All Questions
Tagged with p-adic-numbers ag.algebraic-geometry
19 questions
14
votes
1
answer
1k
views
A quantitative version of Hensel's Lemma
I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts the number of lifts of a $\mathbb{Z}/p\mathbb{...
- 21.4k
11
votes
1
answer
690
views
Mori: p-adic and real hemispheres of the mathematical universe?
I recall having read, some time ago, a beautiful and poetic opening of an article (or was it a book?). From memory, it was by Shigefumi Mori, and talked about the (mathematical) universe consisting of ...
- 2,519
11
votes
0
answers
808
views
Torelli-like theorem for K3 surfaces on terms of its étale cohomology
Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow \...
10
votes
3
answers
1k
views
What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
user178596
10
votes
0
answers
834
views
Elementary aspects of The Fargues-Fontaine curve
To any pair $(E,F)$, where $E$ is a local field and $F$ is a perfectoid field, one can associate a curve $X^{\text{FF}}_{E,F}$, the so-called Fargues-Fontaine curve, which is unique up to Frobenius ...
- 989
9
votes
1
answer
315
views
Tube of a mod p point on a smooth Z_(p)-scheme
Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R \...
- 15.4k
8
votes
1
answer
422
views
p-adic versions of log concavity for graphs (or matroids)
It was recently shown using techniques inspired by algebraic geometry (by Huh and Adiprasito-Huh-Katz) that the chromatic polynomial of a graph (or matroid) has coefficients that satisfy log-concavity....
- 7,746
8
votes
0
answers
585
views
Scholze and Weinstein's $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}_p$
In their Berkeley Lectures, to motivate the introduction of Diamonds, Scholze and Weinstein discuss what should be the definition of $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}...
- 1,479
5
votes
1
answer
485
views
General algebraic result obtained from consideration on $\mathbb{Q}_p$
There are results in field theory which are obtained from, let's say, the complex numbers and then generalized to all algebraically closed fields.
For instance, the fact that a polynomial $P$ admits a ...
- 241
5
votes
0
answers
197
views
Bezout-type theorem for $p$-adic analytic plane curves
Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
4
votes
0
answers
91
views
Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
- 6,622
3
votes
1
answer
180
views
Approximating $p$-adic power series by polynomials
Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
3
votes
0
answers
265
views
Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?
When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
- 7,182
3
votes
0
answers
213
views
Inverse of reduction mod $p$ functor?
I have a very general, and possibly not very precisely stated question, which comes up quite often in my work, and I would be very happy to be able to address. To my dismay, I only have some very ...
- 993
2
votes
1
answer
223
views
Finitely generated $\mathbb{Z}$-algebra embeds into unramified $p$-adic ring
Let $R$ be a finitely generated ring, that is, a $\mathbb{Z}$-algebra of finite type. Assume that $\operatorname{char}(R) = 0$. It follows from Noether's normalization lemma that $R$ can be embedded ...
- 423
2
votes
1
answer
150
views
How to get a ball in the nonvanishing locus of a polynomial in $\mathbb Z_p[x_1,\cdots,x_n]$ canonically?
Suppose $f\in \mathbb Z_p[x_1,\cdots,x_n]$, and consider $D(f):=\{(𝑥_1,…,𝑥_𝑛)∈ℤ^𝑛_𝑝:𝑓(𝑥_1,…,𝑥_𝑛)≠0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
- 785
2
votes
0
answers
148
views
Can there exist different smooth, proper schemes over the p-adics with the same generic fiber? [duplicate]
Can there exist smooth, proper $X_1,X_2/\mathbb Z_p$ such that their generic fibers are isomorphic but their reductions mod $p$ are not? Are there examples if we insist that the special fibers are ...
- 7,746
2
votes
0
answers
478
views
Is there bijective correspondence between $P_n$ and $A_n$?
Let $K \supset \mathbb{Q}_p$ be the $p$-adic field and let $O_K$ be its ring of integers and $M_K$ be the maximal ideal with integral closure $\bar{M}_K$. A power series is invertible if its lowest ...
- 930
1
vote
0
answers
122
views
Cycle with integral coefficients from cycle with $\mathbb Z_l$-coefficients
Let $X$ be a $n$-dimensional ($n>2$) smooth projective variety over $k=\bar k$ of positive characteristic. Take a divisor $D\in Pic(X).$ Suppose we know that $\frac{[D]^2}{2}\in H^4(X,\mathbb Z_l(...
- 155