Representation of prime powers in arithmetical progressions by binary quadratic forms

F

RANZ

H

ALTER

-K

OCH

Representation of prime powers in arithmetical

progressions by binary quadratic forms

Journal de Théorie des Nombres de Bordeaux, tome

15, n

o

_{1 (2003),}

p. 141-149

<http://www.numdam.org/item?id=JTNB_2003__15_1_141_0>

© Université Bordeaux 1, 2003, tous droits réservés.

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

Representation

of

_prime

_{powers in}

arithmetical

progressions

by

binary quadratic

forms

par FRANZ HALTER-KOCH

RÉSUMÉ. Soit 0393 une famille de formes

quadratiques

à deux

va-riables de même

_{discriminant,}

0394 un ensemble de

_progressions

arithmétiques

et m un entier strictement

positif.

Nous nous inté-ressons au

_problème

de la

_{représentation}

des

_puissances

de

nom-bres

_{premiers pm}

_appartenant à une

_progression

de 0394 _par une

forme

_quadratique

de r.

ABSTRACT. Let r be a set of

_{binary quadratic}

forms of the same

discriminant, 0394

a set of arithmetical

_progressions

and m a

_positive

integer.

We

_investigate

the

_{representability}

of

_prime

powers

pm

lying

in some

_progression

from 0394

_by

some form from 0393.

1. Introduction and notations

By

a

_form

_cp we

_always

mean a

primitive integral non-degenerated

binary quadratic form,

that

_is,

_cp =

aX2

₊ bXY ₊

Cy2

_E where

gcd(a,

b,

c)

=

1,

d =

b2 -

4ac is not _{a square,} and _{a &#x3E;} 0 if d 0. We

call d the discriminant of _cp. More

_generally,

_{any non-square}

_integer

d E Z with d - 0 mod 4 or d - 1 mod 4 will be called a discriminant.

Two forms E

_{7G(X, Y]}

are called

(properly)

_equivalent

if

cp(X, Y) =

0 (aX

+ +

_JY)

for some

a, /3, 7, b

E Z such that a6 -

_3-y

= 1.

For _any discriminant

_d,

we denote

by

H(d)

the

(finite)

set of

_equivalence

classes of forms with discriminant d. If cp =

aX2

₊ bXY ₊

Cy2

is _a form

with discriminant

_d,

we denote

by

(~p~

=

[a, b, c]

_E the

_equivalence

class of _p. For _any discriminant

_d,

we call

the

_principal

class of

A form _cp is said to

_represent

_(properly)

an

integer q

E

_Z,

if _q =

cp(x, y)

for _some

x, y E Z such that

gcd (x,

y) =

1.

_Equivalent

forms

represent

integers,

q

[p]

cp

representing

q.

This _paper is addressed to the

_{representation}

of

_prime

powers in

arith-metical

_{progressions.}

To be

_precise,

we shall derive criteria for a form _cp

(or

its class to

_represent

all

_prime

powers

(of

fixed

exponent)

pm

E b + aZ

for

_{given coprime positive integers}

a and b. If m = 1 and if genera _are

considered instead of individual forms or

classes,

the

problem

is solved

by

Gauss’ _genus

_theory.

A first result for individual classes was

_{proved by}

A.

_Meyer

_[9]

_using

Dirichlet’s theorem. A

_complete

solution for m = 1

and fundamental discriminants was

_{presented by}

T. Kusaba

[8]

(using

class

field

_theory).

The case of

_arbitrary

discriminants

(for

m = 1 and

2)

was settled

_by

P.

_Kaplan

and K. S. Williams

[7]

(using

_elementary

methods

and

_Meyer’s

_theorem).

In

_fact,

_{in this paper} we shall consider more

_generally

a set r c

_1-l(d)

(for

some discriminant d E

_Z)

and a set A C of arithmetical

progressions

(for

some distance a &#x3E;

2),

and we shall deal with the

_problem

whether every

prime

power

p’~

(with

fixed

exponent

m)

satisfying

E

A is

_{represented by}

some class C E r. We shall

_throughout

make use of

class field

_theory,

and in order to do _so, we will also formulate Gauss’ _genus

theory

in a class field theoretic

_setting.

In section

_2,

we

_gather

the _necessary facts from genus

_theory

and class

field

_theory

in a form which is suitable for our _purposes. In section

_3,

we

formulate and prove the main results of this paper.

2. Class field

_theory

_{and genus}

_theory

The main references for this section are

[2]

and

[1],

but see also

[3]

and

[4].

For a discriminant

d,

we set

Let

_{C+ (Rd)}

be the Picard group of

Rd

in the narrow sense

(that

_is,

the group

of invertible fractional ideals modulo fractional

_principal

ideals

_generated

by totally

positive

elements).

If _cp =

aX2

+bXY+cY2

_E

Z[X, Y]

is _a form

with discriminant d and a &#x3E;

_0,

then

is a

_primitive

invertible ideal with norm

(Rd :

c~)

= _a.

Every

class C E

_1-l(d)

contains a form _p = ₊ bX Y ₊

Cy2

with a &#x3E;

_0,

and the

assignment

cp t-t _c~ induces a

_bijective

_map

For an invertible ideal a of

_Rd

we denote

by

[a]

E

C+ (Rd)

its class

_(in

the

narrow

sense).

Gauss’

composition

is the group structure on

_1-l(d)

for which

0d

is an

isomorphism,

and

Id

= is the unit element of

71 (d).

For a class C E

_1-l(d)

and _q E

_N,

we have C -3 _q if and

_{only if q}

=

N(a)

for some

primitive

ideal a E For a form _cp =

aX2

₊ bXY ₊

Cy2

_E

Z[X, Y],

we denote

_by

_§3

=

aX2 -

bXY ₊

Cy2

its

conjugate

(or opposite)

form,

and for a

_quadratic

surd a = u +

vU2

E we denote

_by

a = _{v, -}

vU2

its

(algebraic)

conjugate.

Conjugation

induces

inversion,

both on

H (d)

and

G+(Rd)

(that

_means,

[cp]

= for

every form cp, and

~a~

=

[a]-’

for

every invertible ideal

a).

For every class C E

_3-l(d),

C and

C-1

represent

the same

_integers.

A

prime

power

p"’~

with

p f

d is

represented by

some class C E

_1-l(d)

if and

only

_{if (~)}

= 1. In this _case,

pRd

=

pp

for some

_prime

ideal _p of

_Rd

such

that

_p,

and if C = then C and

C-1

_are

precisely

the classes

from

_{1-£ (d)}

_{representing p"’~.}

e

Associated

with a discriminant

_d,

there is an abelian field extension

together

with an

_isomorphism

having

the

_following

_properties:

1’

_Kd/Q

is a Galois extension which is unramified at all

_primes

_{p ~}

doo and whose Galois _group is

_{given by}

the

_splitting

_{group extension}

where and T acts on

by

2. For a class C E

_1-l(d)

and a

_prime

_p with _p

j’

d,

we have C - _p if and

only

if

_ad(C)

is the Frobenius

_automorphism

of some

prime

divisor

of p

in

Kd.

Let

_Ka

be the maximal

_absolutely

abelian subfield of

_Kd.

Then

Gal(Kd /K) )

= and there is _an

isomorphism

The field

_Kd

is called the

_ring

class

_field,

the field

_K~

is called the genus

fields,

the cosets

C1-l(d)2

are called the _genera and

_1-l(d)2

is called the

_principal

_genus of discriminant d.

explicit generation

Kd

given

[5]

p1, ... , pt

be the distinct odd

_prime

divisors of

_d,

set

Then we obtain

we define the reduced discriminant d* associated with d

by

For a E

N,

we denote

by

the field of a-th roots of

_unity

and

_by

the Artin

_isomorphism

for

_oa)

_/Q,

that

_is,

for a

_prime

p t

a,

8a(p

+

_aZ)

is the Frobenius

_automorphism

of the

prime

divisors of _p in

If d is a

_{discriminant,}

then d*

I

d,

hence

~~~~ ,

and d* is the

smallest

_{positive integer}

divisible

_by

all

_prime

divisors of d and

_satisfying

K~

C

Qt~* ~ .

If m E Z and

gcd(m, d) =

1,

we consider the Kronecker

symbol,

defined

_by

if m = where e _E

{0,1},

{3

_E

No,

is odd and

(d )

is the Jacobi

_symbol

_(for

details see

[6],

Ch.

5.5).

The Kronecker

symbol

(~)

depends only

on the residue class m + d*Z E

_{(Z /d*Z) ,}

and

is a

quadratic

character with the

following

_property:

If C E

_{~-l (d),}

k E

_Z,

_{gcd(k, d)}

= 1 and C -

k,

then

(~)

= 1.

Indeed,

observe that C --~ 1~

_implies

C =

~k, b, c~

for _some

_{b, c}

_E

_7~,

and since

d = b~ - 4kc,

it follows that

_{(~) =}

_{(-r) = 1.}

We define

"

If d* - 0 mod

_4,

we define

and if d* - 0 mod

_8,

we define

Then the vector of _{genus characters}

is defined

_by

its

_components

as follows.

By

its very

definition,

for a

_{prime p}

~’

d the Frobenius

_{automorphism /3d*}

(p+

d*Z) I Kd

is

_uniquely

determined

_by

_{its genus} character values

_cpd(p

+

d*Z)

E We have

and

From the class field theoretic

_{description of Kd,}

_Kd

and

_Q(d.)

we derive

immediately

the

_following

two assertions 3. and 4. which are

usually

quoted

as the main theorems of Gauss’ _genus

theory.

3. The _group

consists of all residue classes _{p +} d*Z E

_{generated by primes}

_p

such that

_{p f}

d and C -~ _p for some C E

_1-l(d).

4. The _{map :}

_(71ld*71)X

-~

1-l(d)/1-l(d)2,

defined

_by

is a _group

_epimorphism,

and

Ker(wd)

=

Ker(cpd).

In

If p

prime,

p f

1-l(d)

p,

Consequently,

the genus

representing

p

depends only

on the residue class

p + d*Z.

Meyer

[9]

proved

that if a class C E

_1i(d)

_represents

some

_{prime p}

{

d in a

coprime

arithmetical

_progression,

then it

_represents

_infinitely

_many

_primes

from this

_progression.

We shall need the

_following

refinement of this result.

Proposition

1. Let d be a

_{discriminant, a, b}

E

_{gcd(a, b)}

= 1. Let

Co

E

_1-l(d)

be a class

_representing

some

_prime

_po E b -f- a7G with po

~’

d. 1. The set

_of

all

_primes

_p E

b+aZ

represented by Co

has

_positive

Dirichlet

density.

2. Let S2 C

_1í(d)

be the set

_of

all classes

_{representing primes}

p E b + aZ

with _p

_~’

d. Then

_f21-l(d)2

= S2

(that

_means, S2 consists

of full

genera).

Proof.

We may assume that d*

a, gcd(d, b)

= 1

and (4)

= 1

(otherwise

we

replace

a

_by

ad* and consider all residue

classes

₊

ad*Z,

where (§)

= 1

and

- b mod

_a).

Since we have

For a

_prime

_p E

_{b + aZ,}

we have

Co 2013~

_p if and

only

if is the

Frobenius

_automorphism

for a

prime

divisor

of p

in

_Kd.

_By

Cebotarev’s

theorem,

this set has

_positive

Dirichlet

_density.

A class C E

_{1l ( d)}

_represents

some

_prime

_p E b + aZ if and

_only

if

Qd(C)IKd

=

ad(Co)IKJ.,

and since

_Gal(Kd/Ka)

=

ad(1l(d)2),

this is

equiv-alent to C E

Co1-l(d)2.

Hence we obtain Q = =

fl1i(d)2.

p

3. The main results

For a discriminant

d,

we denote

by

H2 (d)

the

_{2-Sylow subgroup}

and

_by

the odd

_part

of

_3-L(d),

so that

1-l(d)

=

1£2 (d)

_x

1-l’(d).

Theorem 1. Let d be a

discriminants,

m an odd

_{positive integer}

and Q C

1-l(d)m

a set

_of

classes

_satisfying

_n1-l(d)2m

= Q. Then there is _a subset

A c

Xd

such

_{that, for}

_every

_prime

_p

_{

_{d, if p"L}

+ d*Z E 0

then C -

_{P"’’ for}

some C E S2.

Proof.

Suppose

Q =

no,

where

_Sto

C and set A = _C

Xd.

Let _p be a

_prime

such that _p

{

d and

_p"z

+ d*Z E A. Since m is

odd,

we

obtain _{p +} E

_Xd

and

_wd(p

+

_d*Z)

=

cvd(p"’’

₊

d*Z)

= for _some

class

_{Co E S2o.}

Hence there exists some A E

_1-l(d)

such that

COA2 --7

p, and

if C = then C - and C _E = SZ. D

The

_assumption

_nll(d)2m

= S2 made in Theorem 1 is very restrictive. But as we shall see in Theorem

_2,

it is _necessary. We

_investigate

its effect in

the

_special

case Q =

(simple)

Proposition

2 remains true if we

_replace

1-l(d)

_by

_any finite abelian

group.

Proposition

2. Let d be a

discriminant,

m an odd

positive integer

and C E

_1-l(d)

a class

_satisfying

{C,C-1~~-l(d)2"~ _

{C,C-1}.

Then we have

~’(d)"’~ _ (I),

and either

_?-C2(d)2

=

{1}

_or

C~

= I and

3~2(d) _ (C)

_x

~-l2(d),

where

_~L2(l.~)2

=

{1}.

Proof. By assumption,

and since =

1-l2(d)2

_x

~-l’(d)"’~,

_we obtain

tl’(d)m

=

{1}

and

~3-l2(d)2~

2.

_Suppose

that

_1í2(d)2

=

(A 2)

for some A _E

~-l2(d)

with

A4

_{= I, A2 ~ I.}

Then

CA2

=

CA 2m E

{C,

C-1},

hence

CA2

=

C-’

and

therefore

C4

= I. 0

Now we formulate our main results

(Theorems

_2,

3 and

4)

which will be

proved

in a uniform _way later on.

Theorem 2. Let d be a

discriminant, let

a and m be

_{positive integers,}

and

let r C

_H(d)

and A C be _any subsets.

_Suppose

that

_for

_every

prime

p

(except

possibly

a set

of

Dirichlet

density

zero)

the

following

holds:

If

_(~)

= 1 and

p"z

_{+ E}

~,

then C -~

pm for

_some C _E r.

Let S2 be the set

_of

all classes C E

_representing

some

_prime

_power

pm

such that _p

_f

d and

_P"’W-

a7G E

A,

and assume that r C f2. Then

where

r-1

=

{C

_E

3-l(d)

I c-1

_E

r}.

In

particular,

S2 consists

of full

cosets

modulo

_{1-£ ( d) 2m .}

From a

qualitative point

of

_view,

Theorem 2 asserts that either r is

large

or

1-l(d)2m

is small. This will become

_plain

in Theorem

4,

when we

will consider the case

irl

= 1. The

subsequent

Theorem 3

generalizes

[7],

Theorem 1.

Theorem 3. Let

_assumptions

be as in Theorems 2. Let K E and

k E 7G be such that K ~ k and

_gcd(k,

_ad)

= 1.

Then

_for

_every

_prime

_p

_satisfying

_(~)

= 1 and ₊ aZ _E there

exists some C E 1"2 such that

Theorem 4. Let

_assumptions

be as in Theorem

_2,

and _suppose in addition

that r

_{f Cl}

consists

_of

a

_single

class. Then the

following

holds.

1.

_{I n}

I =

_{11£ ( d) 2m}

_I

2.

2.

_Suppose

that m =

2tm’,

where _t &#x3E; 0 and rrt’ _E ~‘I is

odd,

and let

3.

_Suppose

in addition that m =

_1,

and let

A’

be the _set

_of

all residue

classes _{p +} d*Z E

Xd

of

primes

p such that p -I- aZ E 0. Then we have

= 1 and _C

Remark.

_Kaplan

and Williams

_[7]

considered the case m =

1,

r =

{C}

and A =

{b

+

_a7G}

for some b E Z such that

_{gcd(a, b)}

= 1.

They

assumed

moreover that a is even and that every

prime

_p E b + aZ with _p

_{

d is

represented by

C. Then _every

_{prime p}

E b + aZ with p

f

d

satisfies (~)

= 1.

Therefore it follows that C

Q(a) ,

hence

Q(d*)

C

Q(a)

and d*

_I

_a,

since a is even.

Proof of

the Theorems. Let

Ao

C

_{(Z/aZ) ’}

be the set of all residue classes

p + aZ

of

_{primes P}

such

_{that (4)}

= 1 and

p’n

+ aZ _E ~. We

may assume that

A =

Am

C

_{(Z / aZ) x}

_(the

other residue classes of A are of no

interest).

Let

ro

be the set of all classes C E such that C’"L E I~e Since r c

by assumption,

we have r

rm.

For the same _reason, S2 =

00’,

where

S2o

is the set of all classes C E

_{1£ (d)}

_representing

some

_prime

_p

satisfying

_p

f

d

and E

Do.

Now

_S2o

= consists of full

genera

by Proposition

1,

and therefore S2 =

no

=

flo1-l(d)2m

=

n1£(d)2m.

If C E

S2,

then C =

Co

for _some

Co

_E

S2o

and

(by

Proposition

1)

the set of all

_primes

p such that p + aZ E

_Ao

and

_{Co -+ p}

has

_positive

Dirichlet

_density.

Hence the set of all

primes

p such that

pm

+ aZ E 0

and C -3 p has

positive

Dirichlet

density,

too. Therefore there exists some

C’

E r

_representing

a

_prime

_power which is also

represented

by

C,

hence C E

_{C’,

C’-i}

c r U

r - 1,

and Q c r u

r-1

follows. The other inclusion is

obvious,

since r and

r-1

_represent

the same

_prime

_powers. This

_argument

completes

the

_proof

of Theorem 2.

For the

_proof

of Theorem

_3,

let _p be a

_prime

satisfying

(~)

= 1 and

pm

+ aZ E Let _po be a

_prime

satisfying

_{p -}

kpo

mod ad. Then

and

_{pm -}

_k’p’

mod a

_implies

_{po +} aZ E

Ao,

whence

_{Co -}

Po for some

Co

E

ro.

Let

_Ci

E

_71(d)

be such that _p. Then

_Cm

_pm,

and therefore

C1

=

KCOA2for

some A _E

N (d) 2,

which

implies

Or

=

Kmc,

It remains to _prove Theorem 4.

_Suppose

that r = and C =

Cr.

Then

and therefore

2~~ts-t-1~~),

where

_((r))

=

ma,x{r, 0},

the assertions 1. and 2. of Theorem 4

follow.

If in addition m =

_1,

then

clearly

IWd(Â’)1

= 1.

Also,

for

every

prime

p

with

_(~)

=

1,

the residue class _{p +} d*Z is

uniquely

determined

by

_{p +} aZ.

Therefore

Cebotarev’s

theorem

_implies

C

Q(a)

0

References

[1] H. COHN, A Classical Invitation to Algebraic Numbers and Class Fields. Springer, 1978.

[2] D. A. _Cox, Primes _of the _form x2 +

ny2.

J. Wiley, 1989.

[3] F. _{HALTER-KOCH, Representation of primes by binary quadratic forms of} discriminant -256q and -128q. Glasgow Math. J. 35 _(1993), 261-268.

[4] F. HALTER-KOCH, A Theorem _{of Ramanujan Concerning Binary Quadratic} Forms. J. Num-ber _Theory 44 _(1993), 209-213.

[5] F. HALTER-KOCH, Geschlechtertheorie der _{Ringklassenkörpers.} J. Reine _Angew. Math. 250

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