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COLLISION OF ORBITS ON AN ELLIPTIC SURFACE
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 21 July 2026, pp. 1-16
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- Article
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Let C be a smooth projective curve defined over ${\overline {\mathbb Q}}$
, let $\pi :\mathcal {E}\longrightarrow C$
be an elliptic surface and let $\sigma _{P_1},\sigma _{P_2},\sigma _{Q}$
be sections of $\pi $
(corresponding to points $P_1,P_2, Q$
of the generic fibre E of $\mathcal {E}$
). We obtain a precise characterisation, expressed solely in terms of the dynamical relations between the points $P_1,P_2,Q$
with respect to the endomorphism ring of E, so that there exist infinitely many $\lambda \in C({\overline {\mathbb Q}})$
with the property that $[m_{i,\lambda }](\sigma _{{P_{i}}}(\lambda ))=\sigma _{Q}(\lambda )$
(for $i=1,2$
) for some nonzero integers $m_{1,\lambda },m_{2,\lambda }$
, on the smooth fibre $E_\lambda $
of $\mathcal {E}$
.
