Disparity in Selmer ranks of quadratic twists of elliptic curves. (English) Zbl 1300.11063

Ann. Math. (2) 178, No. 1, 287-320 (2013); erratum ibid. 198, No. 2, 879-880 (2023).
Let \(E\) be an elliptic curve defined over a number field \(K\) and, for any quadratic character \(\chi\) of \(\mathrm{Gal}(\overline{K}/K)\), let \(E^\chi\) be the associated twisted curve. The paper deals with the parity of the ranks of the 2-Selmer groups \(\mathrm{Sel}_2(E^\chi/K)\) as \(\chi\) varies. The main theorem considers the set \[ C(K,x):=\{\chi\,:\,\chi\;\text{is\;ramified\;only\;at\;primes\;} \mathfrak{q}\;\text{with\;} N^K_{\mathbb{Q}}(\mathfrak{q})\leq x\} \] (for a positive real number \(x\)) and describes the proportion of odd 2-Selmer ranks for large enough \(x\) with the formula \[ \rho(E/K):=\frac{|\{\chi\in C(K,x)\,:\,\dim_{\mathbb{F}_2}\mathrm{Sel}_2(E^\chi/K)\;\text{is\;odd}\}|}{|C(K,x)|}= \frac{1-\delta(E/K)}{2} \;,\tag{1} \] where \(\delta(E/K)\in [-1,1]\cap \mathbb{Z}\left[\frac{1}{2}\right]\) is a finite product of explicit local factors \(\delta_v\) (depending on \(K_v\) and on the reduction type of \(E\) at \(v\) when \(v|2\Delta(E)\infty\)). In particular, if \(K\) has a real embedding, then \(\delta(E/K)=0\) and the proportion is \(\frac{1}{2}\) as predicted by a conjecture of D. Goldfeld [in: Number theory, Proc. Conf., Carbondale 1979. Lect. Notes Math. 751, 108–118 (1979; Zbl 0417.14031)], but the authors provide also an explicit example of a curve \(E\) for which the values of \(\delta(E/K)\) are dense in \([-1,1]\) as \(K\) varies among the finite extensions of \(\mathbb{Q}(\sqrt{-2})\) unramified at 5 (Example 7.11).
The proof relies on the study of metabolic spaces, i.e., a pair \((V,q)\) of a finite dimensional \(\mathbb{F}_p\)-vector space \(V\) and a quadratic form \(q:V\rightarrow \mathbb{F}_p\) such that \(q\) defines a nondegenerate pairing \((\,,\,)_q\) and there is a (Lagrangian) subspace \(X\subset V\) with \(X=X^\perp\) and \(q(X)=0\). The authors consider a 2-dimensional vector space \(T\) with local quadratic forms \(q_v\) which define metabolic spaces \((H^1(K_v,T),q_v)\) for any \(v\) and, using Lagrangian subspaces to define local conditions at finitely many primes containing the ramified ones, associate to this collection a Selmer group \(\mathrm{Sel}_p(K,T)\) contained in \(H^1(K,T)\). Then the authors use a result of B. Poonen and E. Rains [J. Am. Math. Soc. 25, No. 1, 245–269 (2012; Zbl 1294.11097)] to generalize Theorem 1.4 of B. Mazur and K. Rubin [Ann. Math. (2) 166, No. 2, 579–612 (2007; Zbl 1219.11084)] to \(p=2\), and with that they compute the parity of the 2-rank of \(\mathrm{Sel}_2(K,T)\) and of its twists (defined in Section 4) and prove a formula like (1) for this general setting. Specializing at \(T=E[2]\) and \(q_v=\) the local Tate pairing, they prove that \(\mathrm{Sel}_2(K,E[2])\) (and its twists) correspond to the classical 2-Selmer group of \(E\) (and of the twisted curves \(E^\chi\,\)), and obtain the above formula for \(\rho(E/K)\).
In the final section the authors give a similar formula for the parity of \(\mathrm{Sel}_p(K,T)\) (and its twists) for \(p> 2\). In this case the specialization at \(T=E[p]\) provides Selmer groups for the abelian varieties \(E^\chi:=\mathrm{Ker}\{\mathrm{Res}^{K^\chi}_K(E)\rightarrow E\}\) of dimension \(p-1\) (where \(\chi\) is a character of order \(p\) and \(K^\chi\) is the field fixed by \(\mathrm{Ker}(\chi)\,\)).


11G05 Elliptic curves over global fields
14G25 Global ground fields in algebraic geometry


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