On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

Abstract

We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

Average rank of quadratic twists with a rational point of almost minimal height

“…We note that the author [22,Theorem 1] has recently established an asymptotic formula for the cardinality of this set when 𝛼 ∈ (0, 1∕120). Our main result investigates the average analytic rank of the quadratic twists of 𝐸 as 𝑑 runs over  𝐴,𝐵 (𝛼; 𝑋).…”

“…The cardinality of the set $$\mathcal {D}_{A,B}(\alpha ;X)$$ is the subject of a recent article by the author, where it is shown [22, Theorem 1] that for $$\alpha < 1/120$$, there exists a positive constant $$c(\alpha )$$ such that one has $$\begin{equation} \#\mathcal {D}_{A,B}(\alpha ;X) \sim c(\alpha ) X^{1/2} \log X. \end{equation}$$…”

“…\end{equation*} Let $$L_{\mathbf {w}}$$ denote the Zariski closed subset of $$V_{\mathbf {w}}$$ defined as the union of the lines contained in $$V_{\mathbf {w}}$$. Since the polynomial $$F(x,z)$$ is assumed to be irreducible, meaning that the curve $$E$$ and all its twists $$E_d$$ have trivial 2‐torsion, a recent result by the author [22, Corollary 4.6] states that if a point $$(u_3:u_4:v_3:v_4) \in V_{\mathbf {w}}(\mathbb {Q})$$ lies in $$L_{\mathbf {w}}(\mathbb {Q})$$, then $$(u_3:v_3w_1^2) = (u_4:v_4w_2^2)$$ in $$\mathbb {P}^1(\mathbb {Q})$$. Such points are excluded here and a result of Salberger [24, Theorem 0.1] therefore shows that for any $$\epsilon > 0$$, we have …”

“…For $$\alpha > 0$$, we define the set $$\begin{equation*} \mathcal {D}_{A,B}(\alpha ;X) = \lbrace d \in \mathcal {S}(X) : \eta _d(A,B) \leqslant d^{1/8+\alpha } \rbrace . \end{equation*}$$We note that the author [22, Theorem 1] has recently established an asymptotic formula for the cardinality of this set when $$\alpha \in (0,1/120)$$. Our main result investigates the average analytic rank of the quadratic twists of $$E$$ as $$d$$ runs over $$\mathcal {D}_{A,B}(\alpha ;X)$$.…”

“…Let 𝐻 denote the usual exponential Weil height on ℙ 3 (ℚ) and set𝑎 𝑄,𝐰 (𝑘; 𝛼; 𝑋) = # { (𝑢 3 ∶ 𝑢 4 ∶ 𝑣 3 ∶ 𝑣 4 ) ∈ 𝑉 𝐰 (ℚ) ∶ 𝐻(𝑢 3 ∶ 𝑢 4 ∶ 𝑣 3 ∶ 𝑣 4 ) ≪ 𝑋 1∕4+2𝛼 ∕𝑘Let 𝐿 𝐰 denote the Zariski closed subset of 𝑉 𝐰 defined as the union of the lines contained in 𝑉 𝐰 . Since the polynomial 𝐹(𝑥, 𝑧) is assumed to be irreducible, meaning that the curve 𝐸 and all its twists 𝐸 𝑑 have trivial 2-torsion, a recent result by the author[22, Corollary 4.6] states that if a point (𝑢 3 ∶ 𝑢 4 ∶ 𝑣 3 ∶ 𝑣 4 ) ∈ 𝑉 𝐰 (ℚ) lies in 𝐿 𝐰 (ℚ), then (𝑢 3 ∶ 𝑣 3 𝑤 2 1 ) = (𝑢 4 ∶ 𝑣 4 𝑤 2 2 ) in ℙ 1 (ℚ). Such points are excluded here and a result of Salberger [24, Theorem 0.1] therefore shows that for any 𝜖 > 0, we have 𝑎 𝑄,𝐰 (𝑘; 𝛼; 𝑋) ≪ 𝜖…”