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Results tagged with p-adic-numbers
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user 2481
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
4
votes
1
answer
300
views
Transcendence of a ratio of p-adic logarithms
Let $p, \ell_1, \ell_2$ be distinct prime numbers, and $x_1, x_2 \in \overline{\mathbf{Q}}^\times$.
If
$$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$
does it follow that …
- 37k
9
votes
Accepted
$(\varphi, \Gamma)$-modules of finite height
Proposition: If $D$ is not of finite height, then nor is $D \otimes \delta$ for any $\delta$ of rank 1.
Proof: It suffices to show the contrapositive: if $D$ is of finite height so is $D \otimes \de …
- 37k
9
votes
Accepted
$p$-adic L function of an odd Dirichlet character
Theorem. Let $p > 2$ be prime, and let $\chi$ be a Dirichlet character (non-trivial, and of prime-to-$p$ conductor, for
simplicity). Choose $t \in \mathbb{Z} / (p-1)\mathbb{Z}$ such that
$(-1)^t = \ch …
- 37k
5
votes
Some questions on the $p$-adic properties of special $L$-values
You might enjoy reading this paper: Avner Ash and Glenn Stevens, "Modular forms in characteristic $\ell$ and special values of their L-functions", Duke Math. J. 53 (1986), no. 3, 849-868. The gist of …
- 37k
14
votes
Accepted
What is the value of $p$-adic $\zeta$-function at positive integer point?
If you use this definition, then $\zeta_p(k)$ is zero at negative even integers $k$, so by a $p$-adic continuity argument, it must also be zero at positive even integers.
What about the odd integers? …
- 37k
6
votes
Accepted
Describing the Gamma-transform explicitly in terms of power series
This is a hard problem (and one which is easily overlooked by the unwary)! Just to be clear, I'll summarize (how I think about) the problem: as a relation between additive and multiplicative Fourier t …
- 37k
10
votes
Accepted
Non-existence of "higher" Artin map
There is no way of reformulating local Langlands for $n > 1$ in terms of such a map.
Local Langlands is a bijection between irreducible smooth representations of $\operatorname{GL}_n(K)$, and $n$-dime …
- 37k
5
votes
p-adic L functions from Selmer groups - how canonical are they?
All of your questions are undermined by the same fundamental issue: you cannot talk about "the" p-adic $L$-function in this generality, because there is no sensible definition of what a $p$-adic $L$-f …
- 37k
9
votes
$p$-adic analogue of modular forms, upper half-plane, and $L$-functions
The subjects of "p-adic L-functions" and "p-adic modular forms" are so closely intertwined that it's virtually impossible to talk about either one without immediately running into the other. Applicati …
- 37k
18
votes
Accepted
What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?
I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{G …
- 37k