35 votes
Accepted

Crux of Dwork's proof of rationality of the zeta function?

There is an excellent book by Neal Koblitz "p-adic numbers, p-adic analysis and zeta-functions" were the Dwork's proof is stated in a very detailed way, including all preliminaries from p-adic ...
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  • 6,507
27 votes
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$p$-adic numbers in physics

For an overview of applications of p-adic numbers in physics I would refer to the Wikipedia and Physics.stackexchange links, and to this nLab entry. Regarding the second question "What is the most ...
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25 votes
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Condensed criterion for sheafiness of adic spaces

Thanks for the question! One interpretation of the conjecture is true. Let me elaborate. The following results are kind of implicit in some discussion towards the end of www.math.uni-bonn.de/people/...
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23 votes

$p$-adic numbers in physics

Since the question on physics.stackexchange that was referred to in the comments above has been closed, let me essentially repeat my answer here together with some updates. I think there are mainly ...
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23 votes

Hensel's proof that $e$ is transcendental

You can read Hensel's original article here. The argument is really simple: from this expansion, we see that $e$ satisfies an equation of the form $y^p=1+pu$ where $u$ is a $p$-adic unit. But the ...
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  • 23.3k
18 votes
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$p$-adic Bott periodicity?

The $p$-completed algebraic $K$-theory of the algebraic closure of $\mathbb{Q}_p$, i.e., $K(\bar{\mathbb{Q}}_p; \mathbb{Z}_p)$, is equivalent to its second loop space, up to an issue about path ...
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  • 7,126
18 votes
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Geometric series in algebraic number fields

This holds precisely for elements which aren't roots of unity. Indeed, by the product formula we have $\prod_v|\alpha|_v=1$, where the product runs over all (finite and infinite) primes. This shows ...
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  • 23.3k
17 votes

Where should I learn about the p-adic L-functions of elliptic curves?

We know how to attach $p$-adic $L$-function to elliptic curves over $\mathbb Q$ only because we know how to attach them to cuspidal modular eigenforms, and we know by Breuil-Conrad-Diamond-Taylor that ...
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  • 25k
16 votes

CM $j$-invariants in $p$-adic fields

Good question. I am leaving some first thoughts now; I hope to have a chance to think more about it later. Because CM elliptic curves have potentially good reduction, the $j$-invariant is an ...
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15 votes
Accepted

Automorphisms of $\mathbb C_p$

If you strengthen the condition $\sigma(\mathbf Q_p) \subset \mathbf Q_p$ to $\sigma$ being the identity on $\mathbf Q_p$, so $\sigma$ is a $\mathbf Q_p$-automorphism of $\mathbf C_p$, then a simple ...
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  • 41.2k
14 votes
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What is the value of $p$-adic $\zeta$-function at positive integer point?

If you use this definition, then $\zeta_p(k)$ is zero at negative even integers $k$, so by a $p$-adic continuity argument, it must also be zero at positive even integers. What about the odd integers? ...
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13 votes
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Cardinality of ${\mathbb{C}_p}$

Not only does $\mathbb C_p$ have the same cardinality as $\mathbb C$, but the larger field $\Omega_p$, the spherical completion of $\overline{\mathbb Q}_p$, also has this cardinality. Further, one can ...
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13 votes
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On the definition of the etale site of an adic space

Great question! The short answer is that Huber simply wanted to be in a setting where everything is (stably) sheafy, and so put some assumptions ensuring this. Note that Huber's work remained somewhat ...
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12 votes

Crux of Dwork's proof of rationality of the zeta function?

See Terry Tao's blog post. A very simple proof of a slightly weaker result is given by Mike Larsen.
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  • 93.7k
12 votes
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Zero of the exponential p-adic

The exponential function satisfies $\exp \left({x + y}\right) = \exp\left(x \right) \exp\left(y \right)$ for $x, y$ in the convergence domain. It also satisfies $\exp \left( 0 \right) = 1$. So if $\...
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  • 650
11 votes
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p-adic Stein spaces

The answer is no. In fact, a theorem of Lazard shows that the open unit disk has a non-trivial Picard group when the ground field is a non maximally complete ultrametric valued field. See Proposition ...
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  • 12.4k
11 votes

What is the value of $p$-adic $\zeta$-function at positive integer point?

I agree nothing much is known, but there are a number of formulas linked to the values at positive integers of p-adic L-functions: see Section 11.3 of my GTM 240 book (sorry for the self-advertisement)...
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  • 8,928
10 votes

Trivialisation of vector bundles on Stein spaces

I assume you meant for the fiberwise rank of $E$ to be constant, say $r>0$ (or at least uniformly bounded above). The answer is "yes" when the Stein space has finite dimension (equivalently, when ...
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  • 1,875
10 votes
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Trivialisation of vector bundles on Stein spaces

In the complex analytic world*, every vector bundle which is globally generated by a finite set of global sections has a finite trivialising cover: Let $X$ be a complex space and let $E$ be a locally ...
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  • 939
10 votes
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Topological dimension of $p$-adic manifolds

$p$-adic numbers are locally compact, Hausdorff and totally disconnected (see this nLab page), hence they are zero-dimensional. This means that---at least naively---topological dimension of $p$-adic ...
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9 votes
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convergent series representation for p-adic complex numbers

The blog post http://sbseminar.wordpress.com/2007/08/21/p-adic-fields-for-beginners gives a good overview of the situation. To briefly summarize (extracted from the above post — any errors are ...
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  • 1,756
8 votes
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Is there a multivariate analog of Dwork's theorem?

Update (Nov, 2019). The multivariate rationality criterion is true, and it follows from the reference to Andre's paper below. However, in my suggested proof scheme there was a point I had overlooked ...
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8 votes
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Nijmegen 1978 $p$-adic analysis proceedings

In the Nijmegen University Repository I only found pages 193-204, Non-archimedean differentiation. Since I presume a copy for private use is OK, I have scanned Amice's and Morita's contributions, you ...
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8 votes

Zero of the exponential p-adic

It’s hard to see in what sense it could be true that the exponential function is “defined over $\Bbb C_p$”, since the logarithm is defined on the whole open unit disk there, and has so very many zeros....
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  • 3,818
8 votes
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Can perfect numbers be seen $p$-adically?

(Not a complete answer but a bit too long for a comment.) There's a fundamental difficulty here in proving the sort of result you envision. If there were some prime $p$ which had to divide every odd ...
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  • 3,357
7 votes
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is there a p-adic implicit function theorem?

There are the lecture notes by Serre (Springer lecture notes 1500) of a course on Lie groups and Lie algebras. He proves the implicit function theorem for analytic functions on a $p$-adic manifold (...
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7 votes

CM $j$-invariants in $p$-adic fields

All accumulation points of $J_p$ in $\mathbb{C}_p$ are roots of degree two monic equations over $\mathbb{Z}_p$, and their approximants are necessarily supersingular at $p$. Moreover, there exist ...
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7 votes
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A multidimensional version of Hensel's lemma? (for more than one polynomial)

See the accepted answer at https://math.stackexchange.com/questions/48419/hensels-lemma-and-implicit-function-theorem for Hensel's lemma for $n$ polynomials in $n$ variables (more general versions are ...
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  • 41.2k
7 votes

Crux of Dwork's proof of rationality of the zeta function?

Sasha has already pointed you to the primary source I used some years ago for my "expository" undergraduate thesis on Dwork's Theorem: Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions, ...
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7 votes
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A definition of a (amalgamated) direct sum

The key point here is that $\pi(\alpha)$ and $\pi(\beta)$ are isomorphic representations: both of them are the algebraic representation $Sym^{k-2}$ tensored with the smooth representation $Ind_B^G (\...
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