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OrdersMiriam Ciavarella
Universit\303\240 degli Studi di Torino
Italy
miriam.ciavarella@unito.itMarina Marchisio
Universit\303\240 degli Studi di Torino
Italy
marina.marchisio@unito.it
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Introduction</Font></Text-field>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 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 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: 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 each 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 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 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 where 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Following Hashimoto [Has95] an Eichler order of level N of B can be expressed as the Z-lattice R(N)=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with 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 a is an integer number satisfying 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. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EiUkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGMS8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYnUSNOcUYnRi9GMkY0RjcvRjVRJ25vcm1hbEYnRkJGQg== 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) .
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Initialization</Font></Text-field>restart:with(numtheory):with(padic):
<Text-field style="Heading 1" family="Arial" layout="Heading 1"><Font family="Arial">Representation of </Font><Font bold="true" style="Text" italic="true" size="18">B</Font><Font family="Arial"> as a pair </Font><Font bold="true" style="Text" italic="true" size="18">{-dN,p}</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Procedure Definition</Font></Text-field>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;
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Examples</Font></Text-field>primo(35,5,5,20);primo(35,3,20,5);primo(35,3,5,10);primo(35,3,5,20);primo(35,3,5,8*35*3);
<Text-field style="Heading 1" family="Arial" layout="Heading 1"><Font family="Arial">Basis for a Eichler Order of level </Font><Font bold="true" style="Text" italic="true" size="18">N</Font><Font family="Arial"> in </Font><Font bold="true" style="Text" italic="true" size="18">B={-dN,p}</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Procedure Definition</Font></Text-field>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;
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Examples</Font></Text-field>EichlerN(35,5,13);EichlerN(35,3,13);
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Integera</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Procedure Definition</Font></Text-field>We give a simple but useful procedure Integera in Maple for computing an integer a such that 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 .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;
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Examples</Font></Text-field>Integera(35,5,13);Integera(35,3,13);
<Text-field style="Heading 1" layout="Heading 1"><Font family="Arial">Basis for a Eichler Order </Font><Font bold="true" style="Text" italic="true" size="18">R(Nq) </Font><Font family="Arial">of level </Font><Font bold="true" style="Text" italic="true" size="18">Nq</Font><Font family="Arial"> in </Font><Font bold="true" style="Text" italic="true" size="18">B={-dN,p}</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Procedure Definition</Font></Text-field>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;
<Text-field style="Heading 2" layout="Heading 2"><Font family="Arial">Examples</Font></Text-field>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
<Text-field style="Heading 1" family="Arial" layout="Heading 1"><Font family="Arial">References</Font></Text-field>[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, Arithm\303\251tique des Alg\303\250bres de Quaternions, Lecture Notes in Mathematics, 800, Springer-Verlag, 1980.
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