A note on 8-division fields of elliptic curves

Abstract

Let K be a field of characteristic different from 2 and let E be an elliptic curve over K, defined either by an equation of the form y 2 = f (x) with degree 3 or as the Jacobian of a curve defined by an equation of the form y 2 = f (x) with degree 4. We obtain generators over K of the 8-division field K(E[8]) of E given as formulas in terms of the roots of the polynomial f , and we explicitly describe the action of a particular automorphism in Gal(K(E[8])/K).

An abelian subextension of the dyadic division field of a hyperelliptic Jacobian

“…The g = 1 case. Lemma 3.1, together with the results of §2 and the fact that K ∞ ⊂ K unr , imply that K ab ∞ is an extension of K 2 = K 1 ( √ γ 1,2 , √ γ 1,3 , √ γ 2,3 ) obtained by adjoining 2 independent 4th roots of products of the elements γ i,j ∈ K 1 (recall from [16] that K 1 is generated over K by the γ i,j 's both when d = 3 and when d = 4). Therefore, in this case, we get via Kummer theory a canonical identification of Gal(K ab…”

“…Remark 1.3. We can also verify part (b) of Theorem 1.1 for the d = 3 case by combined use of the formulas given in [14] and [16]. We illustrate how to see that K 4 (µ 2 ) contains an element whose 8th power is γ 1,2 γ 1,3 γ 2 2,3 as follows (one may use a similar argument for the other generators).…”

“…It is well known that we always have K 1 ⊆ k(α 1 , ..., α d ) and that equality holds except when d = 4 (see [16]). The main purpose of this paper is to provide an explicit description of the maximal abelian subextension of K ∞ /K 1 , which we denote by K ab ∞ .…”

“…Let α 1 , ...,ᾱ d ∈k denote the roots of the polynomial f (x) ∈ k[x] and letγ i,j ∈k be given by formulas in terms of theᾱ i 's analogous to those used to define the γ i,j 's in the statement of Theorem 1.1. a) If g ≥ 2, the extension k(J [8])/k(J [2]) contains the subextension 1γ3,2γ 2 1,2 , ζ 16 ). c) Let σ be any automorphism in the absolute Galois group of k which acts on k(J [16]) as multiplication by −1. Then σ acts on the subfields described above by changing the signs of all generators of the form √ γ i,j and by fixing (resp.…”

“…). c) Let σ be any automorphism in the absolute Galois group of k which acts on k( J [16]) as multiplication by −1. Then σ acts on the subfields described above by changing the signs of all generators of the form √ γ i,j and by fixing (resp.…”