We first show joint uniform distribution of values of Kloosterman
sums or Birch sums among all extensions of a finite field
, for almost all couples
of arguments in
,
as well as lower bounds on differences. Using similar ideas, we then study
the biases in the distribution of generalized angles of Gaussian primes over
function fields and primes in short intervals over function fields, following
recent works of Rudnick and Waxman, and Keating and Rudnick, building on
cohomological interpretations and determinations of monodromy groups by Katz.
Our results are based on generic linear independence of Frobenius eigenvalues of
-adic
representations, that we obtain from integral monodromy information via the
strategy of Kowalski, which combines his large sieve for Frobenius with a method of
Girstmair. An extension of the large sieve is given to handle wild ramification of
sheaves on varieties.
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