  The goal of this paper is to explain how a simple but apparently new fact of
linear algebra together with the cohomological interpretation of L-functions
allows one to produce many examples of L-functions over function fields
vanishing to high order at the center point of their functional equation. The
main application is that for every prime p and every integer g>0 there are
absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD
conjecture holds and which have arbitrarily large rank.
