## Explicit Unlikely Intersections

#### Philipp Habegger (Basel)

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