Bounds for Serre’s Open Image Theorem
نویسنده
چکیده
Consider an elliptic curve E without complex multiplication defined over the rationals. The absolute Galois group of Q acts on the group of torsion points of E, and this action can be expressed in terms of a Galois representation ρE : Gal(Q/Q) → GL2(b Z). A renowned theorem of Serre says that the image of ρE is open, and hence has finite index, in GL2(b Z). We give the first general bounds of this index in terms of basic invariants of E. For example, the index [GL2(b Z) : ρE(Gal(Q/Q))] can be bounded by a polynomial function of the logarithmic height of the j-invariant of E. As an application of our bounds, we settle an open question on the average of constants arising from the Lang-Trotter conjecture.
منابع مشابه
On the Surjectivity of Mod ` Representations Associated to Elliptic Curves
Let E be an elliptic curve over the rationals that does not have complex multiplication. For each prime `, the action of the absolute Galois group on the `-torsion points of E can be given in terms of a Galois representation ρE,` : Gal(Q/Q) → GL2(F`). An important theorem of Serre says that ρE,` is surjective for all sufficiently large `. In this paper, we describe an algorithm based on Serre’s...
متن کاملUniform Results for Serre’s Theorem for Elliptic Curves
Let E/K be an elliptic curve defined over a number field K and without complex multiplication (CM). For a rational prime , let K(E[ ]) be the th division field of E, which we know is a finite Galois extension of K. By a celebrated result of Serre [18], there exists a positive constant c(E,K), depending on E and K, such that Gal(K(E[ ])/K) GL2(Z/ Z) for all ≥ c(E,K). In [18, 19], Serre asked whe...
متن کاملGalois Representations and Elliptic Curves
An elliptic curve over a field K is a projective nonsingular genus 1 curve E over K along with a chosen K-rational point O of E, which automatically becomes an algebraic group with identity O. If K has characteristic 0, the n-torsion of E, denoted E[n], is isomorphic to (Z/nZ) over K. The absolute Galois group GK acts on these points as a group automorphism, hence it acts on the inverse limit l...
متن کاملOn Serre’s Complement to Shih’s Theorem
Using Serre’s proposed complement to Shih’s Theorem, we obtain PSL2(Fp) as a Galois group over Q for at least 614 new primes p. Under the assumption that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for 3 8 of the primes that were not covered by previous results.
متن کاملWEIGHTS IN GENERALIZATIONS OF SERRE’S CONJECTURE AND THE MOD p LOCAL LANGLANDS CORRESPONDENCE
In this mostly expository article we give a survey of some of the generalizations of Serre’s conjecture and results towards them that have been obtained in recent years. We also discuss recent progress towards a mod p local Langlands correspondence for p-adic fields and its connections with Serre’s conjecture. A theorem describing the structure of some mod p Hecke algebras for GLn is proved.
متن کامل