# Questions tagged [p-adic-analysis]

p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.

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### Restriction of smooth representaions of SL(2,Q_p) to the maximal compact

I am reformulating a question I asked earlier with no answer: Consider $SL(2, Q_p)$ and $K$ a maximal compact subgroup. Let $\pi$ be an irreducible spherical representation of $SL(2, Q_p)$ (in the ...

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### Valuation of congruent elements in a local division ring

Let $K$ be a complete local division ring (note $v$ its valuation). For $x,y\in K$ ($y\ne0$), one puts $x^y=yxy^{-1}$. Let $r\in\mathbb N$. Consider $x,y\in K$ and $a,b\in K^*$ such that $v(x-y)\ge r$ ...

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### Zeros of $p$-adic power series and rationality

Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series.
Fix $f \in V\langle t_1,\ldots, t_n\rangle$....

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### Topological dimension of $p$-adic manifolds

What is the topological dimension of a (locally analytic) $p$-adic manifold over a non Archimedean field $K$?
Is the topological dimension of $K^n$, $n$?

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### What role, if any, do Archimedean valuations play in adic spaces?

I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...?
Is there a weird ...

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### Reference request: the dual Coleman family

Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an ...

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### Uniformity of the set of poles of Igusa local zeta functions

Let $Ω_p$ denote the set of the real parts of the poles of the Igusa zeta function of a polynomial $f∈\mathbb{Z}[X_1,…,X_m]$ (assume $f(0)=0$ so that $\Omega_p\ne \emptyset$) at the prime p. From ...

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### Comparisons of log canonical thresholds

Premise
Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the ...

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### Reference Request: Specialization map in Huber's Context

The specialization map $sp:\mathfrak{X}_\eta\to \mathfrak{X}_{red}$ has an important role in rigid analytic geometry. I tried looking in Huber's papers ("Continuous Valuations", "A generalization of ...

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### Binomial coefficients in discrete valuation rings

Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer.
For any integer $d,n\ge 0$, define:
$${\pi^d \choose n} := \...

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### Complete characteristic p perfect Tate rings are uniform?

In Lemma 7.1.6 of his lecture notes on perfectoid spaces, Bhatt states that every complete characteristic p perfect Tate ring $A$ is uniform. In the proof he uses the Banach open mapping theorem on ...

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### Strengthening Fargues' conjecture (a categorification of p-adic local Langlands)

Fargues's conjecture is a geometric categorification of p-adic local Langlands; it is described in Section 4.3 of his pre-print, https://arxiv.org/abs/1602.00999. Can the functor between groupoids be ...

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### Integral element over p-adic power series

Let $p$ be a prime number. and $R[[X]]$ be the ring of formal series with coefficients in a $p$-adic field $R$. Let $\Lambda=\mathbb{Z}_p[[X]]$.
Question 1) Does there exist an explicit description ...

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### History of the relation between $p$-adic measures and power series

In 1964, Kubota and Leopoldt defined the $p$-adic $L$-function by means of some $p$-adic sums (now called the Volkenborn integral which is a $p$-adic distribution). Later, Mazur (in his secret ...

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### Differential topology on arbitrary fields

What do the differential topology theories on arbitrary fields have in common?
Different differential topology theories
There is "ordinary" differential topology on real manifolds, with its rich ...

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### Open subgroups of finite index of p-adic semisimple groups

My set up is the following: I have an affine algebraic group $G$ over a $p$-adic field $F$, we assume that $G$ is semisimple and simply connected. I have an abstract subgroup $H\leq G(F)$ of the group ...

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### For tori $S \subseteq T$, every character of $S(k)$ extends to a character of $T(k)$?

Let $k$ be a $p$-adic field, $T$ a torus over $k$, and $S$ an $k$-subtorus of $T$. If $\chi: S(k) \rightarrow \mathbb{C}^{\ast}$ is a smooth (resp. continuous) homomorphism, then does $\chi$ ...

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### Relation between Gross-Koblitz and Chowla-Selberg formulas

The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-...

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### Nuclear operator between general topological modules over ultrametric Banach rings

In the celebrating paper "Completely continuous endomorphisms of p-adic Banach spaces", Serre established a Fredholm-Riesz theory for compact endomorphisms of Banach spaces over (spherically complete)...

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### $p$-adic exponentials for $p$-adic Lie groups

Let $G$ be a $p$-adic Lie group, $\text{Lie}(G)$ its Lie algebra.
Is there any reasonable notion of exponential map $\text{exp} : \text{Lie}(G)\to G$?

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### p-adic Poincaré Lemma

suppose $X$ is a proper and smooth rigid analytic variety over $\text{Spa}(k)$, with $k$ a non-archimedean field of characteristic zero.
One has the de Rham complex of analytic differential forms on $...

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### Definition and properties of $\mathcal{B}^\dagger$

I am currently trying to read Colmez' "Série principale unitaire pour $Gl_2(\mathbb{Q}_p)$ et représentations triangulines de dimension 2", that you can find here
https://webusers.imj-prg.fr/~pierre....

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### A question of integral on $p$-adic fields $\mathbb{Q_p}$

We assume that $(\pi,V)$ is an admissible, irreducible and infinite-dimensional representation of $GL_2(\mathbb{Q_p})$. In the proof of existence and uniqueness of Kirillov model, the key step is that ...

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### Does $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ imply $x,y\in (\mathcal{E}^\dagger)^*$

Let $\mathcal{R}$ be the Robba ring and $\mathcal{E}^{\dagger}$ the elements of $\mathcal{R}$ that are bounded at 0 (so the coefficients of the powerseries are bounded. Is it true that $x, y\in \...

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### References on topological rings

What is a good book on topological rings and modules?
I'm interested in topological rings and modules typically endowed with non-linear topologies, e.g.. non-linearly topologized normed rings.
I ...

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### Analytic p-adic functions from Mahler coefficients

Let $f: \mathbb Z_{\ge 0} \to \mathbb C_p$ be any function.
My understanding is that Mahler's theorem says
that $f$ extends to a continuous function
$f: \mathbb Z_p \to \mathbb C_p$ if and only if the
...

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### (Non-)Existence of certain invariant distributions on a p-adic space

Following Bernstein-Zelevinski, an $\ell$-space is a Hausdorff, locally compact totally disconnected topological space. For an $\ell$-space $X$, denote $S(X)$ the space of Bruhat-Schwartz functions on ...

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### Smooth intertwining operators

Let $V$ be a crystalline irreducible representation of the absolute Galois group of $\mathbb{Q}_p$ with distinct Hodge Tate weights $(0,k-1), k \in \mathbb{Z}_{\geq 2}$.
Then $V$ is uniquely ...

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### Nick Katz observation: “the rationality of the zeta function!”

In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-...

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### Locally analytic vectors of a quotient space

My question here is in connection with one of my previous question
"A definition of a (amalgamated) direct sum"
Following the notations there, my question is:
Why the locally analytic vectors of $B(...

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### A question on analytic vectors of $p$-adic analytic groups

I have some trouble understanding the proof of the first part of proposition $3.3.23$ of this famous paper by M.Emerton. The proposition says that:
Let $G$ be a affinoid rigid analytic group, $K$ ...

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### Nijmegen 1978 $p$-adic analysis proceedings

Anyone knows if there is a chance of getting a copy of the following:
Proceedings of the Conference on p-adic Analysis.
Held in Nijmegen, January 16–20, 1978. Report, 7806. Katholieke Universiteit, ...

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### Parity in degrees of determinantal varieties

Let $M_{m,n}(\Bbb{C})$ be the space of $m\times n$ matrices with entries in $\Bbb{C}$, and let $U_{k,m.n}(\Bbb{C})\subset M_{m,n}(\Bbb{C})$ be the variety of matrices of rank $\leq k\leq\min(m,n)$. ...

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### A definition of a (amalgamated) direct sum

I am wondering about a definition of a direct sum in page $31$ of this paper by R. Liu.
I am following the notations in page $31$ of the above paper. Let $V$ be a crystalline irreducible ...

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### Characterization of Krasner analytic functions on the complement of $p$-adic integers

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers, $\mathbb{Q}_p$ the field of fractions of $\mathbb{Z}_p$, and $\mathbb{C}_p$ the completion of the algebraic closure of $\mathbb{Q}_p$. Let $v_p$ be ...

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### p-adic representations of $GL_2(\mathbb{Q}_p)$

Let $L$ be a finite extension of $\mathbb{Q}_p$. Colmez defines here
the trainguline representations which are extensions of Robba rings of dimension $1$. Then, in this paper he contructs the ...

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### Cardinality of ${\mathbb{C}_p}$ [closed]

I know, that field ${\mathbb{Q}_p}$ (field of p-adic numbers) has the same cardinality as $\mathbb{C}$. Taking algebraic closure doesn't change the cardinality of infinite field, so cardinality $\...

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### Logarithm function on the Robba ring $\mathcal{R}_L$ and some properties of $\mathcal{R}_L$

I am reading some of Colmez papers about $(\varphi,\Gamma)$-modules over the Robba ring $\mathcal{R}_L$, where $L$ is a finite extension of $\mathbb{Q}_p$. Now I would like to understand this ring ...

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### Compact subgroups of general linear groups over affinoid algebras

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $(A,A^0)$ be a $k$-affinoid algebra, where $A^0$ is the subring of power bounded elements. Suppose given a compact subgroup, L, of $GL_n(A)$, is ...

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### $p$-adic numbers in physics

As far as I know, in modern physics we assume that the underlying field of work is the field of real numbers (or complex numbers). Imagine one second that we make a crazy assumption and suggest that ...

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### Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$.
Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb ...

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### Stern-Stolz in $p$-adic case

I'm trying to figure out if the following statement is trivial or not:
For $b_i \in \mathbb{C}_p$ (the complete $p$-adic field), if $\sum |b_i|_p < \infty$, then the continued fraction $b_0+\...

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### Crystalline extension the $p$-adic cyclotomic character

Let $\epsilon_p$ be the $p$-adic cyclotomic character, $F$ be a real quadratic extension of $\mathbb{Q}$ in which $p$ splits, $\psi$ be an odd character of $G_\mathbb{Q}$ of finite image and with ...

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### extension of the universal cyclotomic character

Let $p$ be a prime number, $\psi:G_\mathbb{Q} \rightarrow \bar{\mathbb{Q}}_p$ be an odd character of conductor $N$ prime to $p$, with finite image and such that $\psi(p)=1$. Let $\mathcal{W}$ be the ...

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### convergent series representation for p-adic complex numbers

The field $\mathbb{C}_p$ of $p$-adic complex numbers is the completion of the algebraic closure of $\mathbb{Q}_p$ with the corresponding extension of the usual non-Archimedean valuation $|\;\;|_p$.
...

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### embedding local characters to global (Hecke) characters

Let $F$ be a number field, and $S$ be a non-empty finite set of places of $F$. Suppose that for each $v\notin S$, we have a character $\chi_v$ of $F_v^\times$ (for my purpose, we can require that $\...

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### Discontinuous subgroups of $PGL_2(\mathbb{Q}_p)$

I'm trying to read about Mumford curves. I've barely begun and I've already encountered a stumbling block. I'm sure this is probably a basic question that an expert could resolve quickly. I would very ...

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### Dwork's proof of rationality of zeta function, crux of his generalization of a result of Borel along the way

In this article by Katz and Tate here, there's a nice account of Dwork's argument for showing the rationality of the zeta function part of the Weil conjectures. Here is an excerpt.
To recapitulate, ...

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### Weaker version of Dwork's rationality of zeta function, what is needed to beef up into a complete proof?

This is a followup to my question here.
Here is a note of Michael Larsen where he gives a very simple proof of a slightly weaker result than Dwork's rationality of the zeta function.
http://mlarsen....

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### Crux of Dwork's proof of rationality of the zeta function?

As the question title suggests, what is the crux of Dwork's proof of the rationality of the zeta function? What is the intuition behind the proof, what are the key steps that the proof boils down to?