On 2-Selmer groups of twists after quadratic extension

2020/11/09 by Morgan, Adam, Paterson, Ross · 1 citation
#11G05 (Primary) 11G10 #11N45 #14H52 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2011.04374

Abstract

Let E/ℚ be an elliptic curve with full rational 2-torsion. As d varies over squarefree integers, we study the behaviour of the quadratic twists E_d over a fixed quadratic extension K/ℚ. We prove that for 100% of twists the dimension of the 2-Selmer group over K is given by an explicit local formula, and use this to show that this dimension follows an Erdős--Kac type distribution. This is in stark contrast to the distribution of the dimension of the corresponding 2-Selmer groups over ℚ, and this discrepancy allows us to determine the distribution of the 2-torsion in the Shafarevich--Tate groups of the E_d over K also. As a consequence of our methods we prove that, for 100% of twists d, the action of Gal(K/ℚ) on the 2-Selmer group of E_d over K is trivial, and the Mordell--Weil group E_d(K) splits integrally as a direct sum of its invariants and anti-invariants. On the other hand, we give examples of thin families of quadratic twists in which a positive proportion of the 2-Selmer groups over K have non-trivial Gal(K/ℚ)-action, illustrating that the previous results are genuinely statistical phenomena.

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