Explicit Unlikely Intersections

Philipp Habegger (Basel)

Frank Adams 2,

An overdetermined inhomogeneous linear system has no
solutions, unless we are in the unlikely situation where enough
equations "line up". In number theory, Pink made a precise conjecture
when a certain set of conditions are unlikely enough to admit at most
finitely many solutions. This conjecture came after precursory work of
Bombieri-Masser-Zannier and Zilber.

In this talk I will consider two very explicit unlikely conditions. Any
$t$, not zero or one, defines an elliptic curve $y^2=x(x-1)(x-t)$. For
infinitely many $t$ the point $(2,\sqrt{2(2-t)})$ has finite order and
for infinitely many $t$ the curve has additional endomorphisms (complex
multiplication). But for only three values of $t$ do both properties
hold simultaneously. Moreover, the curve has complex multiplication for
only three values of $t$ that are in addition roots of unity. Both
results are explicit incarnations of Pink's Conjecture and are part of
joint work with Gareth Jones and David Masser
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