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Bernoulli-Carlitz numbers were introduced by L. Carlitz in 1935, they are theanalogues in positive characteristic of Bernoulli numbers. We prove aconjecture formulated by F. Pellarin and the first author on the non-vanishingmodulo a given prime of families of Bernoulli-Carlitz numbers. We then showthat the "exceptional zeros" of certain $L$-series are intimately connected tothe Bernoulli-Carlitz numbers.

Exceptional Zeros of $L$-series and Bernoulli-Carlitz Numbers