The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems

211 questions
Filter by
Sorted by
Tagged with
100 views

### Topological generators for $\mathrm{SL}_2(\mathbf{Z}_p)$

$\DeclareMathOperator\SL{SL}$ Let $p>3$ and $G$ be an open subgroup of the special linear group $\SL_2(\mathbf{Z}_p)$ over the ring $\mathbf{Z}_p$ of $p$-adic integers. Suppose that $G$ is ...
1 vote
83 views

### What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2 \mathbb{Z}_p$?

Let $p \geq 2$ be prime and $K=\mathbb{Q} (\zeta_p),~ \zeta^{p-1}=1$ with ring of integers $\mathcal{O}_K$. We denote $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ ...
1 vote
41 views

110 views

### A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture

A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
221 views

### How do I extend the $2$-adic absolute value to prove Monsky's Theorem?

In proving Monsky's Theorem, it is required that we define the $2$-adic absolute value on an arbitrary finitely generated extension of $\mathbb{Q}$ say $\mathbb{K}=\mathbb{Q}(\alpha_1,\ldots,\alpha_n)$... 272 views

### $\lim_{b \rightarrow \infty} {^{b}a} \in \mathbb{Q}_p$ for any $a \in \mathbb{Z}^+$?

$\newcommand\tetra{{^{#1}{#2}}}$In a recent discussion on the Tetration Forum (see https://math.eretrandre.org/tetrationforum/showthread.php?tid=1703&page=2), it has been pointed out how my ...
250 views

### $p$-adic analogue of modular forms, upper half-plane, and $L$-functions

In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently ...
1 vote
183 views

145 views

### Can there exist different smooth, proper schemes over the p-adics with the same generic fiber? [duplicate]

Can there exist smooth, proper $X_1,X_2/\mathbb Z_p$ such that their generic fibers are isomorphic but their reductions mod $p$ are not? Are there examples if we insist that the special fibers are ...
135 views

### Image of Kummer map for CM Elliptic curves

Let $K$ be an imaginary quadratic field and let $F$ be a finite extension of $K$. Let $E$ be an elliptic curve over $F$ with CM by $K$. Suppose that $p$ is a prime that splits as $p=\pi\pi^*$ in $K$. ...
407 views

354 views

### Why does the field norm on the field extension $\mathbb C/\mathbb R$ induce a vector space norm?
There is a general result which holds for the rational numbers $\mathbb Q$ (as well as number fields in general): For any completion $K$ of $\mathbb Q$ and any finite extension $L/K$ of ...