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### Relative Leopoldt defect

Let F be a totally real number field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$. Is there a bound of the Leopoldt defect of $M$ ...
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### A $p$-adic sum of reciprocals of powers

Let $p$ be a prime number and $k\geq 2$ an even integer. Consider the following $p$-adic integer: $$S_{p,k} := \lim_{r\to+\infty} \sum_{a=1}^{p^r} \big(\frac{p^r}{a}\big)^k$$ Convergence is easy to ...
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### A multidimensional version of Hensel's lemma? (for more than one polynomial)

The classical Hensel's lemma is stated as follows: Let $f(x) \in \mathbb{Z}_p[x]$ and $a \in \mathbb{Z}_p$ satisfy $$|f(a)|_p < | f'(a) |_p^2.$$ Then there is a unique $\alpha \in \mathbb{Z}_p$...
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Let $q$ be a power of a prime $p$. For every $i\in\mathbb N$, one denotes $D_i=\prod_{\substack{h\in\mathbb F_q[T]\text{ monic}\\\deg h=i}}\limits h$. For $n\in\mathbb N$, write $$n=n_0+n_1q+\cdots+... 1answer 240 views ### Tube of a mod p point on a smooth Z_(p)-scheme Let R be a smooth, integral, finite-type \mathbb{Z}_{(p)}-algebra of relative dimension n and \overline{f} \colon R \to \mathbb{F}_p. Then Hensel's lemma tells us that this lifts to a map R \... 1answer 343 views ### Uniformizer for splitting field of p^{1/p^n} over p-adics Does anyone know an explicit uniformizer for \mathbf{Q}_p(\zeta_{p^n}, p^{\frac{1}{p^n}}) / \mathbf{Q}_p? I was reading the question "adding an n-th root to Q_p" where dke mentions this question but ... 1answer 170 views ### Rings that inject in all p-adic integers Denote p a prime number and \mathbb Z _p the ring of p-adic integers. We have a canonical injective ring homomorphism :\mathbb Z \rightarrow \mathbb Z_p for all p. But \mathbb Z is not the ... 0answers 378 views ### Lemma in Scholze-Weinstein In the paper "Moduli of p-divisible groups" by Scholze and Weinstein (see http://math.bu.edu/people/jsweinst/Moduli/Moduli.pdf), one finds the following claim in Lemma 5.2.7: Lemma: Let K be a ... 1answer 71 views ### p-adic orthogonal groups in four variables Let p>2 be prime. By the classification of quadratic forms, there are 8 pairwise non-equivalent isotropic orthogonal groups in 4 variables. Is there a concrete classification of orthogonal ... 3answers 2k views ### Does the equation 1 + 2 + 3 + \dots = -\frac{1}{12} have a natural p-adic interpretation? Consider the equation$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$"proved" by Ramanujan Euler. One correct way to interpret this is that \zeta(-1) = - \frac{1}{12}, where \zeta(s) = \sum_{n = 1}^{\... 1answer 335 views ### When do two lattices have the same stabilizer in the diagonal torus? This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer) Let T be the diagonal torus ... 1answer 216 views ### Transcendence of a ratio of p-adic logarithms Let p, \ell_1, \ell_2 be distinct prime numbers, and x_1, x_2 \in \overline{\mathbf{Q}}^\times. If$$ \frac{\log_p x_1}{\log_p \ell_1} = \frac{\log_p x_2}{\log_p \ell_2}, $$does it follow that ... 0answers 180 views ### Name of some commutative ring akin to p-adics I need help in identifying the naming convention of some commutative ring described below. Let p be a prime, let k be a positive integer, and let$$\textbf{e} = (e_0,\ldots,e_{k-1})$$be a list ... 2answers 330 views ### What is the value of p-adic \zeta-function at positive integer point? p-adic zeta function is a p-adic interpolation of the Riemann \zeta-function for the values \zeta(1−k), k\ge 1 (see p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz) ... 1answer 285 views ### Why is every l-adic Galois representation conjugate to one over the l-adic integers? [closed] Why is every l-adic Galois representation$$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})$$1answer 131 views ### In what sense is \Omega_p universal? In Chapter 3 of the book ''A Course in p-adic Analysis'' A.M Robert defines the field \Omega_p. He calls the field ''universal'' but doesn't show a universal property. I would have guessed that it'... 0answers 102 views ### Cycle with integral coefficients from cycle with \mathbb Z_l-coefficients Let X be a n-dimensional (n>2) smooth projective variety over k=\bar k of positive characteristic. Take a divisor D\in Pic(X). Suppose we know that \frac{[D]^2}{2}\in H^4(X,\mathbb Z_l(... 0answers 194 views ### Is 1+T a topological generator for Z_{p}[[T]]? [closed] Consider the ring of formal power series \mathbb{Z}_p[[T]] (where \mathbb{Z}_p denotes the ring of p-adic integers) with the topology in which a neighborhood basis for 0 is given by the ideals ... 2answers 976 views ### The formal p-adic numbers The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space \mathbb{Q} with its usual absolute value, or as the ... 2answers 504 views ### Absolutely irreducible p-adic representation of the absolute Galois group of Q_p Let p be a prime number, \mathbb{Q}_p the field of p-adic numbers, G_p the absolute Galois group of \mathbb{Q}_p and V a finite dimensional vector space over \mathbb{Q}_p. Assume we are ... 1answer 301 views ### How does the solenoid structure of \mathbb{A}/\mathbb{Q} lift to PGL(2, \mathbb{A})/ PGL(2, \mathbb{Q})? Some papers I am reading talk about an "adelic" object PGL(2, \mathbb{Q}) \backslash PGL(2, \mathbb{A}) . This has sparked a lot of confusion since I don't know what such a quotient could mean. A ... 1answer 161 views ### Coverings/Cech cohomology of totally disconnected spaces For any topological space X we have a natural functor \text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set}) from the category of coverings of X to the category of functors \pi_1(X) \... 0answers 162 views ### Simultaneously using the real and 2adic norms In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ... 2answers 639 views ### Are the p-adics a direct summand of the direct product of the groups \mathbb{Z}/p^n\mathbb{Z}? The p-adic integers \mathbb{Z}_p can be thought of as a subgroup of the direct product group P = \prod_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z}. Are they a direct summand of this group? That is, is ... 1answer 237 views ### Dihedral extension of 2-adic number field Sorry if the question is too long and maybe elementary. I am reading a paper by Hirotada Naito on "Dihedral extensions of degree 8 over the rational p-adic fields". To generate dihedral extension K(\... 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 ... 1answer 195 views ### Can every finite abelian p-group with duality pairing be written as cokernel of a symmetric matrix over the p-adic integers? Let G be a finite abelian p-group (where p is a prime). Suppose there exists a symmetric bilinear map \delta\colon G\times G\to \mathbb{Q}/\mathbb{Z} such that the induced map g\to\langle g,\;... 0answers 123 views ### Finiteness of the set of \mathbb{Q}_p-rational periodic points The statement I am concerned with is this: Let \varphi : \mathbb{P}^r_{\mathbb{Z}_p} \to \mathbb{P}^r_{\mathbb{Z}_p} be a morphism of degree higher than one. Then the set of \mathbb{Q}_p-... 1answer 1k views ### Definable measure preserving isomorphisms of p-adic semialgebraic sets Hi, Consider a p-adic field K (finite extension \DeclareMathOperator{\bQ}{\mathbb{Q}}of \bQ_p) in Macintyre language \DeclareMathOperator{\cL}{\mathcal{L}} \cL_{\rm Mac}. Let Z be a ... 0answers 78 views ### Weil index computation, p-adic integral The following peculiar p-adic integrals have arisen in my work, and I would be interested if anyone can see how to tackle them. Let F be a p-adic field, \mathfrak{o} its ring of integers, \... 2answers 224 views ### Converting p-adic to decimal [closed] Is it possible to convert irrational p-adic numbers to a standard number? Rationals and negative rationals are relatively straightforward, but is there a way to know that for instance \ldots ... 0answers 99 views ### Computing a projection of a p-adic plane curve Given a prime p and a polynomial equation f(x,y)=0 with rational coefficients, I would like to obtain a precise description of the set of all numbers y\in\mathbb Q_p such that the equation has a ... 0answers 179 views ### Do r-th root Harmonic numbers ever sum to integers? None of the Harmonic numbers H_n = \sum_{k=1}^n 1/k are integers for n>1 (e.g., this MSE question and answer). Q. Define the r-th root Harmonic number H_n^{1/r} = \sum_{k=1}^n 1/{k^{1/r}}... 1answer 455 views ### is there a p-adic implicit function theorem? I am trying to find a good reference for a version of the implicit function theorem over p-adic manifolds. None of the texts I have consulted ( including "p-adic numbers, p-adic analysis, and ... 1answer 356 views ### Iwasawa logarithm and analytic continuation I am reading Number Theory vol. 1 by Henri Cohen (among other things) and I am curious about the Iwasawa logarithm. Let \mathbb{C}_p be the completion of the algebraic closure of \mathbb{Q}_p. ... 0answers 272 views ### Transcendental numbers in the p-adic rationals \mathbb Q_p [closed] I know that there are uncountably infinite transcendentals over \mathbb Q in \mathbb Q_p. What i want to ask is if there is a way to determine whether a transcendental over \mathbb Q lays in ... 2answers 640 views ### Trace of n-th root of unity in cyclotomic extension of p-adic rationals Let n\in\mathbb N and p be any prime. Denote by \mathbb Q_p the p-adic numbers. For a field extension L/K denote by Tr_{L/K} the corresponding trace function. Let \zeta_n be a primitve ... 2answers 349 views ### convergence in Z-hat; modulo prime power The following problem appears in Lenstra's Galois Theory for Schemes (p 14, Ex 1.16). Let b\in\mathbb Z_{\ge0}. Define the sequence (a_n)_{n=0}^\infty by a_0=b, a_{n+1}=2^{a_n}. Prove that ... 0answers 382 views ### What is p-adic Fourier series? Q1: Can we define Fourier series for a function \mathbb{Z}_p\to \mathbb{Q}_p? Q2: There are (in a real case) Bernoulli polynomials which have the most simple Fourier expansion:$$B_n(\{x\})=-\frac{...
This might be a easy question, but I couldn't get the point. Let $F$ be a p-adic field, $\bar{F}$ a separable algebraic closure of $F$. Set $\Omega_F=Gal(\bar{F}/F)$. Use $F_{\infty}\subset \bar{F}$ ...