# Tagged Questions

**-1**

votes

**0**answers

75 views

### Help with extension of f (mentioned below) to f: Zp -> Zp ,continuous function [on hold]

This is a (different version to) question from Serre 'A Course in Arithmetic'.Let p be an odd prime number. $\forall n\geq 1$ (n positive integer), $f$ is defined by:
$$f(n)=(-1)^n\prod_{1\le k\le n ...

**4**

votes

**0**answers

94 views

### Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...

**4**

votes

**1**answer

171 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 $\mathbb{Q}_p$.
...

**6**

votes

**1**answer

240 views

### What are the automorphisms of a perfectoid Tate algebra?

Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in ...

**0**

votes

**0**answers

230 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:
...

**5**

votes

**0**answers

198 views

### 2-adic Logarithm and Resistance of n-dimensional Cube

Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=1}^{n}\frac1{{n-1\choose k}}.$$
The ...

**4**

votes

**2**answers

242 views

### Proving the existence of an integral quadratic form

Theorem 11 (Conway & Sloane, Sphere Packings, Lattices and Groups, 3rd Edition, pp 383, Ch 15). If a system of putative $p$-adic symbols for each $p$ satisfies the determinant, oddity, and ...

**2**

votes

**0**answers

115 views

### A nice rigid analytic model for local systems over an elliptic curve?

For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gauge group acts ...

**2**

votes

**1**answer

219 views

### Trivial p-adic measures

I am looking at p-adic distributions, and in this case p-adic measures. To say that $\mu$ is a distribution means that the arguments of $\mu$ are compact open subsets of $\mathbb{Z}_p$, $\mu$ is ...

**7**

votes

**1**answer

220 views

### Describing the ratio of uniformizers in B_dR

In Conrad and Brinon's notes http://math.stanford.edu/~conrad/papers/notes.pdf, two uniformizers of $B_{dR}$ are produced: one is $\xi := [\tilde{p}]-p$ (bottom of p.58), where $\tilde{p} = (p, ...

**6**

votes

**1**answer

397 views

### A p-adic analogue for a formula of Riemann?

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 ...

**16**

votes

**11**answers

2k 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 ...

**34**

votes

**10**answers

5k 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 ...

**7**

votes

**2**answers

847 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 ...

**12**

votes

**3**answers

1k views

### 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 ...

**10**

votes

**1**answer

554 views

### 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 ...

**0**

votes

**1**answer

343 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 ...

**15**

votes

**1**answer

1k 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 ...

**1**

vote

**2**answers

640 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 $p\not\vert$N. Then we can construct 2 p-adic L-functions corresponding to the 2 roots $\alpha$ and $\beta$ of the equation ...

**15**

votes

**3**answers

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 ...

**5**

votes

**2**answers

843 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 ...

**0**

votes

**0**answers

261 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 ...

**7**

votes

**3**answers

2k 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 $p^\nu$ ...

**7**

votes

**1**answer

517 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 ...

**8**

votes

**1**answer

593 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 ...

**9**

votes

**2**answers

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 ...

**14**

votes

**1**answer

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 ...

**10**

votes

**5**answers

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
$$
f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}.
$$
...

**7**

votes

**1**answer

439 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 ...

**14**

votes

**4**answers

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 ...

**7**

votes

**2**answers

969 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 ...

**15**

votes

**3**answers

1k views

### 2-adic Coefficients of Modular Hecke Eigenforms.

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 ...