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Remarks on mod-ln Representations, l=3, 5 by Brian Conrad; Siman Wong is a Mathematics article available to read on EtoBox.

What is Remarks on mod-ln Representations, l=3, 5 about?

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 ](L ) 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.

Who reads Remarks on mod-ln Representations, l=3, 5?

Author

Brian Conrad; Siman Wong

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0022-314X)

Published

1999

Language

EN

Field

Mathematics (Physical Sciences)