Modular analogues of Brauerʼs characterisation of characters

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.