2017
DOI: 10.48550/arxiv.1702.02325
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$2^\infty$-Selmer groups, $2^\infty$-class groups, and Goldfeld's conjecture

Alexander Smith

Abstract: We prove that the 2 ∞ -class groups of the imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra heuristic. Given an elliptic curve E/Q with full rational 2-torsion and no rational cyclic subgroup of order four, we analogously prove that the 2 ∞ -Selmer groups of the quadratic twists of E have distribution as predicted by Delaunay's heuristic. In particular, among the twists E (d) with |d| < N , the number of curves with rank at least two is o(N ). CONTENTS 1. Introduction 2. Algebra… Show more

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Cited by 18 publications
(36 citation statements)
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“…For an odd p, the right hand side of the formula (1.1) coincides with the conjectured distribution of the p-parts of the class groups of imaginary quadratic fields predicted by Cohen and Lenstra [7]. When p = 2, this coincides with the distribution of (2Cl(K)) [2 ∞ ] (Cl(K) denotes the class group of K) for imaginary quadratic fields K, as conjectured by Gerth [10] and proved by Smith [13].…”
Section: Introductionsupporting
confidence: 80%
“…For an odd p, the right hand side of the formula (1.1) coincides with the conjectured distribution of the p-parts of the class groups of imaginary quadratic fields predicted by Cohen and Lenstra [7]. When p = 2, this coincides with the distribution of (2Cl(K)) [2 ∞ ] (Cl(K) denotes the class group of K) for imaginary quadratic fields K, as conjectured by Gerth [10] and proved by Smith [13].…”
Section: Introductionsupporting
confidence: 80%
“…When n = 1 and F = Q, we have rk E(K d /F ) new = rk E d (F ), where E d is the d-th quadratic twist of E, and Goldfeld conjectures that 50% of these twists have rank 0. This conjecture has been verified in many cases [BKLOS19, KL19,Smi17]. Indeed, in his Ph.D. thesis, Smith proves the conjecture for "most" elliptic curves E over Q [Smi20].…”
Section: Introductionmentioning
confidence: 76%
“…We note that the distribution of the p-Selmer group conjectured by Poonen and Rains is also expected to hold for various special families of elliptic curves, and for p = 2, significant progress has been made toward proving the conjecture for such families: In [20], Heath-Brown determined the distribution of the 2-Selmer group for the family of congruent number curves, and Swinnerton-Dyer (see [39]) and Kane (see [22]) showed that the same distribution holds for any family of quadratic twists of a single elliptic curve with full rational 2-torsion. Considerably more is known about 2-Selmer groups in quadratic twist families; see, for example, the work of Klagsbrun, Mazur, and Rubin [23] computing the proportion of twists of a given elliptic curve with even 2-Selmer rank, and the very recent work of Smith [36] determining (under certain technical conditions) the distribution of the 2 k -Selmer group in quadratic twist families of elliptic curves for every k ≥ 1.…”
Section: Statements Of Main Resultsmentioning
confidence: 99%
“…Again, we check that each of the lower bounds on e 2 in (37) is less than or equal to each of the upper bounds. The inequalities 0 ≤ e 1 − a + b 3 , 0 ≤ e1 2 , and a − b 2 ≤ e 1 − a + b 3 follow from (36). The inequality a − b 2 ≤ e1 2 follows from the first property in (35) combined with (36).…”
Section: Ranks Of Special Elementsmentioning
confidence: 97%
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