Rev.R.Acad. Cienc.Exact.Fis.Nat. (Esp) Vol. 94, N." 3, pp 417-421, 2000 Monográfico: Contribuciones al estudio algorítmico de problemas de moduli aritméticos
POLYNOMIALS OF GALOIS REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES (elliptic curves/galois groups) A M A D E U R E V E R T E R * and
NURIA
VILA**
* Departament de Matemàtiques. I.E.S. Bellvitge. Avda America, 99. E-08907 L'Hospitalet de Llobregat. Spain. E-mail:
[email protected] ** Departament d'Algebra i Geometría. Facultat de Matemàtiques. Universitat de Barcelona. Gran Via de les Corts Catalanes, 585. E-08007 Barcelona. Spain, e-mail:
[email protected]
ABSTRACT We construct polynomials with Galois groups the images of mod p Galois representations attached to elliptic curves. Explicit polynomials are computed for each subgroup of GL2(F3) and GL2(F5) that appears as an image for elliptic curves without complex multiplication and with conductor .
(i)
PwB 3(^0) - GL2(F3). We remark that 1 IB is the modular curve XQ(1 1). The polynomial obtained by using Theorem 2.1, is given in table 23b [3].
(ii)
PI4C3(GQ) =
á 16
1
0
Let P = (x, y) be a non-trivial 3-torsion point, we have the relations VJ/14C
^
{AX+ 1)(12Z-25)(144Z2 +
2304 / = Ax^ - g^x
+ 264X+ 1849). By Theorem 1.1, the polynomial is the quadratic factor.
Let us consider {1, x, x", x^, y, xy, x^y, x^y} as a ^-basis of the vectorial space K{x, y). Then, the characteristic polynomial of (iii)
PMA,3(GQ) =
m^^y : K{x, y) ^ K(x, y) a I—> a • (x -I- y).
VJ/14A
_
2304
is the characteristic polynomial of the matrix
( 1 2 X - 1)(576X^ + 4 8 X - - 596X + 625).
0 0 0
1 0 0
0 1 0 S2
82
48
^3
2
-^3
-82
0
12
3^3
^2
0
si
d 48
12
3^3 7,d
^2^3
12
0 0 1 0 4 0 gi
1 0 0 0 0 0 0
0 1 0 0 1 0 0
3^3
48
gi
0 0 1 0 0 1 0 ^2
2
0 0 0 1 0 0 1 0
By Theorem 1.1, the polynomial is the factor of degree 3. (i^)
which defines a surjective modp Galois representation of GQ(7^), for all p (cf. [5], § 63), we obtain the polynomial with coefficients in Q(T) computed in the table 23b of [3].
3.
POLYNOMIALS FOR PE,piG^), /? = 3, 5
In this section we will give examples of polynomials whose Galois groups over Q are the images PE,P(GQ). In [2, Theorem 3.2] it is determined the Galois group Ga.\{Q(E[p])/Q) for all the elliptic curves E defined over Q without complex multiplication with conductor A^ < 200 and for all primes p. Now, we will give a polynomial for each subgroup of GL2(F3) and GL2(Fg) that appears as Galois group.
=
0
1
Vpl4£ _
^
^
2304
(4X + 25)
(1728X^ - 10800^2 - 521820X - 2679769).
If p^ 3 is surjective, this characteristic polynomial has Galois group p¿- 3(G^) = GL2(F3). In particular, if we take the generic elliptic curve Ej : y^ = 4x^ - Tx - Z
P HE, 3(GQ)
By Theorem 1.1, the polynomial is the factor of degree 3. (V)
Pm3(Go) = (^Q
^j.Since4>r = ( X - ^ ) - ^ 3 ,
with 4^3 an irreducible polynomial over Q of degree 3, we can take the basis {P, Q} of E^^\3] with P = {j2' ^5) and Q = (x, y), where x is a root of ^ 3 . The matricial expression of the image of the representation tells us that any 3-torsion point different from ±P is conjugated with Q. Hence, {(X + # . - ( T G G Q } =
x,±j,.:(x,±y,)6£^«^[3],x,;^
12
So, the decomposition field over Q of Irr(x + y, IS
Û({x,±j,},.,2,3) ^ Q(E-^'^'[3]).
420
Rev.R.Acad.Cienc.Exact.Fis.Nat. (Esp), 2000; 94
A. Reverter et al. But we can check that the polynomial Irr(x + y, Q) is ^
5 ^ 6
+
30845 432
^
(295245X'° + 98415X' - 112193IX** + + 3595428X' + 260253X^ + 54675X= +
397015 , 1296
+ 293544X'' - 693360X-' + 912627X2 -333516X+55049).
37960175 , 735364625 X^ X + 20736 373248
By Theorem 1.1, we can take the factor of degree 10.
47376998675 8957952 (iv)
PIIC.5(G^Q) =
* * 0 1
which has the dihedral group Dg
as 0 Galois group over Q. So, Q{{x¡ ± 3',},= i.2,3) = = Q(E^°'^[3]), and the above polynomial is the one we are looking for. (vi)
p,,c. 3(GQ) = (l
1
VJ/llC^
531441
(45X2-F 4575X 4- ii6279)
(243X5 + 21060X4 - 2063205X^ - 322004880X2 - 13790509365X -
J , in the Fj-basis of £'^«^[3]
- 198101488289) (243X5 - 45765X'' - 15650955X^ -
847
r-\ /175 686 , .343V-7 I Q = { ' ./21 2 ^ /' '^ V 4 9
P=
- 1358064135X2 - 48900953415X - 644288081042).
We have
By Theorem 1.1, since the Galois group of either of the irreducible factors of degree 5 is the Frobenius group FJQ C d j , of order 20, we can take either of these polynomials.
Q(£^'^^[3]) = Q 3 4 3 V - 7 ,
So, the polynomial is I r r ( v ^ + x p 7 , Q) = X* + 20X2 + 16. (b) (i)
(ii)
P99D. 5 ( < ^ Q ) =
0 *
^Pf " = (X + 14)(X + 47)(5X2 - 25X - 241)
PioB. SÍGQ) = GL2(F5). We remark that 20fi is the modular curve Xo(20), and the polynomial is given in table 23b of [3]. PIIS.5(GQ)
1
0
0
*,
1 (3X- 14)(3X-47)(45X2+75X-241) 531441
(X'* - 49X^ + 1221X2 - 7699X +19081) Let {P, Q} be a Fj-basis of £^'^''[5] such that the image of the representation has the previous matricial form. We take P = (-14, 33.J-Ï) and Q = = (jc, 3^), where Q is a 5-torsion point with x a root of one of the factors of ^f^ of degree 4. So, we can choose [5 \2
(SIX'' + 189X' + 1026X2 + 3954X + 9391) (SIX^ + 1323X^ + 10989X2 + 23097X + 19081). By Theorem 1.1, we can take either of the factors of degree 4. (iii)
±1 0
(X" - 7 X ^ + 1 1 4 X 2 _ j 3 j g j ^ 939J)
p^5.
"¥I I B _
(V)
PMA.5(GQ)
'
1
*
0
*
1 (3X - 2)(3X + 1) 531441
33 J 5 10
'3267--
6534,/5 25
Then,
3, J
6534J5 3261-25
Q(£[5]).
Since [Q(£''''^[5]) : Q] = 8, the irreducible polynomial over Q with decomposition field Q^£.99Dp] is the polynomial of degree 8
A. Reverter et al.
Rev.RAcad.Cienc.Exact.Fis.Nat.
, 7914686688 ^ X"- - 13056X^ + X^ 125 16891361683776
x'- +
1674227268777390336 15625 ±1 P99C. 5(
