254 research outputs found

    Bounds for Serre's open image theorem for elliptic curves over number fields

    Get PDF
    For E/KE/K an elliptic curve without complex multiplication we bound the index of the image of Gal(Kˉ/K)\operatorname{Gal}(\bar{K}/K) in GL2(Z^)\operatorname{GL}_2(\hat{\mathbb{Z}}), the representation being given by the action on the Tate modules of EE at the various primes. The bound is effective and only depends on [K:Q][K:\mathbb{Q}] and on the stable Faltings height of EE. We also prove a result relating the structure of subgroups of GL2(Z)\operatorname{GL}_2(\mathbb{Z}_\ell) 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

    Get PDF
    It is known that if p>37p>37 is a prime number and E/QE/\mathbb{Q} is an elliptic curve without complex multiplication, then the image of the mod pp Galois representation ρˉE,p:Gal(Q/Q)GL(E[p]) \bar{\rho}_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p]) of EE is either the whole of GL(E[p])\operatorname{GL}(E[p]), or is \emph{contained} in the normaliser of a non-split Cartan subgroup of GL(E[p])\operatorname{GL}(E[p]). In this paper, we show that when p>1.4×107p>1.4\times 10^7, the image of ρˉE,p\bar{\rho}_{E,p} is either GL(E[p])\operatorname{GL}(E[p]), 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 d1d\geq 1, let I(d)I(d) denote the set of primes pp for which there exists an elliptic curve defined over Q\mathbb{Q} and without complex multiplication admitting a degree pp isogeny defined over a number field of degree d\leq d. We show that, for d1.4×107d\geq 1.4\times 10^7, we have I(d)=\{p\text{ prime}:p\leq d-1\}. $

    Local invariants of isogenous elliptic curves

    Full text link
    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
    corecore