Rational points on conic bundles over elliptic curves

The elliptic sieve and Brauer groups

“…Theorem 1.2 is a quantitative strengthening of a result of the fourth author and Berg [5], which proves under suitable assumptions that for certain conic bundles 𝜋 ∶ 𝑋 → 𝐸, the image 𝜋(𝑋(ℚ)) does not contain a translate of a subgroup of finite index. In [5] the authors only consider elliptic curves which are Galois general in a sense captured by conditions (1)-(4) in their Theorem 3.5, and their results only apply to special conic bundles given by pulling back Châtelet surfaces from ℙ 1 which also have a non-split fibre over a rational point.…”

“…Theorem 1.2 is a quantitative strengthening of a result of the fourth author and Berg [5], which proves under suitable assumptions that for certain conic bundles 𝜋 ∶ 𝑋 → 𝐸, the image 𝜋(𝑋(ℚ)) does not contain a translate of a subgroup of finite index. In [5] the authors only consider elliptic curves which are Galois general in a sense captured by conditions (1)-(4) in their Theorem 3.5, and their results only apply to special conic bundles given by pulling back Châtelet surfaces from ℙ 1 which also have a non-split fibre over a rational point. Our results apply to an overlapping collection of elliptic curves and conic bundles, but the key point is that our conclusion is stronger: a subset of ℤ which contains no arithmetic progression may still have positive density (for example the set of squarefree numbers in ℤ).…”

“…Lemma 5.4 shows that to get any hope of obtaining a version of Theorem 1.6 there needs to be ramification. Still this assumption is not sufficient in general, as was first shown for conic bundles in [5, Section 3]. In this section, we build on this example and generalise the construction to higher order Brauer group elements.…”