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We first show joint uniform distribution of values of Kloosterman sums orBirch sums among all extensions of a finite field $\mathbb{F}_q$, for almostall couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds ondifferences. Using similar ideas, we then study the biases in the distributionof generalized angles of Gaussian primes over function fields and primes inshort intervals over function fields, following recent works of Rudnick--Waxmanand Keating--Rudnick, building on cohomological interpretations anddeterminations of monodromy groups by Katz. Our results are based on genericlinear independence of Frobenius eigenvalues of $\ell$-adic representations,that we obtain from integral monodromy information via the strategy ofKowalski, which combines his large sieve for Frobenius with a method ofGirstmair. An extension of the large sieve is given to handle wild ramificationof sheaves on varieties.

Roots of $L$-functions of characters over function fields, generic linear independence and biases