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The non-isotrivial elliptic-surface rank conjecture

Let π:X→P1\pi:X\to \mathbb{P}^1 be a non-constant elliptic surface with section defined over Q\mathbb{Q}. Define

J(X,π)={b∈P1(Q):rk⁡Xb(Q)>rk⁡MW(X,π)}\mathscr{J}(X,\pi)=\{b\in \mathbb{P}^1(\mathbb{Q}):\operatorname{rk}X_b(\mathbb{Q})>\operatorname{rk}\mathrm{MW}(X,\pi)\}

and

N(X,π)={b∈P1(Q):rk⁡Xb(Q)=rk⁡MW(X,π)}.\mathscr{N}(X,\pi)=\{b\in \mathbb{P}^1(\mathbb{Q}):\operatorname{rk}X_b(\mathbb{Q})=\operatorname{rk}\mathrm{MW}(X,\pi)\}.

Non-isotrivial elliptic-surface rank conjecture. If π\pi is non-isotrivial, then both J(X,π)\mathscr{J}(X,\pi) and N(X,π)\mathscr{N}(X,\pi) are infinite.

A heuristic for this conjecture is given in the paper, but no example of a non-isotrivial elliptic surface with infinite N(X,π)\mathscr{N}(X,\pi) is known.

Sources & referencesView supporting material

Primary source

Jerson Caro and Hector Pasten, “On the fibres of an elliptic surface where the rank does not jump”, arXiv:2210.14181 (2022).

elliptic-surfacesmordell-weil-groupsranks-of-elliptic-curves

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