The fullness conjectures for products of elliptic curves

Let 𝐹 be a number field and E , F 1 , … , F r finite field extensions of 𝐹. For all 1 ≤ i ≤ r , let α i ∈ Br ⁡ ( F i ) . Then there exists a finite field extension K / F , linearly disjoint from 𝐸 and from the F i ’s over 𝐹, and such that α i vanishes in Br ⁡ ( K ⊗ F F i ) for all 𝑖.

Proof

For all 𝑖, let S i denote the finite set of places 𝑣 of F i such that α i , v ≠ 0 in Br ⁡ ( F i , v ) . For all 𝑖 and all v ∈ S i , denote by n i , v the order of α i , v in Br ⁡ ( F i , v ) . Let 𝑆 be the (finite) set of places 𝑣 of 𝐹 for which there exists an 𝑖 and a place w ∈ S i dividing 𝑣. For each v ∈ S , let n v denote the lcm of the n i , w [ F i , w : F v ] for all 𝑖 and w ∈ S i such that 𝑤 divides 𝑣. Let E ′ / F denote the Galois closure of the compositum of the fields 𝐸, F 1 , …, F r , and denote its Galois group by 𝐺. By the Chebotarev theorem, for all g ∈ G , there exists a place v g of 𝐹 outside 𝑆 and not dividing 2 such that E ′ / F is unramified at v g and the Frobenius at v g lies in the conjugacy class of 𝑔. One can assume that v g ≠ v h if g ≠ h . Define S ′ to be the union of 𝑆 and all the places v g for g ∈ G . By the Grunwald–Wang theorem, there exists a (cyclic) field extension K / F such that, for all v ∈ S , all places 𝑤 of 𝐾 above 𝑣, the local degree [ K w : F v ] is divisible by n v , and for all v ∈ S ′ ∖ S , the extension K / F splits completely at 𝑣. Then, by construction, the extensions 𝐾 and E ′ are linearly disjoint over 𝐹, and for all 𝑖, the image of α i in Br ⁡ ( K ⊗ F F i ) vanishes at all places of K ⊗ F F i by a restriction-corestriction argument; hence it vanishes globally. ∎

Let 𝑘 be a number field and 𝒢 an étale (algebraic) gerbe over 𝑘, with connected linear band (or 𝑘-kernel) 𝐿. Let 𝐸 be a finite extension of 𝑘. Then there exists a finite extension K / k linearly disjoint from E / k such that G ⁢ ( K ) ≠ ∅ .

Proof

Consider the class 𝛼 of 𝒢 in the non-abelian cohomology set H 2 ⁢ ( k , L ) and its image α F ∈ H 2 ⁢ ( F , L ) . Let 𝑇 be the 𝑘-torus associated to the 𝑘-band 𝐿 (see [4, 1.7 and 6.1] for instance) and let α ′ ∈ H 2 ⁢ ( k , T ) be the image of 𝛼.

By [4, Proposition 6.5], for any totally imaginary finite field extension K / k , G ⁢ ( K ) ≠ ∅ if and only if α K ′ = 0 in H 2 ⁢ ( K , T ) . So we are reduced to find a totally imaginary finite extension K / k , linearly disjoint from 𝐸 over 𝑘, such that α ′ vanishes in H 2 ⁢ ( K , T ) .

First, there exists a totally imaginary finite field extensions F / k that is linearly disjoint from 𝐸 over 𝑘 (one can work as in Lemma A.1). So we are reduced to find a finite extension K / F , linearly disjoint from E ′ : = E F over 𝐹, such that α ′ vanishes in H 2 ⁢ ( K , T ) .

There exists an exact sequence of 𝑘-tori (a flasque resolution of 𝑇 for instance; see [7, Proposition 1.3]) 0 → S → P → T → 0 such that 𝑃 is quasi-trivial, i.e. isomorphic to a finite product of tori of the shape R k i / k ⁢ ( G m ) for some finite field extensions k i / k . The field 𝐹 is totally imaginary; hence H 3 ⁢ ( F , S ) = 0 by [24, Chapter I, Corollary 4.21]. Therefore, the map H 2 ⁢ ( F , P ) → H 2 ⁢ ( F , T ) is surjective. Let β ′ ∈ H 2 ⁢ ( F , P ) be a lift of the class α ′ ∈ H 2 ⁢ ( F , T ) . Writing the quasi-trivial 𝐹-torus P F as ∏ i = 1 r R F i / F ⁢ ( G m ) for some finite field extensions F i / F , we get that

To conclude, we apply the lemma to the elements ( β i ) 1 ≤ i ≤ r ∈ ⨁ i = 1 r Br ⁡ ( F i ) corresponding to the class β ′ ∈ H 2 ⁢ ( F , P ) and we get a finite extension K / F , linearly disjoint of E ′ and the F i ’s over 𝐹, such that the image of β ′ in H 2 ⁢ ( K , P ) ≅ ⨁ i = 1 r Br ⁡ ( K ⊗ F F i ) vanishes. Then 𝐾 and 𝐸 are linearly disjoint over 𝑘 and the proof is complete. ∎