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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4
votes
1answer
172 views

Deformations of the Riemann zeta function

Consider the Dirichlet series (for fixed $0 < a \leq 1$): $$\zeta_a(s) = \sum_{n\geq 1}\frac{a^n}{n^s}$$ which reduces to the Riemann zeta function for $a=1$. What is known about this function, ...
2
votes
1answer
232 views

Local to global principle for a pair of bilinear equations?

Let $A_{i, j}, B_{i, j}, C, D \in \mathbb{Q}$, and consider the following pair of equations $$ A_{1, 1} x_1 y_1 + A_{1, 2} x_1 y_2 + A_{2, 1} x_2 y_1 + A_{2, 2} x_2 y_2 = C $$ $$ B_{1, 1} x_1 y_1 + B_{...
6
votes
1answer
161 views

Lindemann theorem for Artin-Hasse exponential

Though the Lindemann--Weierstrass theorem is not known in the $p$-adic settings, its "Lindemann" part -- the transcendence of $\exp(a)$ for algebraic $a$ with $0<|a|_p<p^{-1/(p-1)}$ -- was shown ...
2
votes
1answer
133 views

Reduced complete Tate ring which is not uniform?

Recall that a topological ring $A$ is Tate if there is an open subring $A_0$ such that the induced topology on $A_0$ is t-adic for some $t \in A_0$ that becomes a unit in $A.$ One can, given a Tate ...
5
votes
1answer
175 views

Classification of finitely generated modules over non-commutative rings

Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...
4
votes
0answers
64 views

Existence of a “p-adic Mahler measure” or alternatively, the converge of a p-adic sequence

Let $f \in \mathbb Z_p[[t]]^\times$ be an invertible power series and let $\log_p$ be the p-adic logarithm with the normalization that $\log p = 0$. Consider the sequence: $$a_n = \frac{1}{p^{n-1}}\...
1
vote
2answers
138 views

Connection between Volkenborn integral and Haar measure on $\mathbb{Q}_p$

This may be a rather elementary question, but I haven't been able to figure it out on my own, and the literature appears to be eerily silent on the topic. Since $\mathbb{Q}_p$ is a locally compact ...
6
votes
0answers
204 views

Numerical analysis with p-adic numbers

How should one go about doing numerical analysis with $p$-adic numbers? By that I mean, how should one go about implementing numerical integration (using analogues of Newton-Cotes or perhaps Gaussian ...
1
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0answers
73 views

compact $p$-adic Lie group can be embedded into $O_K^n$ or $\text{GL}_n(K)$?

Let $K$ be a local field of charecteristic $0$ and $G$ be a compact $p$-adic Lie group of dimension $n$, then can $G$ be embedded into $O_K^n$ or $\text{GL}_n(K)$ as a closed subgroup? This is a dual ...
5
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0answers
243 views

Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics

Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-...
4
votes
2answers
288 views

The formula for (and computation of) the inverse p-adic mellin transform

So, after scouring the entirety of the internet, I managed to find one (and, so far, only one) source that actually explains how to invert the $p$-adic mellin transform: $$\mathscr{M}_{p}\left\{ f\...
3
votes
1answer
117 views

Asymptotic analysis using the p-adic Mellin Transform?

In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...
1
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0answers
58 views

valuation of a derivative in a completion

Let $q$ be a power of a prime $p$ and $w$ be an irreducible polynomial of $\mathbb F_q[T]$. Denote by $\mathbb C_w$ the completion of an algebraic closure of $K_w$, the completion of $\mathbb F_q(T)$ ...
6
votes
1answer
255 views

Theory of integration for functions from $\mathbb{Z}_{p}$ to $\mathbb{Z}_{q}$ for distinct primes $p,q$

Let $p$ and $q$ be prime numbers. When $p=q$, Mahler's Theorem gives a complete description of $C\left(\mathbb{Z}_{p};\mathbb{Z}_{p}\right)$, the space of continuous functions from $\mathbb{Z}_{p}$ to ...
3
votes
1answer
114 views

How does an analytic space correspond to a $p$-adic Banach space

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $V$ be a Banach algebra over $K$, then what is the $K$-analytic space corresponding to $V$? What is the definition of $K$-analytic space? This is ...
3
votes
0answers
98 views

Jacobian change of variables formula for $p$-adic valued integration?

Let $k$ be a $p$-adic field. It's possible to make sense of the Haar measure $\mu_{\operatorname{Haar}}$ on $k^n$ as a $k$-valued measure and define integrals $$\int\limits_{k^n} f(x_1, ... , x_n) d\...
4
votes
2answers
275 views

$p$-adic series bounded if and only if it has finitely many zeros

Let $L\subseteq\mathbb{C}_p$ be a finite extension of $\mathbb{Q}_p$, $r$ be a positive real number, and $f$ be a series $\sum_{n\in \mathbb{Z}} a_nz^n$ convergent in $D= \{x\in \mathbb{C}_p|0<v(x)\...
2
votes
0answers
142 views

Rigid analytic geometry and Tate curve

I am stuck in the proof of theorem 5.1.4 in the book rigid analytic geometry and its applications on page 126. The authurs define $\Gamma:=G^{an}_{m,k}/<q\gt$ where $k$ is a complete non-...
0
votes
0answers
51 views

Comparison of growth of entire functions in a $p$-adic field

Let $p$ be a prime number, $\Omega_p$ be the spherically complete extension of $\mathbb C_p$. Consider two entire functions on $\Omega_p$: $f(z)=\sum_{n\ge 0}a_nz^n$ and $g(z)=\sum_{n\ge0}b_nz^n$. ...
6
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0answers
87 views

ultrametric Rademacher theorem

The classic Rademacher theorem roughly states that Lipschitz continuous functions are almost everywhere differentiable. However, there are well-known ultrametric counterexamples, see Kobliz's classic ...
2
votes
1answer
113 views

Modulus of growth in $p$-adic spherically complete field of $\mathbb C_p$

Let $F$ be the spherically complete extension of $\mathbb C_p$ and $(a_n)_{n\in\mathbb N}$ be a sequence of $\mathbb C_p$ such that for all $r\in\mathbb R$, one has $$\lim_{n\to+\infty}|a_n|_pr^n=0.$$ ...
7
votes
1answer
190 views

Are maps corresponding to affinoid subdomains flat in the Banach sense?

$\newcommand{\Sp}{\mathrm{Sp}}\newcommand{\abs}[1]{\lvert #1\rvert}\newcommand{\comptensor}{\mathbin{\hat{\otimes}}}$ Let $k$ be a complete non-archimedian field and let $X = \Sp(B)$ be a $k$-affinoid ...
2
votes
0answers
140 views

Fontaine - Wintenberger field of norms and imperfect case

Let $K$ be a complete discrete valued field whose residue field $k_K$ has characteristic $p$ and has the property that $[k_K:k_K^p]=p^d$ for some $d$. Let $t_{\alpha}, 1 \leq \alpha \leq d$ be a set ...
6
votes
1answer
315 views

An example of a morphism of rigid analytic spaces with affinoid base which is proper but does not satisfy $(\dagger)$

Let $k$ be a complete non-archimedean field and let $\varphi \colon X \to Y$ be a morphism of rigid analytic spaces over $k$, where $\newcommand{\Sp}{\operatorname{Sp}}Y = \Sp(B)$ is affinoid. ...
-1
votes
1answer
264 views

p-adic expansion of roots of unity [closed]

Let $w$ be an n-th root of unity, I have two questions 1) What are the conditions on the prime $p$ such that $w\in \mathbb{Z}_p$, and if it is the case what is the p-adic expansion of an n-th root of ...
4
votes
0answers
108 views

Finite dimensional irreps of $p$-adic groups

What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$? One knows such a representations cannot be smooth, so probably the examples will be ...
6
votes
1answer
389 views

Change of variables for $p$-adic integral

Say $p$ is an odd prime. Suppose I have a measure $\mu$ on $\mathbf{Z}_p$. As in II.4.3 in Colmez - Fonctions d'une variable $p$-adique, I can restrict $\mu$ to $1+p\mathbf Z_p$, and there is a ...
2
votes
0answers
71 views

Continuous extension of the derivation in positive characteristic

Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$. Does there exist a derivation on $\Omega$ ...
3
votes
1answer
139 views

Igusa zeta functions of univariate polynomials: $\mathbb{Z}_p$ or $\mathbb{Q}_p$ in this statement

Let $f\in\mathbb{Z}_p[X]$ and let $Z_{f,p}(T)\in\mathbb{Z}_{(p)}(T)$ be the $p$-adic Igusa zeta polynomial (i.e. $Z_{f,p}(p^{-s})$ is the $p$-adic Igusa zeta function in the complex variable $s$, with ...
1
vote
1answer
79 views

Converging sequence of polynomials

Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$ of degree $2$. Does there exist two polyomials $\alpha,\beta\in\mathbb F_q[T]$ (not both zeroes) such that the sequence $(\beta T^{q^{2n}}-\...
3
votes
1answer
262 views

Analytic continuation of a $p$-adic function

Let $(a_n)_{n\in\mathbb N}$ and $(b_n)_{n\in\mathbb N}$ be sequences of $\mathbb Q_p$ such that the function $f:z\in\mathbb Q_p\to\sum_{n\ge0}a_nz^n+b_nz^{n+1}$converges in $\{|z|_p<1\}$. Assume ...
2
votes
1answer
120 views

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 ...
1
vote
1answer
53 views

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 $v(x-y)\ge r$ ...
10
votes
0answers
234 views

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 $f \in V\langle t_1,\ldots, t_n\rangle$....
10
votes
1answer
336 views

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 $K^n$, $n$?
9
votes
0answers
293 views

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 ...
2
votes
1answer
151 views

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 ...
2
votes
1answer
83 views

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 ...
3
votes
1answer
246 views

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 ...
4
votes
1answer
363 views

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 ...
6
votes
1answer
258 views

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} := \...
4
votes
0answers
162 views

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 ...
13
votes
0answers
255 views

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 ...
4
votes
0answers
120 views

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 ...
6
votes
0answers
237 views

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 ...
1
vote
1answer
187 views

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 ...
4
votes
1answer
196 views

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 $\chi$ ...
4
votes
0answers
105 views

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 $L$-...
2
votes
0answers
54 views

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)...
7
votes
2answers
502 views

$p$-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 $\text{exp} : \text{Lie}(G)\to G$?