Galois extensions of Q(T) given by torsion on elliptic curves

Consider an elliptic curve E over Q(T ) with j-invariant equal to T , e.g. the curve defined by the equation

y2 + xy = x3 − 36 T − 1728x−

1 T − 1728 .

(Any other choice of E differs from this one by a quadratic twist only.) By adjoining to Q(T ) the coordinates of the n-division points of E, one obtains a Galois extension Kn ofQ(T ) with Gal(Kn/Q(T )) = GL2(Z/nZ). More precisely, the Galois group of C ·Kn over C(T ) is SL2(Z/nZ), and the homomorphism GQ(T ) −→ GL2(Z/nZ) det−→ (Z/nZ)∗ is the cyclotomic character. Hence the extension Kn is not regular when n > 2: the algebraic closure ofQ in Kn isQ(µn). So the method does not give regular extensions of Q(T ) with Galois group PGL2(Fp), nor PSL2(Fp). Nevertheless, K-y. Shih was able to obtain the following result [Shih1], [Shih2]:

Theorem 5.1.1 There exists a regular extension ofQ(T ) with Galois group PSL2(Fp) if

) = −1,

) = −1, or

) = −1.