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.
318 questions
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Dwork's proof of rationality for curves
I would like to have a reference for a proof of the rationality of the zeta function for a curve by Dwork's method. I would like to know if Dwork's proof simplifies for a curve.
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Does Hasse-Arf theorem help for infinite abelian extension of local field?
Let $K$ be a local field and let
$$K \subset L_1 \subset L_2 \subset \cdots$$
be a tower of finite abelian extensions and $L_{\infty}=\bigcup_nL_n$. Then $L_{\infty}/K$ is also abelian.
Let $G=\...
- 510
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A vanishing $p$-adic hypergeometric series
Just for fun, inspired by the recent post of Z-W Sun on vanishing of
a hypergeometric series, how to prove that
$${}_2F_1(1/6,2/3;5/6;80/81)_5=0$$
where ${}_5$ indicates that it is a $5$-adic sum ?
(...
- 14.6k
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Irreducibility of a degree-$27$ polynomial through Newton polygon or residual reduction
Let $K=\mathbb{Q}_3(t)$ be the finite extension of the $3$-adic number field $\mathbb{Q}_3$, where $t=3^{1/13}$. I want to know if the polynomial $$f(x)=(x^9-t^2)^3-3^4x+3^3t^5 \in K[x]$$ is ...
- 510
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Non-solvable subgroups of the first congruence subgroup of two-dimensional special linear group over $\mathbb{F}_{p}[[T]]$
Let $p$ be an odd prime and $\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Let $\mathbb{F}_{p}$ be the finite field of order $p$ and let $\mathbb{F}_{p}[[T]]$ be the ring of formal power series ...
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Find a ring R such that Spec R is homeomorphic to Spa(Z,Z)
I'm following Scholze-Weinstein's Berkeley notes (https://www.math.uni-bonn.de/people/scholze/Berkeley.pdf). And there is a theorem by Huber (Thm 2.3.3) in the notes that says the adic spectrum $\...
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Pointwise bounded subvariety in a rigid tube
I encountered the following question when studying the cohomology of a char p variety, which is really outside of my area of expertise.
Notation: $\mathbb{F}=\mathbb{F}_p$ bar, $W$ its Witt ring, $K=W[...
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Non-Archimedean disks
Let $K$ be a field complete with respect to a non-Archimedean absolute value $|\cdot|$. To develop analysis in $K$, we need the notion of a disk in $K$. There is nothing mysterious at first glance: ...
- 1,354
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Explicit witness to spherical incompleteness of $\mathbb{C}_p$
A nonarchimedean valued field $K$ is said to be spherically complete if, for any nested sequence $B_1 \supseteq B_2 \supseteq \dots$ of balls in $K$, the intersection $\bigcap_{i = 1}^\infty B_i$ is ...
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Orthogonalization of quadratic forms over a $p$-adic Banach space
Let $X$ be an arbitrary set. Let $H = c_0(X, \mathbb{Q}_p)$ be the $p$-adic Banach space with sup norm. Let $\langle \cdot, \cdot \rangle$ be a symmetric, nondegenerate $\mathbb{Q}_p$-bilinear form on ...
- 1,607
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Radius of convergence of solution to p-adic differential equation
I am working on a problem that seems to reduce to determining (to certain precision) the radius of convergence of a particular solution of a p-adic differential equation.
In particular, we have $f(x) =...
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If a $p$-adic power series vanishes at $\zeta_{p^n}^a-1$ for all $n,a$, is it divisible by $\log(1+T)$?
Let $p$ be a prime number, and let $H(\mathbb{C}_p)$ denote the ring of power series $f(T)\in \mathbb{C}_p[[T]]$ such that $f(T)$ converges in an open ball of radius $1$ about $0$. n.b. that this is ...
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Is the NH hash family $\varepsilon$-AXU?
As context, I'll start with summarizing and simplifying the section of "UMAC: Fast and Secure Message Authentication", by Black et al.(https://www.cs.ucdavis.edu/~rogaway/papers/umac-full....
- 111
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Restriction of a Schwartz function to Bruhat open cell over p-adic field
Let $G=GL_n(F)$ over a p-adic field $F$, use $\mathcal{S}(G)$ to denote the compactly supported locally constant functions on $G$. Let $\Omega_n:=N_nA_n\omega_n N_n$ be the open dense Bruhat cell of $...
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algebraic fundamental group of Raynaud generic fiber
Let $k$ be a perfect field of characteristic $p$. Let $X$ be a quasi-projective smooth variety over the Witt ring $W=W(k)$ ($K=\mathrm{Frac}(W)$). Let $\mathcal X$ be the $p$-adic formal completion of ...
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