[Paper Review] Vanishing of some Galois cohomology groups for elliptic curves

Tyler Lawson, Christian Wüthrich|arXiv (Cornell University)|May 12, 2015

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper completely classifies when the Galois cohomology group $ H^1(G, E[p]) $ does not vanish for elliptic curves $ E/\mathbb{Q} $, identifying exactly three exceptional cases: $ p=3 $ with a unique 3-isogeny, $ p=5 $ with a specific 5-isogeny and twist, and $ p=11 $ for the curve 121c2. It further extends this classification to higher torsion groups $ E[p^i] $, showing that $ H^1(G_i, E[p^i]) $ vanishes except in specific cases involving rational $ p $-torsion or isogenies, with applications to the Birch and Swinnerton-Dyer conjecture and the Grunwald–Wang problem.

ABSTRACT

Let E/Q be an elliptic curve and p be a prime number, and let G be the Galois group of the extension of Q obtained by adjoining the coordinates of the p-torsion points on E. We determine all cases when the Galois cohomology group H^1(G, E[p]) does not vanish, and investigate the analogous question for E[p^i] when i>1. We include an application to the verification of certain cases of the Birch and Swinnerton-Dyer conjecture, and another application to the Grunwald-Wang problem for elliptic curves.

Motivation & Objective

• To determine all cases where $ H^1(G, E[p]) $ does not vanish for elliptic curves $ E/\mathbb{Q} $, with $ G = \operatorname{Gal}(\mathbb{Q}(E[p]) / \mathbb{Q}) $.
• To extend the classification to higher torsion groups $ E[p^i] $ for $ i > 1 $, particularly for $ p > 3 $.
• To apply the cohomological results to verify cases of the Birch and Swinnerton-Dyer conjecture and to the Grunwald–Wang problem for elliptic curves.
• To identify explicit counterexamples to local-global divisibility by $ m = 9 $, correcting and extending prior results.

Proposed method

• Reduction to cases where the Galois group $ G $ does not contain nontrivial homotheties, simplifying the cohomological analysis.
• Use of group-theoretic and Galois representation techniques to analyze $ H^1(G, E[p]) $, particularly focusing on the image of the mod $ p $ Galois representation.
• Application of cohomological vanishing results, including a key vanishing result for $ H^2 $, to reduce the problem to manageable cases.
• Computation of Galois groups $ G_i = \operatorname{Gal}(\mathbb{Q}(E[p^i]) / \mathbb{Q}) $ and analysis of $ H^1(G_i, E[p^i]) $ via structural properties of $ \operatorname{GL}_2(\mathbb{Z}/p^i\mathbb{Z}) $.
• Numerical verification of local divisibility conditions using Frobenius elements and reduction modulo $ \ell $, particularly for $ m = 9 $.
• Construction of explicit counterexamples to local-global divisibility by analyzing the kernel of the localization map in the cohomology sequence.

Experimental results

Research questions

• RQ1 For which elliptic curves $ E/\mathbb{Q} $ and primes $ p $ does $ H^1(G, E[p]) $ fail to vanish?
• RQ2 What are the precise conditions under which $ H^1(G_i, E[p^i]) $ is nontrivial for $ i > 1 $, especially when $ p > 3 $?
• RQ3 How can the vanishing of these cohomology groups be used to verify cases of the Birch and Swinnerton-Dyer conjecture?
• RQ4 Can the cohomological obstruction be used to construct counterexamples to the local-global principle for divisibility by $ m = 9 $?
• RQ5 What is the structure of the localization kernel in the cohomology sequence for $ m = 9 $, and how does it relate to the Galois group $ G_2 $?

Key findings

• The cohomology group $ H^1(G, E[p]) $ is nontrivial only in three specific cases: $ p=3 $ with a unique 3-isogeny, $ p=5 $ with a specific 5-isogeny and twist, and $ p=11 $ for the curve 121c2, each with exactly $ p $ elements.
• For $ p=11 $, there are infinitely many exceptions over $ \mathbb{Q} $, but only finitely many for $ p>17 $, and none when $ p \equiv 1 \pmod{3} $.
• For $ i > 1 $, $ H^1(G_2, E[p^2]) $ vanishes if and only if $ H^1(G_i, E[p^i]) $ vanishes for all $ i \geq 2 $, and this holds except in specific cases involving rational $ p $-torsion or isogenies of degree $ p $ or $ p^2 $.
• Explicit counterexamples to local-global divisibility by $ 9 $ exist: for instance, on curve 243a2, a point $ P $ has $ 3P $ locally divisible by 9 at all $ \ell \neq 3 $, but not divisible by 9 globally.
• For $ p=5 $, there are exactly three subgroups $ G_2 \leq \operatorname{GL}_2(\mathbb{Z}/25\mathbb{Z}) $ with nontrivial localization kernel, each of dimension 2 and order 4.
• For $ p=2 $, there are twelve cases with nontrivial localization kernel, with kernel size $ \mathbb{Z}/2 $ or $ \mathbb{Z}/2 \oplus \mathbb{Z}/2 $, and dimensions of $ M_2 $ ranging from 1 to 3.