Abstract
Let J ( C ) be the Jacobian of a Picard curve C defined over a number field K containing Q ( ζ 3 ) . We consider the family ( ρ ˜ l ) l of l-adic representations defined by the natural action of the Galois group Gal ( K ¯ / K ) on the l-power torsion of J ( C ) . We show that for a Picard curve C with endomorphism ring Z [ ζ 3 ] the images of these representations are full for all but finitely many primes l. We consider the reduction modulo l of the image of ρ ˜ l , that is, the action of Gal ( K ¯ / K ) on the l-torsion of the Jacobian. This gives a representation ρ l into either GL 3 ( F l ) or into a unitary group over F l 2 , depending on the splitting behavior of l in Q ( ζ 3 ) . It is sufficient to show that the image of ρ l is full in order to show that the image of ρ ˜ l is full.
Published Version
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