Abstract:
For the first time modular functions for the proof of transcendence
of numbers were used in 1935 in joint article of K. Mahler and Y. Popken.
They proved that at any complex number $\tau$, $\Im\tau>0$, the set
$E_{2}(\tau)$, $E_{4}(\tau)$, $E_{6}(\tau)$ (Eisenstein series)
contains at least one transcendental number. The first transcendence
result about values of the modular invariant $j(\tau)$ has been
proved in 1937 by Th. Schneider. For the proof he used properties of
Weierstrass's elliptic functions, but this way seemed to him
unnatural and Schneider formulated a problem to find the modular
proof of his theorem. Then Mahler formulated a hypothesis about
transcendence at any $\tau$, $\Im\tau>0$, at least one of two
numbers $e^{ 2\pi i\tau }$ and $j (\tau) $. Now a modular proof of
Schneider's theorem is still not found. It is also open the complete
hypothesis of Mahler -Manin about values of modular invariant and
exponential function $a^{\tau}$ for algebraic $a\ne 0,\, $1. In the
talk we will discuss some results in this area and some attempts to
use others modular and quasimodular functions, about further
advances in this area.
