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Eichler Orders

Miriam Ciavarella
Università degli Studi di Torino
Italy
miriam.ciavarella@unito.it

Marina Marchisio
Università degli Studi di Torino
Italy
marina.marchisio@unito.it

 

 

Introduction

The aim of this worsheet is to give an explicit description of some Eichler orders of an indefinite quaternion algebra B defined over Q, [V80], [CM2007]. Let we denote by  the Legendre symbol. Let we fix a representation of B as a pair {-dN,p},  where d is the discriminant of B (e.i. the product of a finite even number of  primes), N is a positive integer prime to d and p is a prime number such that:

• 

• 

for each

• 

for each odd prime factor q of N.

 

 

Given an indefinite quaternion algebra B of discriminant d, it is always possible to find a prime number p as above, su that B can be represented as {-dN,p}. Unfortunately it does not exist an interval in which we are sure to find it. For this reason we give a procedure primo in Maple which returns, if it exists, a prime number  p as above in  the possible range [MinBound,MaxBound]; if it returns the null value,  we can change the input  data MinBound,MaxBound and try again.

This means that B can be expressed as   where

Following Hashimoto [Has95] an Eichler order of level N of  B can be expressed as the Z-lattice  R(N)=with

where a is an integer number satisfying .  This construction provides a very useful tool for working with Eichler orders. However it has the limitation of not respecting the natural inclusion of an Eichler order of level M  in an Eichler order of level N for N dividing M.

Let q be a prime number not dividing the discriminant of  B; it is well known that there are two natural inclusion maps of  in R(N).  In [CT2007]  we provide a basis for a chain on Eichler orders in B. Our last procedure is based on this work and returns two bases: the first one is a basis of  an Eichler order on level Nq in B and the second one is a basis of the other copy of R(Nq) in R(N) .

 

Initialization

> 

restart:

> 

with(numtheory):with(padic):

Representation of B as a pair {-dN,p}

Procedure Definition

We give a new procedure primo in Maple for computing a prime p as in the introduction.

Input data are:

Discr:  the discriminant of the quaternion algebra B; it is a positive integer number, which is the product of a finite even number of  primes;

LevelN:  a positive integer prime to d;

MinBound: a positive integer number;

MaxBound: a positive integer number greater then MinBound.

Output data are:

or a prime number p contained in [MinBound,MaxBound], satisfying the conditions as in the introduction;

or a sequence of prime numbers  contained in [MinBound,MaxBound], satisfying the conditions as in the introduction;

or the null value. In this last case we try to change the parameters MinBound and MaxBound.

 

> 

primo:=proc(Discr,LevelN,MinBound,MaxBound)
local d,N,Md,MN,a,b,l,t,p;
if gcd(Discr,LevelN)<>1 then
        print("LevelN must be a number prime to the discriminant")
end if:
if MinBound>MaxBound then
print("MinBound must be < MaxBound")
end if:
d:=Discr/(2^ordp(Discr,2)):
N:=LevelN/(2^ordp(LevelN,2)):
Md:=factorset(d):
MN:=factorset(N):
for p from MinBound to MaxBound do
        if type(p,prime) and modp(p,4)=1 then
                a:=1:
                for l from 1 to nops(Md) do
                        if modp(p^((op(l,Md)-1)/2),op(l,Md))=(op(l,Md)-1) then
                                a:=a*1
                        else
                                a:=a*0
                        end if
                end do:                                                                                            b:=1:
                if LevelN>1 then
                        for t from 1 to nops(MN) do
                                if (modp(p^((op(t,MN)-1)/2),op(t,MN)))=1 then
                                        b:=b*1
                                else
                                        b:=b*0
                                end if
                        end do
                end if:                                                                                            if a*b=1 and gcd(LevelN*Discr,2)=1 then
                        print(p):
                end if:
                if a*b=1 and gcd(Discr,2)<>1 and modp(p,8)=5 then
                        print(p):
                end if:
                if a*b=1 and gcd(LevelN,2)<>1 and modp(p,8)=1 then
                        print(p):
                end if :
        end if
end do
end proc;


(3.1.1)

Examples

> 

primo(35,5,5,20);

(3.2.1)
> 

primo(35,3,20,5);

(3.2.2)
> 

primo(35,3,5,10);

> 

primo(35,3,5,20);

(3.2.3)
> 

primo(35,3,5,8*35*3);

(3.2.4)

Basis for a Eichler Order of level N in B={-dN,p}

Procedure Definition

We give a new procedure EichlerN in Maple for computing a basis of an Eichler Order of level N (=LevelN) of B. This computation is based on a result of Hashimoto [Has95].

Input data are:

Discr:  is the discriminant of the quaternion algebra B; it is a positive integer number, which is the product of a finite even number of  primes;

LevelN:  a positive integer prime to d, which is the level of the Eichler order;

p: a prime number satisfying the conditions as in the introduction. We can find it using the procedure primo;

Output data are  the elements of a basis of the Eichler order of level N in B.

 

> 

EichlerN:=proc(Discr,LevelN,p)
local e1,e2,e3,e4,a;
if gcd(Discr,LevelN)<>1 then
        print("Insert a level prime to the discriminant")
else
        for a from 1 while modp(a^2*Discr*LevelN+1,p)<>0 do
                a:=a+1:  
        end do:
        e1:=1:
        e2:=(1+j)/2:
        e3:=(i+k)/2:
        e4:=(a*Discr*LevelN*j+k)/p:
        print({e1,e2,e3,e4}):
end if
end proc;


(4.1.1)

Examples

> 

EichlerN(35,5,13);

(4.2.1)
> 

EichlerN(35,3,13);

(4.2.2)

Integera

Procedure Definition

We give a simple but useful procedure Integera in Maple for computing an integer a such that  .

Input data are:

Discr:  is the discriminant of the quaternion algebra B; it is a positive integer number, which is the product of a finite even number of  primes;

N:  a positive integer prime to d;

p: a prime number satisfying the conditions as in the introduction. We can find it using the procedure primo;

Output data an integer number a such that  .

 

> 

Integera:=proc(Discr,N,p)
local a;
if gcd(Discr,N)<>1 then
        print("Insert N prime to the discriminant")
else
        for a from 1 while modp(a^2*Discr*N+1,p)<>0 do
                a:=a+1:  
        end do:
        print(a):
end if
end proc;


(5.1.1)

Examples

> 

Integera(35,5,13);

(5.2.1)
> 

Integera(35,3,13);

(5.2.2)

Basis for a Eichler Order R(Nq) of level Nq in B={-dN,p}

Procedure Definition

We give a new procedure EichlerNq in Maple for computing a basis of an Eichler Order of level Nq of B. This computation is based on a recent result of Ciavarella-Terracini [CT2007].

Input data are:

Discr:  is the discriminant of the quaternion algebra B; it is a positive integer number, which is the product of a finite even number of  primes;

LevelN:  a positive integer prime to d, which is the level of the Eichle order;

p: a prime number satisfying the conditions as in the introduction. We can find it using the procedure primo;

a: an integer number such that  . We can find it using the procedure Integera;

q: a prime number, different from  p and  not dividing the discriminant of B.

Output data are a basis of the Eichler order R(Nq) in B and a basis of the twisted copy of R(Nq) in R(N).

 

> 

EichlerNq:=proc(Discr,LevelN,p,a,q)
local e1,e2,e3,e4,sol,x,y,c1,c2,f1,f2,f3,f4,g1,g2,g3,g4,c,cp,c4,AB;
e1:=1:
e2:=(1+j)/2:
e3:=(i+k)/2:
e4:=(a*Discr*LevelN*j+k)/p:
if gcd(Discr,LevelN)<>1 then
        print("Insert a level prime to the discriminant"):
        return:
end if:
if type(q,prime)=false then
        print("q must be a prime number"):
        return:
end if:
if q=p then
        print("Insert a prime number q different from p"):
        return:
end if:
if gcd(q,Discr)<>1  then
        print("Insert a prime number q not dividing the discriminant of the quaternion algebra"):
        return:
end if:
if gcd(q,Discr*p)=1 and modp(p^((q-1)/2),q)=(q-1) then
        for x from 0 to (q-1)  do
                for y from 0 to (q-1)  do
                        if modp(x^2-p*y^2+Discr*LevelN,q)=0 then
                                break:
                        end if:
                end do:
                if modp(x^2-p*y^2+Discr*LevelN,q)=0 then
                        break:
                end if:
        end do:
        c1:=(y-x) mod q:
        c2:=p^(-1) mod q:
        f1:=e1:
        f2:=-c1*e2+e3:
        f3:=-2*c2*(a*Discr*LevelN-x)*e2+e4:
        f4:=q*e2:
        g1:=e1:
        g2:=c1*e2+e3:
        g3:=-2*c2*(a*Discr*LevelN+x)*e2+e4:
        g4:=q*e2:
        print(B(R(LevelN*q))={f1,f2,f3,f4}):
        print(B(twistR(LevelN*q))={g1,g2,g3,g4}):
end if:
if gcd(q,Discr)=1 and modp(p^((q-1)/2),q)=1 then
        c:=(p-sqrt(p))/2 mod q: cp:=(p+sqrt(p))/2:
        f1:=e1:
        f2:=e2:
        f3:=e3-c*e4:
        f4:=q*e4:
        g1:=e1:
        g2:=e2:
        g3:=e3-cp*e4:
        g4:=q*e4:
        print(B(R(LevelN*q))={f1,f2,f3,f4}):
        print(B(twistR(LevelN*q))={g1,g2,g3,g4}):
end if:
end proc;


(6.1.1)

Examples

> 

(6.2.1)
> 


(6.2.2)
> 



(6.2.3)

References

 

[CT2007] Miriam Ciavarella and Lea Terracini, Some Explicit Constructions of Integral Structures in Quaternion Algebras, PrePrint, 2007.

 

[CM2007] Miriam Ciavarella and Marina Marchisio, Quaternion Algebras, Maplesoft Maple 11  [http://www.maplesoft.com], 2007.

 

[Has95] Ki-ichiro Hashimoto, Explicit form of quaternion modular embeddings, Osaka J. Math., 32  n. 3, 533-546,1995.

 

[V80] Marie-France Vigneras, Arithmétique des Algèbres de Quaternions, Lecture Notes in Mathematics, 800, Springer-Verlag, 1980.


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