THE MOD p REPRESENTATION THEORY OF p-ADIC GROUPS
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چکیده
1.1. The p-adic numbers. A rational number x ∈ Q× may be uniquely written as x = ab p n with a, b and n nonzero integers such that p ab. We define ordp(x) = n, |x|p = p−n, |0|p = 0. |·|p defines an absolute value on Q, satisfying the stronger ultrametric triangle equality |x+ y|p ≤ max(|x|p, |y|p). We define Qp to be the completion Q with respect to this metric and we use the same notation | · |p for the extension of | · |p to Qp; (Qp, | · |p) is a complete valued field. Put Zp = {x ∈ Qp | |x|p ≤ 1}, it is a local discrete valuation ring with maximal ideal pZp. The collection (pZp)n≥0 of compact open subgroups forms a fundamental system of neighbourhoods of 0.
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