Arithmetic invariant theory

Bhargava, Manjul; Gross, Benedict H.

doi:10.1007/978-1-4939-1590-3_3

Cited by 35 publications

(97 citation statements)

References 34 publications

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“…All possible values in this range occur for some s, even if we impose conditions (1) and (2). If d > 2, then we can choose c ∞ (φ s ) so that c(φ s ) = c ∞ (φ s )c 3 (φ s ) ∈ {3, 1/3} and hence T 1 (φ) is non-empty.…”

Section: Further Applications To Ranks Of Elliptic Curvesmentioning

confidence: 99%

“…Even in this case, very little is known. Smith [42], building on work of Kane [22], has recently proved that in the very special case where E [2](Q) = (Z/2Z) 2 , the average rank is at most 1/2. To date, this is the only case where the average rank of A s (F ) has been proven to be bounded.…”

Section: Introductionmentioning

confidence: 99%

“…1 The proofs of these results are accomplished through a computation of the average size of a family of Selmer groups, namely, the φ-Selmer groups associated to a quadratic twist family of 3-isogenies φ s : A s → A ′ s of abelian varieties. There have been a number of recent results on the average sizes of Selmer groups in large universal families of elliptic curves or Jacobians of higher genus curves (see, e.g., [2,3,5,6,39,40,44]) obtained via geometry-of-numbers methods. It has been a recurring question as to whether such methods could somehow be adapted to obtain analogous results in much thinner (e.g., one-parameter) families.…”

Section: Introductionmentioning

confidence: 99%

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3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Bhargava¹,

Klagsbrun²,

Oliver³

et al. 2019

Duke Math. J.

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For an abelian variety A over a number field F , we prove that the average rank of the quadratic twists of A is bounded, under the assumption that the multiplication-by-3 isogeny on A factors as a composition of 3-isogenies over F . This is the first such boundedness result for an absolutely simple abelian variety A of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field.In dimension one, we deduce that if E/F is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If F is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have 3-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -which states that the average rank in the quadratic twist family of an elliptic curve over Q should be 1/2 -and the first progress towards the analogous conjecture over number fields other than Q.Our results follow from a computation of the average size of the φ-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny φ.

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Section: Further Applications To Ranks Of Elliptic Curvesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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3-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field

Bhargava¹,

Klagsbrun²,

Oliver³

et al. 2019

Duke Math. J.

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“…More precisely, we prove the following theorem: Theorem 1. 4 The proportion of monic even degree hyperelliptic curves having genus g ≥ 4 that have exactly two rational points is at least 1 − (48g + 120)2 −g .…”

Section: Bhargava and Gross Showedmentioning

confidence: 99%

Rational points on hyperelliptic curves having a marked non-Weierstrass point

Shankar¹,

Wang²

2017

Compositio Math.

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In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over Q having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over Q having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mordell-Weil group of the Jacobians of these hyperelliptic curves.Finally using an equidistribution result, we modify the techniques of [19] to conclude that as g tends to infinity, a proportion tending to 1 of these monic even-degree hyperelliptic curves having genus g have exactly two rational points-the marked point at infinity and its hyperelliptic conjugate.

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“…In an earlier version of this paper, we constructed h in Lemma 2.4 in the form h = φ b using some matrix computations. The current proof, which minimizes computations, was inspired by [BG11,§5].…”

Section: Summary Thus Ifmentioning

confidence: 99%