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### 2-adic Logarithm and Resistance of n-dimensional Cube

Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is $$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=1}^{n}\frac1{{n-1\choose k}}.$$ The ...
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### Simultaneously using the real and 2adic norms

In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...
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### Finiteness of the set of $\mathbb{Q}_p$-rational periodic points

The statement I am concerned with is this: Let $\varphi : \mathbb{P}^r_{\mathbb{Z}_p} \to \mathbb{P}^r_{\mathbb{Z}_p}$ be a morphism of degree higher than one. Then the set of $\mathbb{Q}_p$-...
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Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+... 0answers 78 views ### Weil index computation, p-adic integral The following peculiar p-adic integrals have arisen in my work, and I would be interested if anyone can see how to tackle them. Let F be a p-adic field, \mathfrak{o} its ring of integers, \... 0answers 96 views ### Unexpected isomorphisms between “unrelated fields” I read in the post Why worry about the Axiom of Choice ? that the existence of isomorphisms between \overline{\mathbb{Q}_p}, p any prime, and \mathbb{C}, makes some worry about the Axiom of ... 0answers 94 views ### Relative Leopoldt defect Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number p and M be a quadratic totally real extension of F. Is there a bound of the Leopoldt defect of M ... 0answers 118 views ### Class field theory, Ideles class Let H be a totally complex Galois extension of \mathbb{Q} and g:G_H \rightarrow \bar{\mathbb{Q}}_p be a continuous morphism. By class field thoery we have \mathrm{Hom}(G_H, \bar{\mathbb{Q}}_p)\... 0answers 221 views ### p-adic Lie theory It is well known that exponential map in C^{n\times n} will cover all non-sigular matrix GL(n,C), which is a basic fact in Lie group and lie algebra theory, whether it is true for p-adic cases. ... 0answers 179 views ### Name of some commutative ring akin to p-adics I need help in identifying the naming convention of some commutative ring described below. Let p be a prime, let k be a positive integer, and let$$\textbf{e} = (e_0,\ldots,e_{k-1})$$be a list ... 0answers 102 views ### Cycle with integral coefficients from cycle with \mathbb Z_l-coefficients Let X be a n-dimensional (n>2) smooth projective variety over k=\bar k of positive characteristic. Take a divisor D\in Pic(X). Suppose we know that \frac{[D]^2}{2}\in H^4(X,\mathbb Z_l(... 0answers 99 views ### Computing a projection of a p-adic plane curve Given a prime p and a polynomial equation f(x,y)=0 with rational coefficients, I would like to obtain a precise description of the set of all numbers y\in\mathbb Q_p such that the equation has a ... 0answers 382 views ### What is p-adic Fourier series? Q1: Can we define Fourier series for a function \mathbb{Z}_p\to \mathbb{Q}_p? Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion:$$B_n(\{x\})=-\frac{...
Let $R$ be a ring and $A$ and $R$-algebra. A pro-infinitesimal thickening of $R$ is a pair $(D, \theta)$ such that $\theta: D \rightarrow R$ is surjective and $D$ is separated and complete for the \$I:=...