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CM-points on straight lines

Posted in
Speaker: 
Yuri Bilu
Affiliation: 
U Bordeaux 1/MPI
Date: 
Wed, 02/07/2014 - 15:00 - 16:00
Location: 
MPIM Lecture Hall
Parent event: 
Dynamics and Numbers
Parent event: 
Number theory lunch seminar

Let $\tau$ be an imaginary quadratic number with ${\mathrm{Im}\tau>0}$ and let $j$ denote the $j$-invariant function. According to the classical theory of Complex Multiplication, the complex number $j(\tau)$ is an algebraic integer. A CM-point in $\mathbb{C}^2$ is a point of the form $(j(\tau_1),j(\tau_2))$, where both $\tau_1$ and $\tau_2$ are imaginary quadratic numbers. In 1998 Yves André proved that a non-special (the notion will be defined during the talk) irreducible plane curve ${F(x_1,x_2)=0}$ may have only finitely many CM-points. This was the first non-trivial contribution to the celebrated André-Oort conjecture. Relying on recent ideas of Lars Kühne, we obtain a very explicit version of this result for straight lines defined over $\mathbb{Q}$: with ``obvious'' exceptions, a CM-point cannot belong to such a line. Kühne himself proved this for the line ${x_1+x_2=1}$. A joint work with Bill Allombert and Amalia Pizarro-Madariaga.

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