Modular analogues of Brauerʼs characterisation of characters

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We give an analogue of Brauerʼs characterisation of characters of finite groups for the trace functions defined by modular representations. We then “lift” this result to get a p-adic analogue for the Brauer characters. Similar analogues hold for Brauerʼs induction theorem, and Artinʼs induction theorem is then a special case. We also establish an analogue of Greenʼs “necessity” theorem.