Just want to clarify the relation between the decremented exponential and with respect to base change by giving the formula. We know that they are affine conjugates:

where .

We also have thatswhy:

So this is the relation

where .

Update: This formula can be generalized to a base change from to , i.e. from a base to one of its fixed points:

where and .

where .

We also have thatswhy:

So this is the relation

where .

Update: This formula can be generalized to a base change from to , i.e. from a base to one of its fixed points:

where and .