Arithmetic surjectivity forces recurrence of locally soluble fibers

Definitions and mathematical statement

Viray's AIM question concerns a fibration $\phi:X\to E$ over an elliptic curve of positive Mordell--Weil rank over ${\mathbb{Q}}$, whose generic fiber is smooth and geometrically irreducible. For $p\in E({\mathbb{Q}})$, put $X_p=\phi^{-1}(p)$, and define \[ Z=\{p\in E({\mathbb{Q}}):X_p({\mathbb{Q}}_v)\ne\varnothing\text{ for every place }v\}. \] What can be said about $Z$, and can $0<|Z|<\infty$ occur? The original does not give full properness, regularity, or fibration conventions. Browning's appended comment singles out conic bundles as an interesting case.

The motivation in the source is that constructions attributed to Poonen, Skorobogatov--Harpaz, and Colliot-Thélène--Pal--Skorobogatov fail the Hasse principle without the known obstructions applying, and use maps to curves $C$ with $0<|C({\mathbb{Q}})|<\infty$. This is the source's 2014 context, not a current exhaustive claim about obstructions.

The attached report proposes a number-field recurrence criterion. Let $E/k$ have positive rank, and suppose a fixed finite set of places $S$ satisfies \[ X(k_v)\longrightarrow E(k_v)\text{ is surjective for every }v\notin S. \] Let $p_0\in E(k)$ admit, for every $v\in S$, a local lift $x_v\in X(k_v)$ with $\phi(x_v)=p_0$ at which $\phi$ is smooth. The report asserts that for every nontorsion $P\in E(k)$, there are infinitely many distinct points in \[ \{p_0+nP:n\in{\mathbb{Z}}\}\cap Z \] accumulating at $p_0$ in the $S$-adic topology. Thus it claims recurrence in each chosen nontorsion cyclic Mordell--Weil coset, not only infinitude somewhere in $E(k)$. The local-surjectivity condition is uniform outside one fixed $S$; varying exceptional sets with the base point would be a different hypothesis.

It concludes that a smooth proper family with geometrically integral fibers over $E$, in the intended number-field setting, has either no everywhere locally soluble rational fibers or infinitely many. The generic smoothness condition in the original is weaker than this all-fiber assumption.

Combining recurrence with the Loughran--Skorobogatov--Smeets arithmetic-surjectivity criterion, the report further claims that a finite nonempty $Z$ containing a point over which $\phi$ is smooth forces a non-pseudo-split codimension-one fiber on some smooth modification. A fiber is pseudo-split in the relevant sense when each element of the residue-field Galois group fixes some multiplicity-one geometric irreducible component; this does not require a single component fixed by the whole group. The precise modification and hypotheses of the cited criterion are not supplied, so the conclusion is conditional on that criterion's applicable scope. No finite nonempty example is constructed here.

Short English statement

Can only finitely many nonzero rational fibers over a positive-rank elliptic curve be locally soluble everywhere? The report gives a recurrence criterion under uniform local surjectivity and smooth local lifting, then reduces certain finite examples to a pseudo-splitness failure.

Sources

• AIM workshop "Rational and integral points on higher-dimensional varieties," Problem 4, attributed to Viray, with Browning's conic-bundle comment: https://aimath.org/pastworkshops/ratlhigherdimvarproblems.pdf. The supplied verification dates the PDF May 28, 2014 and refers to page 2 when correcting the OCR 6= to $\ne$; no PDF was accessed here.
• Local canonical registry aim-algebraic-number-theory-notes.json, notes[23], tag problem; permanent UnsolvedMath ID $20000372$, source code catalogue entry. Source motivation names Poonen, Skorobogatov--Harpaz, and Colliot-Thélène--Pal--Skorobogatov. Original remarks and literature field are otherwise empty.
• Supplied research attempt 1 invokes a Loughran--Skorobogatov--Smeets criterion and proposes the recurrence reduction. No full authoritative bibliographic references or complete modification hypotheses are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.