# Tag Info

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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### $p$-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 ...
• 147k
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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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### $p$-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 ...

### 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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### $p$-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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### 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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### 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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### 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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### 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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### 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? ...
• 30.9k
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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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### 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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### 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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