Exceptional zeros of L-series and Bernoulli-Carlitz numbers

Bruno Anglès (1) , Tuan Ngo Dac (2, 1) , Floric Tavares Ribeiro (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 CNRS - Centre National de la Recherche Scientifique

Bernoulli-Carlitz numbers were introduced by L. Carlitz in 1935, they are the analogues in positive characteristic of Bernoulli numbers. We prove a conjecture formulated by F. Pellarin and the first author on the non-vanishing modulo a given prime of families of Bernoulli-Carlitz numbers. We then show that the " exceptional zeros " of certain L-series are intimately connected to the Bernoulli-Carlitz numbers.

Domaines: Mathématiques [math]

Bruno Anglès, Tuan Ngo Dac, Floric Tavares Ribeiro. Exceptional zeros of L-series and Bernoulli-Carlitz numbers. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2019, 19 (3), pp.981-1024. ⟨10.2422/2036-2145.201706_006⟩. ⟨hal-01404891⟩