# Questions tagged [p-adic]

The p-adic tag has no usage guidance.

**16**

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**2**answers

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### 3-adic valuation of a sum involving binomial coefficients

Let $$a(n) = \sum_{0 \leq k \leq n} {n \choose k}{{n+k} \choose k},$$ and define
$b(n) = \nu_3 \bigl(a(n)\bigr)$, where $\nu_3$ is the $3$-adic valuation. About twenty years ago or so, I discovered (...

**8**

votes

**2**answers

703 views

### The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in \...

**2**

votes

**1**answer

593 views

### History of Fargues-Fontaine curve

In this paper, Pierre Colmez wrote about some history of the Fargue-Fontaine curve. In this schedule of London Number Theory Study Group, Fargues was said to give a talk on November 15th on " Where ...

**12**

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**0**answers

259 views

### $p$-Adic or arithmetic variants of Khovanskii's “low complexity $\Rightarrow$ tame topology” theory

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ...

**2**

votes

**0**answers

210 views

### Control theory for Kitagawa's $\Lambda$-adic modular symbols

Let $p$ be a prime, $\Gamma=1+p\mathbb Z_p$ and $\Lambda=\mathbb Z_p[[\Gamma]]$ the Iwasawa algebra. Let $\kappa\colon\Gamma\rightarrow\mathbb Z_p^\times$ be the inclusion. For a character $\...