Collquium: Jinbo Ren, University of Virginia, Transcendental number theory in algebraic geometry
Event Description:
Abstract: The classical Lindemann-Weierstrass theorem asserts that if
n algebraic numbers are linearly independent over$ \mathbb{Q}$, then
their exponentials are algebraically independent over $\mathbb{Q}$.
This theorem generalizes the fact that $e$ is a transcendental number.
More generally, the (very difficult) Schanuel's conjecture predicts
that if n complex numbers $a_1, a_2,...,a_n$ are linearly independent
over $\mathbb{Q}$, then the field $\mathbb{Q}(a_1,...,a_n,
e^{a_1},...,e^{a_n})$ has transcendence degree at least $n$ over
$\mathbb{Q}$. This assertion is very strong, for example, the
transcendence of $\pi$ is one of its simple consequences.
In my colloquium talk, I will explain how to formulate the analogues
of these classical conjectures in the context of algebraic tori,
abelian varieties and Shimura varieties. I will also discuss their
applications in Diophantine geometry.