  We study the prime number race for elliptic curves over the function field of
a proper, smooth and geometrically connected curve over a finite field. This
constitutes a function field analogue of prior work by Mazur, Sarnak and the
second author. In this geometric setting we can prove unconditional results
whose counterparts in the number field case are conditional on a Riemann
Hypothesis and a linear independence hypothesis on the zeros of the implied
L-functions. Notably we show that in certain natural families of elliptic
curves, the bias generically dissipates as the conductor grows. This is
achieved by proving a central limit theorem and combining it with generic
linear independence results that will appear in a separate paper. Also we study
in detail a particular family of elliptic curves that have been considered by
Ulmer. In contrast to the generic case we show that the race exhibits very
diverse outcomes, some of which are believed to be impossible in the number
field setting. Such behaviors are possible in the function field case because
the zeros of Hasse-Weil L-functions for those elliptic curves can be proven to
be highly dependent among themselves, which is a very non generic situation.
