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34
votes
10answers
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 ...
19
votes
8answers
3k 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 ...
17
votes
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 ...
16
votes
11answers
3k 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 ...
16
votes
3answers
2k 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., ...
15
votes
3answers
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 ...
15
votes
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 ...
15
votes
1answer
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 ...
14
votes
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 ...
14
votes
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 ...
14
votes
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 ...
13
votes
1answer
700 views

P-adic C* algebras

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 ...
12
votes
3answers
2k 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 ...
12
votes
3answers
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 ...
12
votes
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 ...
11
votes
1answer
495 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 ...
10
votes
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 $$ f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}. $$ ...
10
votes
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 ...
10
votes
1answer
557 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 ...
8
votes
1answer
234 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 ...
8
votes
1answer
595 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 ...
7
votes
3answers
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
4answers
1k 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 ...
7
votes
3answers
484 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 ...
7
votes
2answers
977 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 ...
7
votes
1answer
482 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?
7
votes
2answers
849 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 ...
7
votes
1answer
524 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 ...
7
votes
1answer
221 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, ...
7
votes
1answer
442 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 ...
6
votes
1answer
250 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 ...
6
votes
1answer
406 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 ...
6
votes
0answers
245 views

Differences in tree picture of ${\bf Q}_p$, $\overline{{\bf Q}_p}$, ${\bf C}_p$, $\Omega_p$

I was discussing the tree picture of ${\bf Z}_p$ and ${\bf Q}_p$ and mentioned that the idea can be extended to ${\bf C}_p$, with the caveat that the tree is no longer locally finite (as the value ...
5
votes
1answer
377 views

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 ...
5
votes
1answer
623 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. ...
5
votes
2answers
854 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 ...
5
votes
1answer
219 views

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 ...
5
votes
0answers
200 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 ...
5
votes
0answers
422 views

a naive question about p-adic local monodromy theorem

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 ...
5
votes
0answers
203 views

Lower bound for p-adic distance between roots

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 - ...
4
votes
4answers
1k 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 ...
4
votes
1answer
249 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 ...
4
votes
1answer
185 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 $\mathbb{Q}_p$. ...
4
votes
2answers
247 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 ...
4
votes
0answers
97 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
0answers
157 views

irrationality of the p-adic exponential

I would like to illustrate my lecture on p-adic numbers with some elementary results. I proved that the series $e=\sum_{n\ge0}\frac{p^n}{n!}$ converges in $\mathbb Q_p$ for every prime $p$. Now I ...
4
votes
0answers
447 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 ...
3
votes
1answer
1k 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 ...
2
votes
1answer
302 views

Terminology-history of p-adic representations

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.
2
votes
1answer
222 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 ...