any)}

order of the field:

_{informally speaking,}

while this set does not form a _group under the

operation

multiplication

with

subsequent

reduction - the associative law does not hold - it behaves "almost" like

counterparts,

and that real

_quadratic

function fields exhibit an

infrastruc-ture much like that of real

_quadratic

number fields. As in the number field

case, this infrastructure can be used to

_compute

the

_regulator

and the ideal class number of these fields

_~6j.

While there are

only

three

_types

of number fields of unit rank one

-real

_{quadratic, complex cubic,}

and

_{totally complex quartic}

fields -

there

are function fields of

_{arbitrarily high degree}

that have a

_{representation}

as a unit rank one extension of some field of rational functions over a

finite

_{field; presumably,}

_many or

perhaps

all of these fields exhibit some

kind of infrastructure. This is

_certainly

the case for

_purely

cubic function

field

_{representations}

of unit rank one, the function field

analogue

of

purely

cubic number fields. Ideal arithmetic and the infrastructure in

_purely

cubic number fields were

_{investigated by}

Williams et al. in

_{[11, 12, 10],}

and much of the work in this _paper was

_{guided by}

these sources. As in the case of

quadratic

function

_fields,

the infrastructure can be used to

_compute

the

regulator

and the ideal class

_number,

and hence the order of the group of rational

_points

of the Jacobian of a

_purely

cubic function field.

For a

general

introduction to function

_fields,

we refer the reader to

_{~7J .}

Purely

cubic function fields are discussed in detail in

[5].

Let k =

Fq

be

a finite field of

_{order q}

whose characteristic is not

_equal

to 3. Denote

_by

and the

_ring

of univariate

_polynomials

and the field of rational

functions, respectively,

over k in the indeterminate x. Let D =

D(x)

_E

be a cubefree

_polynomial,

write D =

GH2

with

G, H

_E

squarefree

and

coprime.

We choose a cube root _p of D in some

_algebraic

closure of

Then the field K =

k(x, p)

is a

_purely

cubic

_function

_field;

it is the function

field of the

_plane

_{curve y3 -}

_D(x)

= 0 over k and is an extension of

degree

3 over

k(x).

We assume that the

_leading

coeffcient

sgn(D)

of D is a cube in k* =

k 1 101;

this _can

_always

be achieved

by replacing

k

by

_a suitable cubic extension of l~ if _necessary.

The

_ring

_{of integers functions}

or maximal order of is the

integral

closure C~ of

_k[x]

in .K. C~ is a

k[x]-module

of rank 3 that is

generated by

the

integrals

basis

_11,

_p,

_c~}

where c~ = so c~ is a cube root of

G2H.

The discriminant of is A =

-27G2H2.

- The unit group of

k’/k(x),

i.e.

the _group of units O* of the

_{ring 0,}

is an Abelian _group with torsion

_part

1~*;

it is

_equal

to k*

_{if deg(D)}

is not a

multiple

of 3 and infinite otherwise.

In the latter _case, the unit rank of is the rank of this _{group; it is}

independent

set of

_generators

of the torsionfree

_part

of C~* is a

_system

of

fundamental

units of

If C~* is

_infinite,

then it is

_possible

to choose _p in the field

of Puiseux series over

_k;

nonzero elements in have the form

a -

EM -

a-ixi.

The

_degree

valuation on

ex-tends

_canonically

to

_(and

hence to

_K)

via

_deg(a)

= m. We also set

lal

=

qdeg(a)

and

[aj =

Emo

a-ix’

(with 0

= 0 and

0).

Elements in K and in 0 are

_represented

in terms of the

_integral

basis

~1,

p, of If a = _{a +}

bp

_{+ cc,v E}

K~,

denote the

_conjugates

of _cx

by

a’ =

a+b¿p+c¿2w

and a" = where t is a fixed

_primitive

cube

root of

_unity.

Since i E k if and

_only

if q =- 1 mod

_3,

it follows that

_a’,

a" E

K _{if q -} 1 mod

_3,

whereas

_K,

but a’a" E

_K,

if

_{q - -1}

mod 3. In the latter _{case, set}

_{deg(a’) = deg(a’a") /2}

and

_la’l

=

qdeg(a’)

=

The norm of a is

N(a) -

= a3

+

b3GH2

+ c3G2H -

3abcGH E The above introduction enables us to _compare

purely

cubic function

fields with their number field

_analogues

and

_point

out similarities as well as differences between the two. We recall that a

purely

cubic number

field has the form K -

_{~( 3 D)}

with D =

GH 2

where

_{G, H}

E Z are

squarefree

and

_coprime.

While is

_always

a cubic

_extension,

_regardless

of the

_generator,

a

_purely

cubic function field K can have

representations

as an extension of some rational function field that are not

_cubic,

although

K/k

will

_always

have transcendence

_degree

1. Once a

purely

cubic

_{representation}

has been

_fixed,

the

_integral

basis maximal order

_C~,

discriminant

_A,

and the definitions of

_conjugates

and norm are

essentially

the same as in the number field

_setting.

_However,

while

_purely

cubic number fields are

complex

cubic fields and thus

always

have unit rank

one, the

corresponding

function fields can have unit rank

_{0, 1,}

or 2. The case of unit rank 1 is similar to the number field situation in many

_respects,

generally exhibiting large regulators

(where

the

_regulator

is half the

_degree

of the fundamental unit of

_positive

_degree),

small ideal class

_numbers,

and

an infrastructure on the set of reduced

_principal

fractional ideals that is

discussed in this paper.

Embedding K

into the field of Puiseux series is

akin to

_embedding

a

_purely

cubic number field into the

_{reals; however,}

the

valuation [ . I

is

_discrete,

i.e. nonarchimedian.

_{Consequently,}

the results on

ideal reduction and the infrastructure are somewhat

_simpler

and cleaner in

the function field

_setting.

The discussion of

_ideals,

their

_{decomposition}

into

_{prime ideals,}

ideal

bases,

and ideal

_{multiplication}

in Sections 2 - 5 is

_essentially

the same as

for number

_fields,

and the results in these sections hold for

_purely

cubic function fields of _any unit rank. In Sections 6

_(on

ideal

_reduction)

and 7

(on

the infrastructure in the set of reduced

_principal

_ideals),

we will restrict

every

integral

Every

f

is

_{generated by}

at most two elements

_{9, /J}

E

_K;

that

_is,

_f

=

(a0

₊

80

1

E

_(~~.

_{Write f}

_{= (B, ~).}

_{If f}

is

_generated

_{by only}

one

_{element 0,}

then

f

is

_{principal; write f}

=

(9).

Nonzero fractional ideals _are

k(x~-modules

of

rank

_3;

if

_{IA, p, vl}

is a of a fractional ideal

f, write f

=

[A, v] .

The discriminant

_{of f}

is the rational function

it is

_independent

of the choice of

_k(x)-basis

_{of f}

up to a factor that is a square in k*.

Henceforth,

all ideals

_(fractional

and

_integral)

are assumed to be nonzero,

so the term "ideal" will

_always

be _synonymous with "nonzero ideal" . The

product

of two fractional

_{ideals fl}

=

(81, 01)

and

f2

=

(82,

is the frac-tional ideal

_flfa

=

~Ble2, eu2, ~ie2, ~n2)~

Two nonzero fractional ideals are

_equivalent

if

_they

differ

_by

a factor that is a

_principal

fractional

_ideal;

this is

_easily

seen to be an

equivalence

relation. The set of

equivalence

classes is a finite Abelian _group under

_{multiplication}

of

_{representatives,}

the

ideal class _group of

_K/k(x);

its order h’ is the ideal class number of

_K/k(x).

An

_integral

ideal is

_primitive

if it is not contained in _any nontrivial

principal integral

ideal

_{( f )}

with

_f

E

_k[x].

The

_unique

monic

_polynomial

of minimal

_degree

contained in a

primitive

integral

ideal a is denoted

_by

L(a);

it is the

_greatest

common divisor of all

_polynomials

in a and can

_always

be

included in a

k[x]-basis

of a

(see

Section 3 of

[5]).

Similarly,

if a fractional

ideal f

contains

_1,

then 1 can

_always

be included in a

k~x~-basis

of

_f.

Primitive ideals and fractional ideals that contain 1 are in one-to-one

cor-respondence

as follows: to a

_primitive

ideal a =

(L(a),

a, /3~

corresponds

the

unique

fractional

_{ideal fa}

=

(L(a)-’)a

=

1, a/L(a), //L(a).

Conversely,

let f

=

(l, ~c, v~

be a fractional ideal

where p

=

(mo

₊

ryP +

m2w)/d

and

v =

(no

+ _{nip +}

n2w)/d

with _{mo, ml, m2, no, nl,}

n2, d E

k[x],

d

monic,

and = 1. Then

to f

corresponds

the

unique

prim-itive

_integral

ideal

c~

=

df

=

[d,

dv].

The

polynomial

d =

d(f)

is

unique

and is the denominator

_{of f.}

We have

_d(f)

=

L(af)

and

_L(a)

=

d(ta)-The norm of a fractional

_{ideal f}

=

[1,

(mo

_{+ +}

(no

+ _{nip +}

(mo,

Ml, m2, no, nl,

n2, d E

k[x]

jointly

coprime)

is

N(f) _

a(mln2 -

k(x)

where a E k* is chosen so that

N(f)

is monic.

some b _{E k*} and

_{N ( f l f 2 )}

= for fractional ideals

_{f i , f 2}

of O. If

a is an

integral

ideal,

then

L(a) ~

If in

addition,

a is

_primitive,

then

N(a) ~

L(a)2.

3. Prime ideals

Voronoi

_[8]

found bases of all

_prime

ideals of a

_purely

cubic number

field,

their _powers, and certain

_products

of their _powers. His

_results,

_easily

adapted

to the function field

_setting,

are stated here without

proof:

Theorem 3.1. Let P E

_k[x]

be an irreducible

polynomials.

Then the

prin-cipal

ideal

_(P)

_splits

into

_prime

ideals in C~ as

follows:

1. then

_(P)

=

p3

where p =

[P, p, w]

and

p2

=

p is called a

_type

1

_prime

ideal.

2.

_{If P}

_H,

then

_(P)

=

p3

where p =

[P,

_p,

c~~

and

p2

=

[P,

p,

pW].

p is

called a

_type

2

_prime

ideal.

3.

_If

_{P f}

_GH,

D is a cube

_{mod P,}

and

qdeg(P) ==

-1 mod

_3,

then D has a

unique

cube root X mod P in

k[x].

In this _case,

(P) =

_{pq where}

or

_{equival entl y,}

X 1

- mod PH

and

for

i E N. Note that

_{Xi, Y}

E

_k[x],

Xi3 =-

D mod

P’

and

Xi -

0 mod H

for

all i E

_{N. p}

and _q are called

_type

3

prime

ideals.

4.

If

P

_{

_GH,

D is a cube

mod P,

and

qdeg(P) =-

1 mod

3,

then D has

three distinct cube roots

_{X, X’, X"}

mod P in

_I~(x~.

In this _case,

_{(P) =}

pp’ p"

where

for i, j

E ~T.

_Analogous

_congruences hold

_for

_Xi

and

_Xi"

_(with

corre-sponding

A~

and

_A",

_{respectively),}

and similar bases can be

found for

i E _{N. p and q} are called

_type

4

_prime

ideals.

5.

_If

D is not a cube mod

_P,

then

(P) = p

is inert.

called a

_type

5

prime

ideal.

Note that

_{if q ==}

1 mod

_3,

then K does not contain _any

_type

3

_prime

ideals.

4. Canonical bases and ideal

_{multiplication}

In this

_section,

we introduce two

_special

_types

of

_k[z]-module

bases

(which

we call

_"triangular"

and

_{"canonical" ,}

respectively)

that lend

them-selves well to

_computation.

For

_{primitive ideals,}

such bases

_always

_exist,

and the two

_types

of bases will turn out to be the same. We describe how

to find such a

basis,

determine containment and

equality

of ideals

_using

triangular

bases,

and

_compute

the

_product

of two

_coprime

ideals

_using

tri-angular

bases. We also show that a nonzero is a

primitive

ideal

if and

_only

if it has a

_triangular

basis that is also

_canonical,

in which case

all of its

_triangular

bases are canonical. Several of the results in the next two sections as well as their derivations are

_analogous

to the number field

case discussed in

_[12],

so we omit some of the details here.

We define a basis of a

k~x~-module

in O to be

_triangular

if it is of the form

Let

_{{s, c~ /3}}

be a

triangular

basis of a

primitive

ideal a with a =

s’(u+p)

and /3

= Then _{sp E} a

_implies

s’ ~ s;

_similarly,

sw E a

_implies

s" ~ I

s. Since a is

_primitive,

we must have

_{gcd(s’, s")}

= 1.

Furthermore,

s =

sgn(s)L(a),

and ss’s" =

sgn(ss’s")N(a).

Triangular

bases

_provide

an _easy means for

_comparing

modules and

prim-itive ideals.

Lemma 4.1.

_{Let al =}

_{and a2 =}

_[52?-S2(~2+

1. _{al C a2}

_if

and

_{only if}

2.

_If

_al and _a2 are

primitive ideals,

then _al = _a2

if

and

only if

s 1 =

aS2,

si

=

a’s’ 2~

s" - 1

"

(a &#x3E; a’ &#x3E; a"

~&#x26;*)?

&#x3E;

Every

primitive

ideal in 0 has a

triangular

basis which can be

easily

be

found:

Lemma 4.2. Let a =

[L(a),

_p,

v]

be a

_primitive

ideal where _J.L =

mo +

= w2th

k[x].

Then a

has a

triangular

basis which can be obtained as

follows.

Set

Then

_triangular

basis

_{of a.}

Proof.

Since

_{N(a) ~}

_I

_L(a)~,

we have

s’s" ~

I

s. Let U =

(mon2 -

nom2)/s",

Y -

_a’mo

+

_b’no,

and W =

a’ml

₊

b’nl.

Then

U, V, W

E

l~~x~,

and if

a

= (n2p -

m2v)/s"

= U ₊

s’p

and

(3

=

a’j,L

₊ b’v = V ₊

W p

₊

s"w,

then

~ s, cx, ~3 }

is a basis of a. Since

_{~3 p}

E _a, we have s"

I

W H;

similarly, (3w

implies

s"

_I

_V,

and s

_I

WGH.

_{By expressing}

_ap and aw in terms of _{s, a,}

and

_,~,

we see that s’ ~ U.

Now

_{suppose s’}

and s" had an irreducible common divisor _{p E}

Then

_{p 2}

_{1 I}

_{W GH,}

so _{p ~}

I

W and hence

(p) ~ I

a,

contradicting

the

_primitivity

of a. So

gcd(s’, s") -

1 and t as

_given

above exists. Now

is,

a, ,C3 -

is the desired

_triangular

basis. D

We note that

_by

_part

2 of Lemma

_4.1,

all other

_triangular

basis

_(up

to constant factors in the basis

_elements)

are

given by

be two

_primitive

ideals

_given

in terms

_of

triangular

bases with

_gcd(‘

is a

_triangular

basis

_of

_ala2 where

Proof.

Let

_~s3,

+

_p),

+ w3 p +

c~) ~

be a

_triangular

basis of _{a1 a2}

and assume that are monic for i =

1, 2, 3.

Since _{al a2}

₉

_{al, a2,}

by

part

1 of Lemma 4.1

_I

_S37

_{sis2 ,}

_I

_s3,

_sls2

_I

_s3-

_Examining

_N(ala2)

shows that these divisibilities are in fact all

equalities.

The _congruences for

We call a

_triangular

basis

Is, s’(u+p), s"(v+wp+w)}

of a

k[x]-submodule

of O canonical if and

_only

if the

_following

conditions hold.

where _sG =

gcd(s, G)

and

sH =

gcd(s,

H).

It is

_easily

seen that all the

prime

ideals of

_types

1-4 and their

primitive

powers have canonical bases. Canonical bases

satisfy

a number of additional

divisibility

conditions:

Lemma 4.6. Let be a canonical basis

of

some

Then

Proof.

(4.5)

and

_(4.6)

are immediate _consequences of

(4.1) - (4.4). (4.7)

is

obtained

_{by multiplying}

_(4.2)

_{by w and subtracting}

_(4.4).

_Multiplying

_(4.2)

by

(4.3)

produces s

I (Huw -

Hv -

_{u2)(v - Hw2).}

From

_{(4.4), s I}

GH -Hvw - uv +

Huw 2.

_Multiplying

the latter

_by

2Hw - u and

taking

sums

yields s ~

+

Hw2) -

2GHw -

_{u(Hw3 -}

_G)),

so both

SISH

and

SH / s"

divide

_{v(v +Hw2) - 2GHw - u(Hw3 - G).}

Since

_by

_(4.1),

_S/SH

and

_sH/s"

are

coprime,

(4.8)

follows.

Multiplying

(4.4)

by

H,

(4.2)

by u

+

Hw,

and

taking

differences

_generates

_(4.9).

_Finally,

_(4.10)

follows

_directly

from

_(4.3)

and

_(4.6).

D

Canonical bases characterize

_{k[x]-inodules}

as ideals:

Theorem 4.7. A a in a

_primitive

ideal

if

and

only if it

has a

_triangular

basis that is canonical. In this _{case, every}

_triangular

basis

of

a is canonical.

Proof.

Let a be a in 0. If a is a

_primitive

_ideal,

then a has a

tri-angular

basis

_{Is, s~(~u-f-p),}

_by

Lemma 4.2.

_By

Theorems 3.1

and

_4.4,

_gcd(s/sGsH,

_GH)

=

1,

and the fact that a is closed under

multi-plication by

p and

implies

the rest of

(4.1)

and

(4.2) - (4.4).

Conversely,

containing

1,

all the above results can

_immediately

be

applied

to the latter: Theorem 4.8.

1.

_{Every fractional}

ideal

_containing

1 has a canonical

_basis,

i. e. a basis

of

5. Ideal

_squaring

In this

_section,

we describe how to find a canonical basis of the _square

of a

_{primitive ideal,}

_given

a canonical basis of the

original

ideal. We

give

a more detailed

proof

of the

_following

lemma as it is

slightly

different from

its number field

_equivalent

due to the nature of our

underlying

finite field

irreducible divisor

_{of s,}

but not s~. Then

_{(-1 :f: a)u}

mod _p, so

Now

_by

_(4.2),

Huw’ -

u2 -

Hv’

mod p, so

_{since j}

Thus,

gcd(2v’

-f- H(w’)2, s)

= 1.

In

_practice,

it is _easy to find a suitable

f by

trial and error.

f

= 0 _or

f

E k* is almost

_always

sufficient.

We now have all the tools to

_compute

canonical bases of ideal squares.

Theorem 5.2. Let be a canonical basis

of an

ideal a. Set

Proof.

Write a = _ala2a3 where _al is the

product

of

type

1

_{prime ideals,}

_a2 is the

_product

of

_type

2

_{prime ideals,}

and _a3 is the

_product

of

_type

3 or

Corollary

5.3. Let, be a canonical basis

tional ideal with canonical

_{basis I}

Proof.

Let be as in the

_previous

Theorem. ’

fractional ideal with canonical basis

-

-Example

5.4. Let

_k,

_G,

H be as in

Example

4.3 and let

We have

and the

_given

basis is canonical. We wish to

compute

a canonical of the

_primitive

ideal

_(F)f~

with

and

_replace

w

_by

thereby achieving

6. Reduced bases and ideal reduction

For the remainder of this _paper, we

_only

consider the situation where

q - -1

mod

_3,

so K has unit rank

_1,

and

_{char(k) &#x3E;}

5. We use the notation

of

_[5].

Let 0 = 1 + _mp + nw E K

_{with l,}

_{m, n E}

_k(x).

We define

where t

_{(g k)}

is a

_primitive

cube root of

_unity.

Then

If 0, 0

E K and a, b E

_k(~),

then

_Ça8+bØ

+

_b6,p,

_similarly

for the other

quantities

of

_(6.1).

_{Also 6a}

= = 0

and (a

= 2a.

For a fractional

ideal f

and an element 0 E

_f,

set

Lemma 6.l.

_{Let f}

be a

fractional

ideal

containing

1. Then

Nf(1)

is

finite.

An element 0 in a fractional

ideal f

is a minimum

in f

if

Nf(B)

=

k0;

that

is,

_Nf(8)

contains

_only

constant

_multiples

of

_{0. f}

is reduced if 1

_{E f}

and

1 is a minimum

_{in f,}

i.e. = k. An

integral

ideal a is reduced if a is

primitive

and

_(L(a)-’)a

is

_reduced,

or

equivalently,

L(a)

is a minimum in a.

Every

ideal

equivalence

class contains at least one and at most

_finitely

many reduced

representatives.

If f

is a fractional ideal and 0 E

_K*,

then it is _easy to infer from the definition of a minimum that 0 is a minimum

in f

if and

_only

if

_(B-1)f

is reduced. In

_particular,

an element 0 is a minimum in O if and

_only

if the fractional

_principal

ideal

_(0-1)

is reduced.

We summarize some

properties

of reduced fractional and

integral

ideals:

Lemma 6.2.

1.

_{If f}

is a

fractional

ideal

containing 1,

then

I

2.

_{If f}

is a reduced

_{fractional ideal,}

then

2. See Theorem 4.5 of

_[5].

3. See

_Corollary

4.6 of

_[5]

for the first

_inequality.

The second

_inequality

follows from

_Idl-1

4. Let 0 E

_Nf(1)

and set

_{A (0) = ((0 -}

_~(~).

Then

I

1. Let be a

of f.

Since

do2

E

_f,

there exists

a 3

by

3 matrix M with entries in

k(x~

such that

Taking

determinants and _squares on both sides

_yields

d2A(B) =

det(M)20(f),

therefore 1 &#x3E; 2 &#x3E;

_det(M)~.

So

_det(M)

=

A(0)

=

0,

implying 0

= 0’ = 0" and hence 0 _E k.

D

Let f

be a fractional ideal and let 0 be a minimum in

_f.

An

element §

_{E f}

is the

_{neighbor of}

0

_{in f if 0}

is also a minimum in

_f,

101

_1p1,

and for no

~~f,~)~)~~

and

_1"p’1

_([5]

uses the

_terminology

"minimum

adjacent

to

_B").

_By

Theorem 5.1 of

_{~5~, ~}

_always

exists and is

_unique

_up

to nonzero constant factors.

According

to

_[5],

the troronoi chain

_(0n)neN

of successive minima in 0 where

₀₁

= 1 and is the

neighbor

of

On

in ()

yields

the

entirety

of

minima in 0 of

_{nonnegative degree.}

This chain is

_{given by}

the recurrence

=

pn0n

where _{An is} the

neighbor

of 1 in the reduced fractional

prin-cipal

ideal fn =

(On-1)

(n OE N) .

The first nontrivial unit c =

Op+l

(p

_E

N)

encountered in this chain is the fundamental unit of

_Klk(x)

of

_positive

de-gree

(unique

up to nonzero constant

_factors).

Since the recurrence for the

Voronoi chain

_implies

= for _{m E}

No

and _n E

N,

{fl,

f2, ~ ~ ~ ,

fP}

is the

_complete

set of reduced

_principal

fractional ideals in K. We call _p the

_period

of E.

We will see later on that a _{process very} similar to the

_computation

of the Voronoi chain can be used to obtain from a nonreduced fractional ideal an

_equivalent

reduced one. For this _purpose, we introduce the

_concept

of a reduced basis of a

(reduced

or

nonreduced)

fractional ideal

f;

that

is,

a

Voronoi

_([9],

see also _pp. 282-290 of

_[2],

and

_[12]

for the

_purely

cubic

ver-sion)

essentially

described how to obtain the

_equivalent

of a reduced basis

of a fractional ideal in a cubic number field. A function field version for

reduced ideals was first

_given

as

_Algorithm

7.1 in

_[5].

_Here,

we

_give

a more

general

version of the method which includes the nonreduced case.

Algorithm

6.3.

Input: A, f.,

where

_{1,

_A,

_v}

is a basis of some fractional ideal

_f.

Output:

p, v where

{1, ~, v I

is a reduced basis of

f.

Algorithm:

3.2.

_Replace

I 1 1 3.3. If _

lq, 1,

replace

where a = _E k*. 4. While

_I

_1,

_replace

While

_I

&#x3E;

_1,

_replace

If f

is

_{reduced, then p}

is the

_neighbor

of 1

_{in f by}

Theorem 7.5 of

_[5],

so

_repeated

_application

of

_Algorithm

_6.3,

_beginning

and

_ending

with

_input

ideal f

= O

generates

the fundamental unit c of

_Furthermore,

it can be shown that in this

situation,

the conditions in

_step

_3.1,

the first

loop

of

_step

_4,

and

_step

6 cannot _occur, so these

_steps

can be omitted

(see [4]).

The _process of

_computing

from the reduced fractional

_{ideal f,,}

_(n

E

_N)

a

reduced basis of the next reduced fractional ideal = where

~c~ is the

_neighbor

of 1

_{in fn}

is referred to as a

_baby

_step.

Example

6.5. For illustrative _purposes, we

_compute

the fundamental unit E of an extension with an

_unusually

short

_period.

_{Let q}

be _any odd

prime

power

with q &#x3E;

7 and

q - -1

mod 3 and let

D(x)

=

G(x) _

~6 - ~

and

_H(x)

= 1. Then _p

= ~2

₊

O(x-3)

and w

= p2

= x4

₊

O(X-1).

We call

Algorithm

6.3 on the

_{input p}

= _p and v =

p2.

After

_step

2,

we

_{have p}

= -p2

and v = _p, and _we

proceed

_to

_step

3.2. We have

-~2,

so we

_{obtain p}

= _p and v =

-x2p

₊

p2.

Since

In.1

=

lpl

=

q2 &#x3E;

1

and

_117vl

_q4

we enter the second while

_loop

of

_step

4.

_Here,

=

L - x2 _ pj =

-2x2,

_so after the first iteration

Ec =

-x2p- p2

and

v = _p. Now

[

=

Ix2p - p2~

I

_1,

_{so we}

go on to

_step

5. We see

that - L(ttJ/2

=

-~4

~2~2,

_{so we}

have p

= -(x4

+~Zp+ p2)

and v =

x2/2

₊

p. The

inputs

to our next call of

_Algorithm

6.3 are =

~-~2

+ and =

(-~4 - ~2p

₊

The

_following

table shows the

_{complete computation}

of the Voronoi chain up to E. For

_simplicity,

certain constant factors

_(such

as -1 and

_1/2)

of the basis elements have been removed.

_{Here, f}

_i is the

_input

ideal of the i-th round of the

_algorithm.

At this

_point,

the

_input

ideal has denominator

_1,

hence

_f4

= O

and p

= 3.

The Voronoi chain _{up to} the fundamental unit E is

_{given by}

A reduced basis

_provides

an _easy means for

recognizing

whether or not the ideal

_generated

_by

this basis is reduced:

Theorem 6.6. Let

_{1,

_p,

_v}

be a reduced basis

of

a

fractional

ideal

f .

Then

f

is

_{reduced if}

and

_{only if}

_1p,1

&#x3E; 1 and &#x3E; 1.

Proof. By

(6.2)

and

_(6.3)

_1p,/1

1. If

_1p,1

_1,

then _p E

_Nf(1),

_{so f}

is not reduced.

_Similarly,

if

_maxfivl,

_1,

then

_by

_(6.2)

_lv’l

_1,

so

v E

_{Nf ( 1 )}

and

_{again, f}

is nonreduced.

Conversely,

suppose that

1p,1

&#x3E;

_1,

&#x3E;

_1,

and let 6 = 1 + mp -f- nv E with

_{l, m, n}

E

_k(x~.

_By

_{(6.1) ~~e~,}

1 and

_by

_(6.2)

1. Assume

_Imi

_lnl,

then

_In17vl,

so 1

In17vl

=

+

_{n17vl =}

_1,

_implying

_117vl

_

Inl

= 1. It follows that m = 0 and

H

&#x3E;

_1;

_{also Ill}

=

~~8 -

1. But then 1

= Inl

Invl

=

10 - 11

1,

a

contradiction. So

_Inl.

Suppose m ~

0,

then

_Inçvl

so 1

Iml

lçol

1,

a

contradiction.

_Hence,

m = n = 0

and 111

=

181

1,

implying 1

= 0 _E k. 0

Corollary

6.7. Let

_~1,

_p,

_vl

be a reduced basis

of

a

fractional

ideal

f.

Then

f

is nonreduced

_if

and

_{only if}

1 or

Iv

I

= 1.

Let f

be _any nonreduced fractional ideal and define a _sequence

of fractional ideals as follows.

, ....1 I..

and

_{11, v,,, I}

is a reduced basis of

fn.

Clearly,

all the

fn

are

equivalent.

Here,

the _process of

_obtaining

_f,,+,

_{from f,,}

is also called a

_baby

_step;

the

difference to the reduced case is that

by Corollary

6.7,

the

ideal f,,

is

always

divided

_by

an element of

_nonpositive

_degree,

whereas in the reduced _case, one divides

_by

the

_element

of

positive

degree.

We note that as in the

recursion for the Voronoi

_chain,

we

always

divide

by

_pn

_except

for one

special

case where we do not need

_Algorithm

6.3 to

_produce

a reduced

basis

_(see

_[4]):

Lemma 6.8. Let

_{11, M, vl}

be a reduced basis

of

a nonreduced

ideal f

with

fn

(6.4),

Corollary

that

_[

1 for all i E N that one of

[

and

[

is

always strictly

less

than

_1,

while the other is no

bigger

than 1. Therefore 1 and

yr/y ~

1,

where at least one of the

_inequalities

is strict.

_Furthermore,

_{on E f 1 implies}

so

IN(fl)1

and

IN(fl)ll/2,

where

again

inequality

holds in at least one of the two cases.

We claim that a finite number of

baby

_steps

applied

to a nonreduced

fractional ideal will

_yield

an

equivalent

reduced one:

Lemma 6.9.

_{Let f}

=

_{f 1}

be _a nonreduced

fractional

ideal. Then there exists

m E N‘ such

_{that fm is}

_reduced,

where

_fm

is as in

(6.5).

Proof.

From our above

observation,

[

[

for all n e N. If

no

(n

E

_N)were

_reduced,

then would be an infinite _sequence of

pairwise

distinct elements in

_{contradicting}

Lemma 6.1. D

Theorem 6.10. Let be such that

_fm

is reduced

_{and fn is}

not reduced

for

n _m, as in

(6.5)

for

n E N‘. Then

Proof.

If

_{f 1}

is

_reduced,

then m =

1,

_{so suppose}

f 1

is not reduced and set

dn

=

deg(N(fn))

for n _E N.

By

Lemma

6.8,

On

= _pn for 1 n m -

_2,

so

dn &#x3E;

+ 2 for 2 n m - 2 and

_dm-2

+ 1. Hence

_inductively,

d~-2 &#x3E; dl

+

_{2(m - 3)}

and

_dl

+ 2m -

_5,

so m

_{(5 -}

_dl

+

Since is not

_{reduced, by}

_part

4 of Lemma 6.2

By

part

1 of the same lemma

1,

so

_together,

we obtain or 0

Corollary

6.7,

together

with Lemma 6.8

_gives

rise to the

_following

ideal reduction

_algorithm.

The number of iterations of the while

_loop

in this

algorithm

is

_{given by}

Theorem 6.10.

Algorithm

6.11.

Input:

ft,

v where

_{{1, ~c, v}}

is a reduced basis of a fractional ideal

_fi.

Output:

p, v where

11,1-L,vl

is a reduced basis of a reduced fractional

ideal

_equivalent

to

_{f .}

Algorithm:

1. Set

_{tz=p, v=v.}

, - ,- . _

2.2. From the

_compute

a reduced

using

Algorithm

6.3.

7. The infrastructure of the

_principal

class

According

to Section

_6,

_every reduced

_principal

fractional ideal is gen-erated

_by

the inverse of an element of the Voronoi chain

(8n)nEN.

- For

fn

=

(8;1)

with

1

n _p

(where

_p is the

period

of the fundamental unit

E),

we define the distance of

_f,~

to be

_8(fn)

=

6n

=

deg(8n).

Then the dis-tance is a

_nonnegative

function on the set of reduced fractional ideals that

strictly

increases with n and is

_easily

seen to

_satisfy

the

_properties

for _{any n} E

_{{2,3,... p~,}

_where,

as

usual,

is the

neighbor

of 1 in

f n-1.

Here,

the last

_inequality

follows from the fact that

_by

Theorem 7.6 of

_[5].

It follows that for all n E N:

Let fi

=

(Oil)

and

_fj

= be _two reduced

principal

fractional ideals

(I

i, j

p)

such

_{that 6;}

_{-f- 8~}

_deg(E).

Then the

_product

ideal

_{fi fj}

is

_generally

not

_{reduced; however,}

there is a reduced

principal

fractional

ideal

_fm

"close to"

_it,

i.e. +

_6j,

and from

_(7.2)

m z I +

_j.

Shanks first observed this behavior for the set of

_principal

ideals of a real

quadratic

number field and coined it the

_{infrastructure}

of the

_principal

class

_[3].

More

exactly:

Theorem 7.1.

_{Let fi}

_{and fj}

be two reduced

_principal

_fractional

ideals with

= 1 and

Ji

+ 8~

deg(E).

Then there exists a reduced

principal fractional

ideal fm

which we denote

by fi

_fj

such that

6m

=

ai

Proof. Let f

=

jjj.

By

Lemma

_6.9,

there

_{exists 1b =}

_{1j;m E f}

such that

fm

=

(1j;-l)f

is reduced for some m E

_{f 1, 2,. p}.}

Then

_f"!1

so

A has even

degree),

so 8 ~ 2 -

deg(0).

D

Theorem 7.2.

_{Let fi}

be a reduced

principal

fractional

ideal

with 8i

deg(E)/2.

Then there exists a reduced

fractional

principal

ideal

fm

which we denote

_by

_fi

*

_fi

such that

8m

=

2Ji

+ 8 where 0 &#x3E; 8 ~

3(l -

deg(A)/2).

is reduced and 0 &#x3E;

_{deg(~) &#x3E;}

2 -

_deg(A).

Then

_6m

=

2c5i

+ c5

where 6 =

deg(~) 2013 deg(F)

satisfies the bounds of the Theorem. D

By

(7.2),

distances can be as

large

as

8(p) 1.

Since

by

Theorem 6.5 of

[5],

p =

O(q(deg(,::l)/2)-2),

the

quantities 6

in Theorems 7.1 and 7.2 are

gener-ally

logarithmically

small relative to the distances of the initial fractional

ideal(s).

In other

_words,

the ideal

_fm

is

_essentially

where one would

_expect

it to

_be,

_namely

_6m

is _very close

_{to b2}

+

_{respectively,}

26i. Furthermore,

fr",

can be obtained

_quickly:

Corollary

7.3.

1. Let

_fi,

_fj,

_f,",,

be as in Theorems 7.1. Then the number

of baby

_steps

required

to

_{compute fm from}

_fifj

is at most

_L3(deg(A)

+

_4)/8J.

2.

_{Let fi}

be as in Theorem 7.2. Then the number

_{of baby}

_steps

required

to

compute f."L from f

=

(p-l )fr is at

_most

[3(deg(A) +4)/8J.

Proof.

From the

_proof

of Theorem

_7.1,

we have

2 - deg(A)

in the situation of Theorem

_7.1;

_similarly,

_deg(N(f))

=

2deg(N(fi))

₊

3 deg(F) &#x3E;

2 -

_deg(A)

in the case of Theorem 7.2. The

corollary

now

follows from Theorem 6.10. 0

Note that we did not

_specify

how to

_multiply

two distinct fractional

ideals fi

_{and fj}

whose denominators are not

_coprime.

It is

_possible

to

develop multiplication

formulas for this

_{situation; however,}

the details are very tedious.

Instead,

we

_compute

a reduced fractional

principal

ideal _very

close

_{to fi}

*

fj

as follows.

_Begin

_by

_finding

the first reduced fractional ideal

fi-n

(0

n

i)

such that

_d(fj))

= 1. In many

applications,

such as the

_computation

of the fundamental unit

(or

the

regulator),

the

infrastructure is used in such a _way

that fi

is

fixed,

and

usually,

some or

all of the ideals

_{f i}

=

0, f 2, ... , fZ-i, fi

are

_precomputed

and

_stored,

so it is easy to find our desired ideal

- - - - -. - , - - , - - - . =

1 For two functions _{f (n), g(n)} defined on _N, we _say that _{f (n)} = _8(g(n)) if there exist _positive

constants _{c, d} such that _{cg(n) f (n) dg(n)} for _{sufficiently large n} E N, i.e. if _{f (n)} = O(g(n))

so

by

(7.1), -

ndeg(A)/2

and hence -n &#x3E;

J &#x3E;

2

-(n

+

₂₎

so *

fj

is within n

_baby

_steps

of the ideal

_fi

*

_fj.

n is

_generally

_very

small;

most of the

_{time, n}

= 0 or 1 will be sufficient.

Given two reduced fractional

_{ideals fi}

and

_fj,

it is now _easy to

_compute

the reduced fractional ideal

_fi

*

_fj,

or at least one that is

only

a few

baby

steps

short of

_fi

* This _process is called a

_giant

_step.

Note that a

_giant

step

does not use distances

explicitly.

Algorithm

7.4.

Input:

Reduced bases of two reduced

_principal

fractional

_{ideals fi}

and

_f~.

Output:

If

_fi

=

fj:

a reduced basis of a reduced

principal

fractional

ideal f

and

5 =

8(f) - 26(fi)

with 0 &#x3E; 8 &#x3E;

3(1 -

deg(A)/2).

a reduced basis of a reduced

principal

fractional ideal

f ;

also n E

N and 6 =

6(j) - 8( f i) -

with -n &#x3E; J &#x3E; 2 -

(n

₊

2) deg(A)/2

(n

= 0 if and

only

if

gcd(d(fi),

d(fj))

=

1).

Precomputed:

Reduced bases of a list of ideals

_{fi-i, ~ ~ ~}

with

_m

i-1

_{sufficiently large}

_(required

_only

_{if fi}

_{=1= fj}

and

_gcd(d(fi),

_{d(f~)) ~}

_1).

Algorithm:

1. 2. while 2.1.

_{Replace n by n}

+ l. 2.2.

_{Replace f i by}

3.

_Compute

canonical bases

_{of fi}

_{and fj}

_using

Lemma 4.2.

4. If

_{fi #}

_compute

a canonical basis

of f

_{= fifj}

using

Theorem 4.4.

If fi

_{= fj,}

compute

a canonical basis

of f

=

(F)ft

_using

Theorem

5.2,

where F is

_given

as in

_Corollary

5.3.

_Replace

J

_by

6 -

_deg(F).

5.

_Compute

a reduced basis

{1, ~, v}

of f using Algorithm

6.3. 6. While

_deg(p)

0

6.1.

_Replace

J

_{by J}

+

_deg(p,).

6.2.

_{Replace f by}

_(i.e.

_compute

the basis

_{11, M-1,}

_vIL-1

_1).

6.3.

_Compute

a reduced basis

11,

_p"

v}

of

f.

7. If

_deg(v)

_deg(17v)

= 0

7.1.

_{Replace J by}

J +

_deg(v).

baby

Thus, giant

travel

_through

the set of reduced fractional ideals than

_baby

_steps.

This fact can be

exploited

to

_compute

the fundamental unit E of

The naive _way to

_{compute E}

_(and

the method

_employed

in

_[5])

is to

_apply

baby

steps

to the ideal

_fl

= 0 until a unit is

_encountered,

thus

_obtaining

E =

Op+l

after

p

baby

steps.

Instead,

one can

apply approximately

_~

baby

steps

to tl

to find an ideal

_{f m}

with m ~ and

_subsequently

execute m

_giant

_steps

_{g, = g2 - gl} *

_{f ~, ... ,}

_{gm = gm-,} *

_jz,

each

resulting

in a distance

_jump

of

_{approximately}

m. Then the total advance

in distance is

_roughly

_{p ~}

_deg(E).

This reduces the run time from

order _{p to} order and it is

_likely

that further

_improvements

are

_possible;

for

_example,

clever search

_techniques

find the fundamental unit of a real

quadratic

function field in The

_difficulty

here is that one

needs to know a

_{good approximation}

of _p ahead of time.

Similar methods

_generate

the ideal class numb er h’ and hence the order h

_{= h’ deg(E)/2}

of the _group of k-rational

_points

of the Jacobian of

_K/k;

work on

finding h

is

currently

in _progress. We

point

out that h

is

_independent

of the transcendental element x and hence the

_particular

purely

cubic

_{representation}

of it is a true invariant of K.

We

_expect

that our results in

[5]

and in this _paper extend to

_arbitrary

cubic function fields - function fields of

curves

F(x, y) -

0 of

degree

3 in _y -

of unit rank 1 and characteristic different from 3. While the characterization of these extensions

_according

to unit rank will not be as

beautifully simple

as the one

_given

in Theorem 2.1 of

[5]

for the

_purely

cubic _case, much of the arithmetic may be

similar, particular

if one uses a basis of the form

_{1, p, w}

with _pw E as described

_by

Voronoi in the

number field case

(see

[2,

_pp.

108-112~ ) .

Furthermore,

it may be

possible

to use elements of

_Algorithm

6.3 in the case of unit rank

_2,

where the

two fundamental units

_correspond

to two different

_embeddings

of K into We also mention that

_purely

cubic function fields of unit rank 0

are

_currently

_being

_{investigated by}

M.

_{Bauer, currently}

a Ph. D. student

at the

_University

if Illinois at

_{Urbana-Champaign,}

who _gave an efficient

algorithm

for

_finding

the

_unique

reduced

_{representative}

in every ideal class

if D is

_squarefree,

i.e. the curve

representing

is

nonsingular

[1].

Finally,

it is as

_yet

unclear how to define and

_compute

a reduced basis in

the case of even characteristic.

Preliminary investigations

_suggest

that an

approach quite

different from the one

_given

in Section 6 is

_needed;

work on

not

_yet

been

_explored;

their arithmetic is

_likely

somewhat different from their

_counterparts

of characteristic different from 3.

Acknowledgments.

The author is indebted to Mark Bauer for

_pointing

out an error in the initial version of Theorem 3.1 and to Andreas Stein for some useful ideas that were

_incorporated

into Section 4. I also thank two anonymous referees for their careful

_reading

and

_{helpful suggestions}

for

improvement

of this _paper.

References

[1] M. _BAUER, The arithmetic of certain cubic function fields. Submitted to Math. _Comp.

[2] B. N. DELONE, D. K. FADDEEV, The theory of irrationalities of the third degree. Transl. Math. Monographs 10, Amer. Math. Soc., Providence _{(Rhode Island),} 1964.

[3] D. SHANKS, The _{infrastructure of} a real quadratic field and its applications. Proc. 1972 Number Theory Conf., Boulder _(Colorado) 1972, 217-224.

[4] R. SCHEIDLER, Reduction in purely cubic function fields of unit rank one. Proc. Fourth

Algorithmic Number Theory Symp. ANTS-IV, Lect. Notes Comp. Science 1838, Springer, Berlin, 2000, 151-532.

[5] R. SCHEIDLER, A. _STEIN, Voronoi’s _algorithm in _purely cubic _congruence function fields of

unit rank 1. Math. Comp. 69 _(2000), 1245-1266.

[6] A. STEIN, H. C. _WILLIAMS, Some methods for evaluating the regulator of a real _quadratic

function field. Exp. Math. 8 _(1999), 119-133.

[7] H. _{STICHTENOTH, Algebraic function fields} and codes. Universitext, Springer-Verlag, Berlin,

1993.

[8] G. F. _VORONOI, Concerning algebraic integers derivable from a root of an _{equation of} the third _{degree (in Russian).} Master’s Thesis, St. _{Petersburg (Russia),} 1894.

[9] G. F. _VORONOI, On a _{generalization of} the _{algorithm of} continued fractions (in Russian). Doctoral _{Dissertation,} Warsaw _(Poland), 1896.

[10] H. C. _WILLIAMS, Continued fractions and number-theoretic _{computations.} Rocky Mountain

J. Math. 15 _(1985), 621-655.

[11] H. C. WILLIAMS, G. CORMACK, E. _SEAH, Calculation of the regulator of a _pure cubic field.

Math. Comp. 34 _(1980), 567-611.

[12] H. C. WILLIAMS, G. W. DUECK, B. K. SCHMID, A rapid method of evaluating the regulator and class number of a pure cubic field. Math. Comp. 41 _(1983), 235-286.

Renate SCHEIDLER

Department of Mathematics and Statistics

University of _Calgary 2500 University Drive N.W.

Calgary, AB T2N lN4 Canada