  Let K be an algebraically closed field of characteristic p>0 and let Sp(2m)
be the symplectic group of rank m over K. The main theorem of this article
gives the character of the rational simple Sp(2m)-modules with fundamental
highest weight as an explicit alternating sum of characters of Weyl modules.
One obtains several formulae for the dimensions of these simple modules, what
allows us to investigate the asymptotic behavior of these dimensions, for a
given p, when the rank m is growing towards infinity. One also gets the simple
Weyl modules with fundamental highest weight, and the article ends with an
application to the modular representation theory of the symmetric group.
