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\(3\)-isogeny Selmer groups and ranks of abelian varieties in quadratic twist families over a number field. (English) Zbl 1442.14084
The quadratic twist \(A_s\) of an abelian variety over a number field \(F\) is determined by a class \(s\in F^*/F^{*2}\). Geometrically \(A_s\) is the same as \(A\) (they are isomorphic over \(F(\sqrt{s})\)) but arithmetically they can vary considerably. Because of the absence of complications arising from moduli, the families \(A_s\) are natural ones to study from the point of view of arithmetic, but even very basic information such as the statistics of the Mordell-Weil ranks are essentially unknown. It is expected, at least in some cases, that the average rank should be bounded: for elliptic curves \(E\) over \(\mathbb{Q}\) it is expected to be \(\frac{1}{2}\) and it is known to be at most \(\frac{1}{2}\) if \(E[2](\mathbb{Q})\) is maximal, i.e. is the Klein 4-group.
This paper improves the position dramatically by proving that \(A_s(F)\) has bounded average rank as long as \(A\) has a 3-selfisogeny: slightly more generally, if there is an isogeny \(A\to A\) that factors as a composition of 3-isogenies over \(F\). In particular this is the case if \(E\) is an elliptic curve over \(\mathbb{Q}\) with a 3-torsion point, so this provides examples of families without full 2-torsion for which the average rank of \(A_s\) is bounded, as well as many examples in dimension \(>1\).
The main invariant actually studied in the paper is not in fact the Mordell-Weil rank directly, but the 3-Selmer groups mentioned in the title. If \(\phi\colon A\to A'\) is an isogeny defined over \(F\) there is an associated Selmer group \({\operatorname{Sel}}_\phi(A)\) (which reduces to the usual \(n\)-Selmer group in the case where \(A\) is an elliptic curve and \(\phi\) is mutiplication by \(n\)), and there is also an isogeny \(\phi_s\colon A_s\to A'_s\). The main technical result is an estimate of the average size of \({\operatorname{Sel}}_{\phi_s}(A_s)\) as \(s\) varies in a (mildly restricted) subset \(\Sigma\subset F^*/F^{*2}\), ordered by a height. The point is that the size of \({\operatorname{Sel}}_{\phi_s}(A_s)\) should, in view of the results of [M. Bhargava et al., J. Lond. Math. Soc., II. Ser. 101, No. 1, 299–327 (2020; Zbl 1458.11091)], be determined by an Euler product, but the local factors turn out to be equal to \(1\) away from \(3\), \(\infty\) and the conductor of \(A\), and this allows a direct computation of the average size of \({\operatorname{Sel}}_{\phi_s}(A_s)\), which is even a rational number.
Several corollaries are listed, some of them conditional on conjectures about the 3-rank of the Tate-Shafarevich group. The precise statements need some care, but one, which gives the general flavour, says (unconditionally) that if \(E\) is an elliptic curve over a totally real field \(F\) and admits a 3-isogeny, then the proportion of the \(E_s\) having Mordell-Weil rank zero is positive.
The proofs rest on a correspondence between elements of \({\operatorname{Sel}}_\phi\) and binary cubic forms over \(F\), whose origins can be traced back to [E. S. Selmer, Acta Math. 85, 203–362 (1951; Zbl 0042.26905)] but which is new even for elliptic curves. One may identify \({\operatorname{SL}}_2(F)\) orbits of \(F\)-valued binary quadratic forms (elements of \(\operatorname{Sym}^2(\mathbb{Z})^3(F)\)) of discriminant \(4D\) with the kernel of the norm map \(K^*/K^{*3}\to F^*/F^{*3}\), where \(K=F[t]/(t^2-D)\). But this kernel (replacing \(D\) with \(-3D\)) is also isomorphic to \(H^1({\operatorname{Gal}}(\overline{F}/F), A[\phi])\) for a 3-isogeny \(\phi\), and this classifies \(\phi\)-coverings of \(A\).
14G05 Rational points
11G10 Abelian varieties of dimension \(> 1\)
Full Text: DOI Euclid
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