254 research outputs found
Bounds for Serre's open image theorem for elliptic curves over number fields
For an elliptic curve without complex multiplication we bound the index
of the image of in
, the representation being given by the
action on the Tate modules of at the various primes. The bound is effective
and only depends on and on the stable Faltings height of .
We also prove a result relating the structure of subgroups of
to certain Lie algebras naturally
attached to them.Comment: Final version, accepted for publication in Algebra and Number Theor
Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan
It is known that if is a prime number and is an elliptic curve without complex multiplication, then the image of the mod Galois representation of is either the whole of , or is \emph{contained} in the normaliser of a non-split Cartan subgroup of . In this paper, we show that when , the image of is either , or the \emph{full} normaliser of a non-split Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For , let denote the set of primes for which there exists an elliptic curve defined over and without complex multiplication admitting a degree isogeny defined over a number field of degree . We show that, for , we have I(d)=\{p\text{ prime}:p\leq d-1\}. $
Local invariants of isogenous elliptic curves
We investigate how various invariants of elliptic curves, such as the
discriminant, Kodaira type, Tamagawa number and real and complex periods,
change under an isogeny of prime degree p. For elliptic curves over l-adic
fields, the classification is almost complete (the exception is wild
potentially supersingular reduction when l=p), and is summarised in a table.Comment: 22 pages, final version, to appear in Trans. Amer. Math. So
- …