## 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 $$\mathrm{Ker}(\chi)\,$$).

### MSC:

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

### Keywords:

elliptic curves; parity; quadratic twists; Selmer groups

### Citations:

Zbl 0417.14031; Zbl 1294.11097; Zbl 1219.11084

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### References:

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