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### What is the Plancherel Measure for $\textrm{SL}_3(\mathbb{Q}_p)$?

I am looking for a description of the Plancherel Measure of $\textrm{SL}_3(\mathbb{Q}_p)$. Has this been calculated yet? I've search many places for it, but I've only found results on real/complex ...
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### Books on analytic functions on Banach spaces over a non-Archimedean field

I'm looking for good textbooks on analytic functions on Banach spaces over a non-Archimedean field. If you know one(s), please let me know.
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This might be naive question but I was wondering whether a p-adic analogue of the following (shockingly) beautiful formula $$\zeta(s)\Gamma(s) = \int_0^\infty \frac{t^{s-1}}{e^t-1} dt$$ (vaild for $\... 0answers 661 views ### Newton Method in$p$-adic case The Newton Method over$\mathbb{R}$has the property that the precision is doubled (under some continuous differentialbe assumption) in each iteration. For the ring$\mathbb{Z}_p$of$p$-adic integers,... 1answer 195 views ### The set of$p$-adic numbers of some fixed$n^ {\rm th}$-power residue Hi, Consider the set$\lambda P_n \subset \mathbb{Q}_p$, where$\lambda \in \mathbb{Q}_p^{\times}$and$P_n$is the set$\lbrace x \in \mathbb{Q}_p \mid \exists y \in \mathbb{Q}_p x= y^n\rbrace$. Is ... 11answers 4k views ### 'Important' applications of p-adic numbers outside of algebra (and number theory). Surely,$\mathbb{Z}_p$and$\mathbb{Q}_p$(and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily ... 10answers 7k views ### Elementary results with p-adic numbers I'm giving a talk for the seminar of the PhD students of my math departement. I actually work on Berkovich spaces and arithmetic geometry but, of course, I cannot really talk about that to an audience ... 2answers 880 views ### P-adic representations Hi, I am reading about p-adic representations from Fontaine's book which can be found at http://staff.ustc.edu.cn/~yiouyang/research.html. On page 145 where they prove Proposition 5.24 which is ... 1answer 260 views ### Partitioning a compact open set into balls in an ultrametric space Consider a$p$-adic field$K$with the standard topology inherited from the usual$p$-adic norm$\mid \cdot \mid$. Consider the ultrametric space$X=K^n$with the topology inherited from the norm$\| \...
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Where appears for the first time the term Hodge-Tate representation. Can i find somewhere explanation of the terminology Hodge-Tate, Derham etc. for representations and Fontaine's rings.
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### When are roots of power series algebraic?

Let $K$ be a field and consider a power series $f(T) \in K[[T]]$. Under what conditions (on $K$ and/or on $f$) can we conclude that if $\alpha$ is a root of $f(T)$ then $\alpha$ is in fact algebraic ...
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I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
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### Can local duality for elliptic curves be proven with “big rings”?

From Exercise 5.14, Ch. V of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", I learned that the local duality for elliptic curves over $p$-adic fields can be proven for Tate curves ...
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### Is anything known about flow along vector fields over complete normed fields?

In Bourbaki "Variétés différentielles et analytiques" there is a statement (without proof) that for a vector field on smooth manifold over complete normed field (characteristic zero) there is a local ...
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### Continuous extensions reals and to p-adic numbers

Assume $f\colon \mathbb Q\to \mathbb Q$ is a function which admits continuous extensions $f_0\colon\mathbb R\to \mathbb R$ and $f_p\colon \mathbb Q_p\to \mathbb Q_p$ for each prime $p$. Is ...
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The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator. it is known that the classical local monodromy theorem (i.e. for ...
Let $f$ be a formal power series with coefficients in the ring of integers of a finite extension of ${\mathbb Q}_p$. Is there a simple algorithm to compute a positive lower bound for $|\alpha - \beta|... 1answer 2k views ### Field with one element example? $$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$ This formula holds for the unit ball in$\mathbb{Q_p}$. This formula also holds for$\mathbb{R}$when$p=1$. Should one expect $$\... 1answer 365 views ### Relating p-adic Valuations of Elements in \mathbb{C} and \mathbb{C}_p Let K = \mathbb{Q}(\theta), where \theta is a root of an irreducible polynomial g \in \mathbb{Z}[t]. Fix a rational prime p. Let \theta^{(1)}, \ldots, \theta^{(n)} be the roots of g in \... 1answer 528 views ### The Galois representation of a p-divisible group is crystalline Can someone explain (or give a reference) why the Galois representation attached to a p-divisible group over the ring of integers of a p-adic ring is Crystalline? 0answers 322 views ### reference for p-adic Stein spaces Hi, I'm looking for a reference in english for p-adic Stein spaces. The usual referneces I come across are all in german. Thanks 1answer 2k views ### Two-variable p-adic L-functions of elliptic curves Suppose K is an imaginary quadratic field (with class number 1, for simplicity), p \ne 2 a prime split in K, and K_\infty the \mathbb{Z}_p^2-extension of K. If E / \mathbb{Q} is an ... 3answers 557 views ### Connectifications? Like many of my questions, this question is actually aimed at p-adic analysis. One of the main obstacles in doing analysis p-adically ist that the \mathbb{Q}_p is totally disconnected. From ... 3answers 1k views ### Are there 'analytic' p-adic modular forms. The most elementary way to define p-adic modular forms is via limits of classical modular forms. More precisely f \in \mathbb{Z}_p[[q]] is called a p-adic modular form if there are modular forms ... 4answers 2k views ### Locally constant functions with compact support = smooth ? Hello, I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions. Let F be a p-adic field. Then (from what I understand) C_c^{\infty}(F) is the ... 2answers 720 views ### Functional equations relating to p-adic L-functions Let f be a modular form of weight k for \Gamma_0(N). Let us assume that p\not\vertN. Then we can construct 2 p-adic L-functions corresponding to the 2 roots \alpha and \beta of the equation x^... 3answers 1k views ### Non-vanishing of p-adic L-functions In Non-vanishing of L-series of modular forms (easy case?) it was answered that for a cuspidal newform f of weight strictly greater than 2, then L(f,1) is non-zero. (Here the L-series is ... 8answers 4k views ### p-adic integrals and Cauchy's theorem A short version of my question is: Is there a p-adic theory of integration? Now let me expand a little further. In introductory texts such as Koblitz' book p-adic numbers,.. a bunch of p-adic ... 1answer 2k views ### Why use Teichmuller representatives? In p-adic mathematics, what is the advantage of using Teichmuller representatives over using just the numbers 0,1,2,...,p-1 ? In either case, the norm is the same. In either case, all the points are ... 0answers 1k views ### The radius of convergence of the p-adic exponential function. As every number theorist learns, the radius of convergence of exp(x), defined by the usual power series in a neighborhood of zero, is$$\rho = p^{-1/(p-1)}.$$This is typically proven by computing ... 2answers 1k views ### Direct proof of special case of Hasse's theorem for elliptic curves Consider the elliptic curve y^2 = x^3 + x over \mathbb{F}_p, where p \equiv 1 \pmod 4. If memory serves correctly, the number of points (excluding the point at infinity) is p - a where a is ... 1answer 336 views ### Does it exist a p-adic L function which interpolates the values of the complex one at positive integers? I known that there are classical ways to construct p-adic L functions for Dirichlet characters through p-adic integrals. We fix a character \chi modulo Np^r with N and p coprime and an ... 3answers 3k views ### What is the p-adic valuation of a number? There seem to be two conflicting definitions for p-adic valuation in the literature. Firstly, for any non-zero integer n, we have \nu=\nu_p(n) is the greatest non-negative integer such that p^\nu ... 1answer 625 views ### Can an etale (phi, Gamma) module be an extension of non-etale ones? This question is about p-adic representations of \mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p) and (\varphi, \Gamma)-modules. By theorems of Fontaine, Cherbonnier-Colmez and Kedlaya, the ... 1answer 679 views ### Fields of definition for p-adic overconvergent modular eigenforms If we consider the action of the U_p operator on overconvergent p-adic modular forms, then we can get some information about the field over which the eigenforms are defined by looking at the ... 4answers 2k views ### Extension of valuation Fix a prime number p. Suppose that I have a valuation v_p: \mathbb{Q} \to \mathbb{Q} on the rationals \mathbb{Q}. That is, v_p( p^n(\frac{a}{b})) = p^{-n} where each of a,b is not divisible ... 1answer 509 views ### Are centrally extended p-adic groups defined over F_1? Let G be a semisimple algebraic group. Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor G : Rings → Groups by the second algebraic K-... 1answer 262 views ### Classifying continuous characters X \to \mathbb{Z}_p^*, X=\mathbb{Z}_p^* or (1+p\mathbb{Z}_p)^{\times} ? Question : are the continuous characters of the form \eta : \mathbb{Z}_p^* \to \mathbb{Z}_p^*, or \eta : (1+p\mathbb{Z}_p)^{\times} \to \mathbb{Z}_p^* (i.e., on the principal units in \mathbb{... 2answers 2k views ### p-adic L-functions For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's p-... 1answer 687 views ### bibl. q.s on Dwork's “p-adic cycles”, Mazur's “p-adic variations”: Matthew Emerton mentioned recently the relevance of Dwork's "p-adic cycles". As I wonder if I should read that, reviews of it are ambiguous, I'd be happy on remarks and possible further bibl. hints. ... 3answers 3k views ### Dwork's use of p-adic analysis in algebraic geometry Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much ... 1answer 1k views ### Stark's conjecture and p-adic L-functions Not long back I asked a question about the existence of p-adic L-functions for number fields that are not totally real; and I was told that when the number field concerned has a nontrivial totally ... 3answers 3k views ### An unfamiliar (to me) form of Hensel's Lemma In his very nice article Peter Roquette, History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355, Fields Inst. Commun., ... 5answers 1k views ### When does a p-adic function have a Mahler expansion? Let f: \mathbb{Z}_p \rightarrow \mathbb{C}_p be any continuous function. Then Mahler showed there are coefficients a_n \in \mathbb{C}_p with$$ f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}.$$... 1answer 475 views ### how do you evaluate the p-adic modular form E_p-1 in the region |j|<1 background/motivation let Ek denote the modular form of level one and weight k with q-expansion given by$E_k(q)=1- \frac{2k}{b_k}\sum_n \sigma_{k-1}(n)q^n$where σi is the divisor sum and bk ... 4answers 2k views ### Special values of$p$-adic$L$-functions. This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes. My understanding is that nowadays there are conjectures which essentially ... 3answers 1k views ### Free subquotient of Galois representations coming from Hida theory Let$\mathbf{T}$be the reduced nearly ordinary Hecke algebra of level$N$of Hida theory for$\operatorname{GL}_{2}$over$\mathbb{Q}$(or more generally over a totally real field$F$). Then$\mathbf{...
Suppose that $N$ is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level $\Gamma_0(N)$. For such an eigenform $f$, the coefficients generate (an order in) the ring of ...