Let \(E\) be an elliptic cuve over \(\mathbb{Q}\) given by an integral Weierstrass equation, and \(P\) a rational point on \(E\) of infinite order, which is contained in the connected component of the identity of \(E(\mathbb{R})\). In the paper under review, the authors prove (Theorem 1.1) that there is a constant \(\omega(E,P)>0\) such that the number of integers \(n\) with \(|n|\leq B\) for which the \(y\)-coordinate of \(nP\) is a sum of two squares is of order \(B/(\log B)^\omega\) depending on \(E\) and \(P\), thus showing that most multiples of \(P\) do not have \(y\)-coordinate a sum of two squares. The analogous statement for integers is a famous theorem of Landau and Ramanujan.
Theorem 1.1 is proved using a conic bundle over \(E\) (a conic bundle is a surjective morphism of varieties all of whose fibres are isomorphic to plane conics). In Theorem 1.2, the authors show that for \(E, P\) as above, and \(\pi\colon X\rightarrow E\) a conic bundle such that \(\pi^{-1}(mP)\) is irreducible and isomorphic to two lines over an imaginary quadratic extension for some \(m\in\mathbb{Z}\), there exists a constant \(\omega(X,E,P)\) such that the number of integers \(n\) with \(|n|\leq B\) for which \(nP\) is contained in \(\pi(X(\mathbb{Q}))\) is of order \(B/(\log B)^\omega\) depending on \(X, E, P\). In other words, for almost all multiples of \(P\) the associated conic has no rational point. This is an analogue of a theorem of Serre stating that almost all plane conics over \(\mathbb{Q}\) have no rational point when ordered by the size of their coefficients [J.-P. Serre, C. R. Acad. Sci., Paris, Sér. I 311, No. 7, 397–402 (1990; Zbl 0711.13002)], which was later generalized by Loughran and Smeets to other families of varieties over \(\mathbb{P}^n\) [D. Loughran and A. Smeets, Geom. Funct. Anal. 26, No. 5, 1449–1482 (2016; Zbl 1357.14028)]. Theorem 1.2 is also a strengthening of [J. Berg and M. Nakahara, Math. Z. 300, No. 3, 2429–2449 (2022; Zbl 1493.14041) ], in which Berg and Nakahara prove for an overlapping collection of elliptic curves and conic bundles that the image \(\pi(X(\mathbb{Q}))\) does not contain a translate of a subgroup of finite index.
To prove Theorem 1.2 the authors prove a more general result (Theorem 1.6), which is stated in the framework of Brauer groups. For \(E\) as above and \(P\) a rational point on \(E\) of infinite order, such that there is an \(m\in\mathbb{Z}\) with \(b\in\mbox{Br }\mathbb{Q}(E)\) ramified at \(mP\), Theorem 1.6 gives an explicit upper bound depending on \(E,P,b\) for the number of integers \(n\) with \(|n|\leq B\) for which the evaluation of \(b\) at \(mP\) equals 0 in Br \(\mathbb{Q}\), under certain technical conditions on an elliptic divisibility sequence associated to \(P\). A Dirichlet character is associated to the residue of \(b\) at \(mP\), and the authors show in Section 6 of the paper that the most technical assumption needed in Theorem 1.6 holds in 100% of suitable Dirichlet characters of prime moduli. They also show that Theorem 1.6 has applications to norm form equations (Theorem 1.7). The key ingredients for proving Theorem 1.6 are a version of the Selberg sieve for elliptic curves and elliptic divisibility sequences.