Counterexamples to the local–global divisibility over elliptic curves

Abstract

Let p ≥ 5 be a prime number. We find all the possible subgroups G of GL 2 (Z/pZ) such that there exists a number field k and an elliptic curve E defined over k such that the Gal(k(is isomorphic to the G-module (Z/pZ) 2 and there exists n ∈ N such that the local-global divisibility by p n does not hold over E(k).

On the local-global principle for divisibility in the cohomology of elliptic curves

“…1 , then as in the proof of the preceding lemma, [Ran18] implies that G 1 is generated by a diagonal matrix of order dividing 2 with 1 as an eigenvalue. Otherwise p | #G ′ 1 .…”

“…Since δ ≡ 0 mod p, C δ,p is a Borel subgroup. So in this case [Ran18] implies that G ′ 1 is the subgroup of stricly upper triangular matrices and that G 1 = G ′ 1 or G 1 is generated by G ′ 1 and diag(1, −1) as required. The assumption in the second statement of the lemma implies that G is generated by the matrices…”

“…Combining the above with [Ran18] and explicit lower bounds for the gonality of modular curves [Abr96] we will prove the following.…”

“…Over the past decades there has been substantial interest in the problem of determining conditions on N and k implying that such a local-global principle holds for all elliptic curves over k [DZ01, DZ04, DZ07, PRV12, PRV14, Cre16, LW16,Ran18]. Due to its connection with a question of Cassels [Cas62, Problem 1.3], the analogous question where E(k) = H 0 (k, E) is replaced by the Galois cohomology group H 1 (k, E) has also recieved much attention [C ¸S15, Cre13,Cre16].…”

“…The determinant of the modp representation Gal(k) → Aut(E[p]) ≃ GL 2 (Z/p) → Z/p × is the p-cyclotomic character. Since d < (p − 1)/2 = [Q(µ p ) + : Q],the image of this determinant map is of size > 2 [Ran18,. Theorem 2] shows that the possibilities for the image of the mod p representation are rather limited.…”