An Introduction to p-adic Numbers and p-adic Analysis by Andrew Baker

By Andrew Baker

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Extra resources for An Introduction to p-adic Numbers and p-adic Analysis [Lecture notes]

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We can think of (a, b) as a disc of radius (b − a)/2 and centred at (a + b)/2. This suggests the following definition in Qp . 8. Let f : X −→ Qp be a function where X ⊆ Qp . Then f is locally constant on X if for every α ∈ X, there is a real number δα > 0 such that f is constant on the open disc DX (α; δα ). This remark implies that there are no interesting examples of locally constant functions on open intervals in R; however, that is false in Qp . 9. Let X = Zp , the p-adic integers. 29, we know that for α ∈ Zp , there is a p-adic expansion α = α0 + α1 p + · · · + αn pn + · · · , where αn ∈ Z and 0 αn (p − 1).

3 + 2 × 25 + 4 × 125 + · · · ) 3-7. ; ∑ 1 ; n! ∑ 22n − 1 2n − 1 for p = 2; ∑ (pn+1 ) . pn 3-8. Find the radius of convergence of each of the following power series over Qp : ∑ Xn n! ; ∑ pn X n ; 3-9. Prove that in Q3 , ∑ X pn pn ; ∑ nk X n with 0 ∞ ∑ 32n (−1)n n=1 42n n =2 ∞ ∑ 32n . X n ; ∑ Xn n . 3-10. For n 1, let X(X − 1) · · · (X − n + 1) n! and C0 (X) = 1; in particular, for a natural number x, ( ) x Cn (x) = . n Cn (X) = (a) Show that if x ∈ Z then Cn (x) ∈ Z. (b) Show that if x ∈ Zp then Cn (x) ∈ Zp .

Determine each of the following 5-adic numbers to within an error of norm at most 1/625: α = (3/5 + 2 + 4 × 5 + 0 × 25 + 2 × 125 + · · · ) − (4/5 + 3 × 25 + 3 × 125 + · · · ), β = (1/25 + 2/5 + 3 + 4 × 5 + 2 × 25 + 2 × 125 + · · · ) × (3 + 2 × 5 + 3 × 125 · · · ), γ= (5 + 2 × 25 + 125 + · · · ) . (3 + 2 × 25 + 4 × 125 + · · · ) 3-7. ; ∑ 1 ; n! ∑ 22n − 1 2n − 1 for p = 2; ∑ (pn+1 ) . pn 3-8. Find the radius of convergence of each of the following power series over Qp : ∑ Xn n! ; ∑ pn X n ; 3-9. Prove that in Q3 , ∑ X pn pn ; ∑ nk X n with 0 ∞ ∑ 32n (−1)n n=1 42n n =2 ∞ ∑ 32n .

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