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### What is the precise relationship between primitive Hida families and the connected components of the ordinary locus of the eigencurve?

In the references I've found discussing this question, I have not found any statements that I can understand and that are as precise as I would like. I'm more familiar with Hida families than with the ...
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Let $p$ be a prime, $\Gamma=1+p\mathbb Z_p$ and $\Lambda=\mathbb Z_p[[\Gamma]]$ the Iwasawa algebra. Let $\kappa\colon\Gamma\rightarrow\mathbb Z_p^\times$ be the inclusion. For a character $\... 2answers 744 views ### Witt-vector vectors I've never really made my way in any detail through the Witt-vector construction. I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could ... 0answers 49 views ### Dimensions of fibers of analytic map I must admit that I know nothing about p-adic geometry, so the following question may be completely trivial. Let$V\subset K^n$be an affine algebraic variety. Let$D$be a polydisk, and$F$be an ... 0answers 85 views ### How is the p-adic norm calculated when using universal witt vectors? How is the p-adic norm calculated when using UNIVERSAL WITT VECTORS? Is the p-adic norm calculated in the familiar way, in the sense that we look to the last digit to the right, and the prime number ... 0answers 179 views ### Name of some commutative ring akin to$p$-adics I need help in identifying the naming convention of some commutative ring described below. Let$p$be a prime, let$k$be a positive integer, and let $$\textbf{e} = (e_0,\ldots,e_{k-1})$$ be a list ... 1answer 285 views ### Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed] Why is every l-adic Galois representation $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Q}_{l})$$ conjugate to one over the l-adic integers? $$G_{\mathbb{Q}_p}\rightarrow GL_n(\mathbb{Z}_{l})$$ 0answers 272 views ### Power series which are$p$-adic modular forms for all$p$Suppose that, for some integer$k$, a series$f(q) \in \mathbb Q \otimes \mathbb Z[[q]]$has the property that for every prime$p$,$f(q)$is the$q$-expansion of a$p$-adic modular form of weight$k$... 0answers 174 views ### The operator$\left(q\frac{d}{dq}\right)^s$and fractional derivatives of modular forms Recall the notion of a "nearly holomorphic modular form" introduced by Shimura: A function$f : \mathfrak h \to \mathbb C$is said to be nearly holomorphic of level$\Gamma_1(N)$, weight$k$and ... 1answer 342 views ### Definition of p-adic modular forms I have been reading Hida's book "p-Adic automorphism forms on Shimura varieties" and I don't understand a point. He first describes p-adic modular forms of tame level N as functions on the Igusa ... 0answers 199 views ###$p$-Adic or arithmetic variants of Khovanskii's “low complexity$\Rightarrow$tame topology” theory This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ... 1answer 112 views ### Twisted Padé approximants Let$f$be a continuous function defined on$\mathbb Z_p$. By Mahler theorem, there exists a sequence$(\gamma_k)_{k\in\mathbb N}$of$\mathbb Z_p$such that for every$z\in\mathbb Z_p$f(z)=\sum_{k\... 1answer 158 views ### “frequency” of fields for which the p-adic regulator vanishes (mod p) There is a very nice question which arises in the study of the Discrete Logarithm Problem which I wish to present here. The question, in a general setting, is to specify an empirical expression for ... 2answers 398 views ### Texts about Dwork's work I want to ask about references to papers, that probably exist, which explain the articles of Bernard Dwork starting from "The rationality of the zeta function of an algebraic variety" to "On the ... 2answers 619 views ### Automorphisms of$\mathbb C_p$I am looking for a non-trivial automorphism$\sigma$of$\mathbb C_p$such that$\sigma(\mathbb Q_p)\subset\mathbb Q_p$. If$\mathbb C_p$were spherically complete, then by Hahn-Banach theorem, that ... 0answers 203 views ### p-adic etale cohomology Let$X$be a smooth projective scheme over$\mathbb{Z}_p$, with special fiber$X_s$over$\mathbb{F}_p$, generic fiber$X_{\eta}$over$\mathbb{Q}_p$, and geometric generic fiber$\bar{X_{\eta}}$over ... 1answer 363 views ### p-adic L-functions of modular forms: why the condition$v_p(\alpha)<k-1$? Let$f$be a modular form (cuspidal, new, eigenform) of weight$k$and level$N$. Let$p$be a prime number not dividing$N$. In order to construct a$p$-adic$L$-function$L_p(f, s)$interpolating ... 1answer 140 views ### Can one determining the p-adic lattice just from the values of the quadratic form on a p-group? Given a finite$p$-group$A$, with a non-degenerate quadratic form$q:A\rightarrow \mathbb Q/2\mathbb Z$(that is a map satisfying$q(na)=n^2q(a)$for all$n\in \mathbb Z,a\in A$), an important result ... 1answer 269 views ### computing spaces of$p$-adic modular forms Let$p$be a prime, and$\alpha$a positive integer. How do you compute the space of$p$-ordinary$p$-adic modular forms (in the sense of Serre) of weight 2 on$\Gamma_0(p^\alpha)$? I'm really only ... 2answers 177 views ### differences between character distributions of supercuspidal representations and others Let$G$be a$p$-adic linear reductive group. For an irreducible admissible smooth representation$\pi$of$G$, there is a distribution$\Theta(\pi)$, called the character distribution, attached to$\...
Let $K$ be a complete local field with discrete valuation, and let $L/K$ be a finite Galois extension. Use $G=Gal(L/K)$ to denote the Galois group. In Serre's book 'local fields', chapter 6, a ...