# Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

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### Weil cohomology theories “genuinely” of positive characteristic

One of the reasons why Weil cohomology theories are required to have coefficients in a field of characteristic 0 is that they are supposed to be robust enough to solve Weil conjectures, i.e. to count ...

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213 views

### On two questions of Mazur

Let's consider the proof of Theorem 4.1 in Mazur's Eisenstein ideal paper.
Work over $\mathbb{Q}$, and consider the projection $X_0(N)\rightarrow \tilde{J}$ defined by
$x\rightarrow \mathrm{image}...

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**1**answer

202 views

### Crystalline comparison for rigid-analytic varieties

Let $k$ be a finite extension of $\mathbb{Q}_p$. In this paper, Scholze proves an analogue of de Rham comparison for proper smooth rigid-analytic varieties over $k$. He also says:
...it should be ...

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63 views

### Integral lifts of families of varieties over a finite field

Let $X\rightarrow \mathrm{Spec}\:F_q[[t]]$ be a flat morphism with smooth proper geometrically connected fibers. Suppose the central fiber lifts to a scheme $X'_0$ smooth proper over $W(F_q)$. Is our ...

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**1**answer

279 views

### Integral $p$-adic Hodge theory and the space of comparisons of cohomology theories

Weil cohomology theories can be considered as fibre functors from the category of motives. Given two such functors, we have an affine scheme of invertible natural transformations between them, and ...

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**1**answer

116 views

### Non-abelian Berthelot comparison?

Berthelot's comparison theorem connects the algebraic de Rham cohomology of a $\mathbb{Z}_p$-scheme and the crystalline cohomology of its special fiber. Is there a statement on the level of homotopy ...

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100 views

### Elliptic curves with the same Galois representation

Fix a prime $p$. If two elliptic curves over $\mathbb{Q}$ have the same p-adic Galois representation, then what relatinships do we know between them? Any references are welcome.

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**1**answer

328 views

### In how many ways can one extend the zero section of the affine line with a double origin

Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.
Let $0$ be ...

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76 views

### When does a morphism of schemes induce a morphism of crystalline sites (not topoi)?

Here it is stated that the crystalline site of a scheme is not functorial in general. Is there a non-tautological characterization of morphisms of schemes which in fact do induce morphisms of ...

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174 views

### Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?

In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...

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89 views

### Local-global compatibility and modular curves

I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...

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148 views

### Applications of $h$-topology and $h$-descent

This is a technical problem about applications of Grothendieck topologies. In some recent works, the technique of $h$-topology and $h$-descent is very useful, for an introduction see https://stacks....

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**4**answers

42k views

### What are “perfectoid spaces”?

This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?
Edit: A bit ...

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90 views

### Algorithmically computing Weil cohomology groups

Fix a Weil cohomology theory. If I give you a presentation of a smooth projective scheme over an algebraically closed field, do you have an explicit algorithm for computing its cohomology groups? ...

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530 views

### Inverse Limits in the category of Perfectoid Spaces

First I apologize as the following is likely littered with misunderstanding.
My understanding at least from the discussion in Scholze-Weinstein is that inverse limits in the category of adic spaces ...

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171 views

### Which interesting characterestic zero field $E$ (e.g a pseudofinite field) can support a Weil cohomology?

Let's consider the category of smooth projective varieties over a fixed characteristic $p>0$ algebraically closed field $k$. For a Weil cohomology theory with coefficient field $E$, by definition ...

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282 views

### Current status of independence of Betti numbers for different Weil cohomology theories

Previous problem: Is $\operatorname{dim} H^1$ of an abelian variety the same for any Weil cohomology?
Let $X$ be an smooth projective variety over a field $k$. For any Weil cohomology theory for ...

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251 views

### The Frobenius at the infinite prime?

For simplicity, suppose $X$ is a smooth $n$-dimensional variety defined over $\mathbb{Q}$. Then the etale cohomology of $X$, denoted by $H^i_{\text{et}}(X,\mathbb{Q}_\ell)$, gives a representation of ...

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108 views

### Is there literature on a de Rham analogue of the Mumford-Tate group or ell-adic monodromy group?

Let $X$ be a smooth projective variety over $\mathbb{Q}$. The theory of motives predicts that for each cohomology theory, there should be a distinguished Zariski closed subgroup of $GL(H^k_{\bullet}(X)...

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**1**answer

746 views

### Does anybody do $p$-adic Teichmüller theory?

In "Foundations of $p$-adic Teichmüller theory", Mochizuki describes a theory one of whose goals (according to the author) is to generalize Fuchsian uniformization of Riemann surfaces to the $p$-adic ...

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24 views

### Counting geometrically irreducible components

If we take a finite field and consider irreducible varieties over it, are there any interesting arithmetical statistics problems associated to the number of geometrically irreducible components?

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923 views

### Tower of moduli spaces in Scholze's theory

My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...

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**1**answer

133 views

### Endomorphism rings of ordinary elliptic curves

Let's say $p$ is a prime and $t\neq 0$ is a trace of Frobenius that occurs over $\mathbb{F}_p$. The discriminant of the Frobenius polynomial is
$\Delta:=t^2-4p.$
So we obtain $4p=t^2-\Delta.$ If $E$ ...

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**2**answers

171 views

### Hyperelliptic Jacobians with (or without) CM

Let $C$ be a hyperelliptic curve $y^2 = f(x)
$ defined over $\mathbb{Q}$, where $f(x) \in \mathbb{Q} [x]$ is a polynomial of degree $n=5$ or $6$, and $J = Jac(C)$ its Jacobian.
I know Zarhin's result [...

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154 views

### Grothendieck-Teichmüller conjecture and tropicalization of moduli of curves

Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli ...

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183 views

### Bounded version of linear and quadratic Hasse--Minkowski theorem

The Hasse-Minkowski theorem states that if
$$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$
is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation
$$Q(x_1,\...

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94 views

### Existence of regular hypersurface sections

Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...

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195 views

### Moduli spaces of arithmetic varieties with isomorphic $l$-adic cohomology

Given a positive integer $d$, a rational prime $l$ and a number field $K$, is it sensible to consider the moduli stack of $d$-dimensional varieties over $K$ whose $l$-adic cohomology rings are ...

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164 views

### Does etale homotopy type see the existence of rational points?

Do there exist two smooth projective schemes over $\mathbb{Q}$ that are etale homotopy equivalent and only one of them has a $\mathbb{Q}$-point?

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249 views

### Rigid versus log-rigid cohomology for semistable varieties

If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...

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94 views

### Computing the genus of a plane curve

Let $b(x)=x^4 + 3x^3 + 3x^2 + 2x + 1$, and let $a(x)\in \mathbb Z[x]$ be a separable polynomial. Let $C$ be the plane curve defined by $(y^2+(x+x^2+x^3)a(x))^2-a(x)^2b(x)=0$. I would need to show that ...

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86 views

### Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup ...

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121 views

### Hasse-Weil zeta function of smooth projective toric varieties

Let $X$ be a smooth projective toric variety over a number field $K$ (assume the tori is split). As $X$ is rational, maybe the related Hasse-Weil zeta function can be well-understand, so how much do ...

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131 views

### Lifts of smooth algebras

Let $(R, I)$ be a Henselian pair, with $I$ a finitely generated ideal.
We know that for any smooth $R/I$-algebra $A_0$, there exists a smooth $R$-algebra $A$ such that $A/I\simeq A_0$.
We also know ...

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182 views

### Integral models of perfectoid modular curves

Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...

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**1**answer

124 views

### Identifying elements in the kernel of an explicit endomorphism of a Jacobian variety

I hope this question fits here.
Let $H/k$ be a genus $2$ curve and $J$ its Jacobian variety. Since $J(k)\cong \text{Pic}^0(H)(k)$ we have that its generic point looks like $[(x_1,y_1)+(x_2,y_2)-2\...

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351 views

### Arithmetic symplectic geometry via mirror symmetry?

Homological mirror symmetry in the classical setting relates the bounded derived category of coherent sheaves on a Calabi-Yau manifold to the split-closure of the derived Fukaya category of the mirror ...

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**1**answer

316 views

### Extending section of étale morphism of adic spaces

This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...

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312 views

### The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$.
I know that to construct the Jacobian variety associated to $C$, one ...

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103 views

### Do many homogeneous polynomials help in faster integer root extraction?

Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...

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100 views

### Unique way to topologise finite algebra over Huber ring

Let me start with the following Lemma.
$\textbf{Lemma}$ Let $A$ be a Tate ring, and let $f\colon A\to B$ be a finite $A$-algebra. Then there is a unique way to topologise $B$ turning it into a ...

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88 views

### Maximum number of bounded primitive integer points in a zero-dimensional system

Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...

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66 views

### What is the probability of 'yes' to this likely $coNP$ problem?

Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$.
Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...

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65 views

### Current status of uniform boundness of rational points on higher genus curves

We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others.
How is the conjecture of ...

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**5**answers

1k views

### Connection Between Knot Theory and Number Theory

Is there any connection between knot theory and number theory in any aspects?
Does anybody know any book that is about knot theory and number theory?

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**1**answer

248 views

### Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$

My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of ...

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**1**answer

191 views

### On a refinement of Mordell's conjecture for curves

Let $C$ be an algebraic curve of genus $g \geq 2$, defined over $\overline{\mathbb{Q}} \subset \mathbb{C}$. It is then defined over a finite extension $K$ of $\mathbb{Q}$. We assume that $C(K) \ne \...

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321 views

### Down to earth, intuition behind a Anabelian group [closed]

An anabelian group is a group that is “far from being an abelian group” in a precise sense: It is a non-trivial group for which every finite index subgroup has trivial center.
I would like to know ...

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657 views

### Multizeta function values

Francis Brown Theorem says that $\zeta(a_{1},\dots ,a_{r})$ the multi-zeta value of weight $N=a_{1}+\dots +a_{r}$ is a $\mathbb{Q}$-linear combination of elements of the set $S=\{\zeta(a_{1},\dots, ...

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254 views

### Smooth proper variety over a number field with prescribed bad reductions

Given a number field $K$, and a finite set $S$ containing finite places of $K$. When can we find a smooth proper geomerically connected variety $X$ over $K$ such that $X$ has good reduction outside $S$...