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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

5 votes
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Primes mod 4 and integer polynomials

Here is a way to argue without showing directly that the polynomial must have degree $2$. It was explained to me by Borys Kadets (all further mistakes are, of course, my contribution). Lemma. If a se …
0 votes
1 answer
249 views

Equation in integers of irrational degree

Are there any algebraic irrational numbers in $\{log_xy|x,y\in\mathbb{N},x,y\geq2\}$?
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2 votes
1 answer
517 views

Representation of rationals by quadratic form

In one paper about number theory author stated 2 lemmas Lemma 1. If $p$ is a prime $\equiv3(mod $ $4)$ then $x^2+y^2-pz^2$ represents a non-zero rational number $m$ if and only if $m$ is not of the f …
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4 votes
Accepted

Finite image but not crystalline

If a crystalline representation has finite image when restricted to inertia, then this restriction has to be trivial. Indeed, suppose that $K$ is a discretely valued extension of $\mathbb{Q}_p$ and …
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3 votes
Accepted

maximal unramified extension of Breuil ring in $A_{cris}$

It is false that $\mathfrak{S}^{un}\cap pA_{cris}=p\mathfrak{S}^{un}$. For example, $E(u)\in\mathfrak{S}\subset\mathfrak{S}^{un}$ is not divisible by $p$ in $\mathfrak{S}^{un}$ but gets mapped to $\va …
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7 votes
0 answers
417 views

Failure of integral comparison between crystalline and de Rham cohomology over a highly rami...

Let $K$ be a finite extension of $\mathbb{Q}_p$ with the ring of integers $\mathcal{O}_K$ and the residue field $k$. By a theorem of Berthelot and Ogus(https://link.springer.com/article/10.1007%2FBF0 …
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12 votes
Accepted

Question on the Sato-Tate conjecture

No. If $E_p$ is a supersingular elliptic curve and $p>3$ then trace of Frobenius on $E_p$ is zero, so $\theta_E(p)=\pi/2$. By a result of Elkies any elliptic curve over $\mathbb{Q}$ has supersingular …
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3 votes

The largest number $y$ such that $(x!)^{x+y}|(x^2)!$

$\newcommand{\eps}{\varepsilon}$It seems that for any $y$ the number of such $x$ is infinite. First of all, let's fix a prime $p$ and compute $v_p(\frac{(x^2)!}{(x!)^x})$ -- the exponent of $p$ in th …
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6 votes
Accepted

Does every section of the map Gal$(\overline{k(\!(t)\!)}/k(\!(t)\!))\rightarrow$ Gal$(\overl...

$\newcommand{\Gal}{\mathrm{Gal}}\newcommand{\Z}{\mathbb{Z}}$Fix a compatible system $(t_n)$ of roots of $t$. It provides us with a section of $\rho$ thus giving an isomorphism between $\Gal(\overline{ …
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10 votes
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P-adic functions on annuli

$\def\bQ{\mathbb{Q}}\def\bF{\mathbb{F}}\def\bZ{\mathbb{Z}}$This is false as stated because of the following important difference between $K$ and $\mathbb{C}$: the former is not algebraically closed. F …
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3 votes
Accepted

The kernel from $A_\mathrm{inf}$ to $\mathcal{O}_{\mathbb{C}_K}$

1)Pick a sequence of elements $p^{1/p^n}\in \mathcal{O}_{\overline{K}}$ such that $(p^{1/p^{n+1}})^p=p^{1/p^n}$. The ideal $\ker\phi$ is in fact principal and is generated by the element $p^{\flat}:=( …
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6 votes
1 answer
744 views

Group laws in class field theory

In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field. In the case of a …
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9 votes
Accepted

Exactness of the Weil restriction functor $\mathrm{Res}_{X/k}$

It is not right exact. Assume that $k$ is algebraically closed. If the map $Res_{X/k}B\to Res_{X/k}C$ was surjective as a map of sheaves for the fppf topology, then in particular, the map on sections …
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35 votes
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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 analysi …
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4 votes
1 answer
560 views

Lifting of Frobenius on semi-abelian varieties

Let $A$ be a semi-abelian variety over a field $k$($char\, k=p$). Namely, there is an exact sequence of group schemes $$0\to T\to A\to B\to 0$$ where $T$ is a torus, $B$ an abelian variety. Assume tha …
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