Remarks on mod-ln Representations, l=3, 5

✍ Scribed by Brian Conrad; Siman Wong

• Book ID: 102604565
• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 190 KB
• Volume: 78
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2406

✦ Synopsis

Let l=3 or 5. For any integer n>1, we produce an infinite set of triples (L, E 1 , E 2 ), where L is a number field with degree l 3(n&1) over Q and E 1 and E 2 are elliptic curves over L with distinct j-invariants lying in Q, such that the following conditions hold: (1) the pairs of j-invariants [ j(E 1 ), j(E 2 )] are mutually disjoint, (2) the associated mod-l n representations G L =Gal(L ÂL) Ä GL 2 (ZÂl n ) are surjective, (3) for almost all primes p of L, we have l n | a p (E 1 ) if and only if l n | a p (E 2 ), and (4) the two representations E i l n are not related by twisting by a continuous character G L Ä (ZÂl n ) _ . No such triple satisfying (2) (4) exists over any number field if we replace l by a prime larger than 5. The proof depends on determining the automorphisms of the group GL 2 (ZÂl n ) for l=3, 5 and analyzing ramification in a branched covering of ``twisted'' modular curves.