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Questions tagged [arithmetic-geometry]

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

44
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0answers
2k views

What is Prismatic cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...
38
votes
0answers
2k views

What are the potential applications of perfectoid spaces to homotopy theory?

This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about ...
22
votes
0answers
717 views

Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question. Question. Is there a smooth proper scheme $X\to\operatorname{...
20
votes
0answers
503 views

Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...
19
votes
0answers
711 views

Finiteness of etale cohomology for arithmetic schemes

By an arithmetic scheme I mean a finite type flat regular integral scheme over $\mathrm{Spec} \, \mathbb{Z}$. Let $X$ be an arithmetic scheme. Then is $H_{et}^2(X,\mathbb{Z}/n\mathbb{Z})$ finite ...
18
votes
0answers
750 views

Is the Dieudonne module actually a cohomology group?

One often times thinks of the Dieudonne module $M(X)$ of a $p$-divisible group (say over $k$, a perfect characteristic $p$ field) as being some sort of cohomology theory $$M:\left\{p\text{- divisible ...
17
votes
0answers
452 views

Tropicalization of perfectoid spaces and their tilts

Does tropicalization exist in the world of perfectoid spaces? Since it does for Huber's adic spaces, I thought it might for perfectoid spaces too, yet I can't find any explicit references so far. ...
17
votes
0answers
1k views

Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber (see Conjectures 2 and 3, page 1975) http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf 1) Let $R$ be a strictly henselian ...
16
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0answers
332 views

Are there smooth and proper schemes over $\mathbb Z$ whose cohomology is not of Tate type

Is there an example of smooth and proper scheme $X \to \mathrm{Spec}(\mathbb Z)$, and an integer $i$ such that $H^i(X, \mathbb Q)$ is not a Hodge structure of Tate type? Alternatively: such that $H^...
15
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0answers
396 views

Is the absolute Galois group of the rationals Hopfian?

Is every continuous epimorphism from the absolute Galois group of $\mathbb{Q}$ to itself injective?
14
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0answers
579 views

How to approach the Mazur-Wiles paper on Iwasawa theory?

I would like to read and understand the Mazur-Wiles paper on Iwasawa theory: "Class Fields of Abelian Extensions of $\Bbb Q$". What would be the right way to approach this paper? Currently, my ...
14
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0answers
449 views

Vanishing of rigid cohomology for affine varieties

Let $k$ be a perfect field of positive characteristic and denote by $K$ the field of fractions of the ring of Witt vectors over $k$. Question: If $X$ is an affine variety over $k$, do the rigid ...
13
votes
0answers
304 views

Hensel lemma and rational points in complete noetherian local ring

Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal. If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ ...
13
votes
0answers
333 views

Lifting automorphic Galois representations to arithmetic fundamental groups and their quotients

Suppose $V$ is an algebraic variety over a number field $K$. The absolute Galois group $G_K$ of $K$ acts by outer automorphisms on the étale fundamental group $\pi_1(V_{\bar{K}})$ where $\bar{K}$ is ...
13
votes
0answers
404 views

Topological finiteness of etale fundamental group for arithmetic schemes

By comparison theorem and finite CW complex structure we know every complex variety's etale fundamental group is topological finitely generated (see here). It seems natural to ask similiar things for ...
13
votes
0answers
1k views

Inter-Universal Teichmuller Theory and the Field with One Element

The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...
13
votes
0answers
400 views

comparison of completion and Henselization in class field theory

Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...
12
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0answers
213 views

Abelian varieties (over $\mathbb{Q}$) with large Mordell-Weil rank

Let $A$ be an abelian variety defined over $\mathbb{Q}$ of dimension $g \geq 1$. We shall denote by $A(\mathbb{Q})$ the Mordell-Weil group of rational points in $A$, and denote by $r = r_A$ the rank ...
12
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0answers
688 views

Number field analog of Artin-Tate $\Rightarrow$ BSD?

What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each ...
12
votes
0answers
261 views

Modularity of endomorphism algebras

This question is about comparing Hecke algebras and endomorphism algebras. Let $\mathbf{A}_f$ be the ring of finite adèles of $\mathbf{Q}$ and let $K$ be a compact open subgroup of $\mathrm{GL}_2(\...
12
votes
0answers
263 views

$p$-Adic or arithmetic variants of Khovanskii's “low complexity $\Rightarrow$ tame topology” theory

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ...
12
votes
0answers
1k views

Meaningful review of Moriwaki's “Arakelov Geometry”

I have been asked to write a mathscinet review for Atsushi Moriwaki's Arakelov Geometry book: http://www.ams.org/bookstore-getitem/item=mmono-244 I could do the review the standard way in a day or ...
12
votes
0answers
336 views

Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin L-...
12
votes
0answers
1k views

Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a ...
11
votes
0answers
208 views

Deligne's theorem on finite flat group schemes and generalizations

Recall Deligne's theorem that for a finite flat commutative group scheme $G$ of order $n$, the multiplication by $n$ map $[n]: G \to G$ is the zero map. I have seen the proof a few times but I can't ...
11
votes
0answers
366 views

Reference request: Newton above Hodge

Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...
11
votes
0answers
415 views

Sheaf-theoretic Grothendieck groups

Let $S$ be a scheme, $M\to S$ a commutative monoid object in algebraic $S$-spaces, ie. an algebraic $S$-space such that, functorially on $S$-schemes $T$, $M(T)$ is a commutative monoid with neutral ...
11
votes
0answers
461 views

Can an abelian variety/Q have no non-trivial points over Q_sol?

Let $A/\mathbb{Q}$ be an abelian variety. Must there be a finite solvable extension $K/\mathbb{Q}$ such that $A(K)$ is nontrivial? This follows from the conjecture that the maximal (pro-)solvable ...
11
votes
0answers
180 views

Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...
11
votes
0answers
284 views

Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...
11
votes
0answers
193 views

On the definition of LGP-monoids in IUT III

I have been trying to understand, without success, the definition of "LGP-monoids" on p. 80 of Mochizuki's IUT III and was wondering if anyone could provide some more explanation than what is given ...
11
votes
0answers
710 views

Points of bounded height in a number field

Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...
10
votes
0answers
213 views

Zeros of $p$-adic power series and rationality

Let $K$ be a non-archimedean field with valuation ring $(V,\mathfrak{m})$, and $K\langle t_1,\ldots, t_n\rangle$ a Tate algebra of convergent power series. Fix $f \in V\langle t_1,\ldots, t_n\rangle$....
10
votes
0answers
207 views

What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...
10
votes
0answers
181 views

Does the Tate pairing agree with the Brauer-Manin pairing

Let $X$ be a proper, smooth, geometrically integral variety over a field $k$. Let $A$ be (the identity component of) its Picard variety and let $B$ be (the identity component) of its Albanese variety. ...
10
votes
0answers
136 views

Finiteness of torsion in $\mathcal{K}_2$-cohomology

Let $F$ be a number field, $C$ be a smooth projective curve over $F$ and $\mathfrak{C}$ be a proper regular model. I am interested in $\mathcal{K}_2$-cohomology, i.e., Zariski cohomology of the sheaf ...
10
votes
0answers
293 views

Extension of Messing-Mazur-Oda to general groups

The following may be well-known (or obviously false), but I can't find a counterexample or a reference. Suppose that $k$ is some perfect field (one can assume algebraically closed, if that makes you ...
10
votes
0answers
349 views

Is the compositum of all quadratic extensions of the rationals an ample field?

Let $K$ be the compositum of all quadratic extensions of $\mathbb{Q}$, that is $$K = \mathbb{Q}(\sqrt{d} \ : \ d \in \mathbb{Q}).$$ Is there a (geometrically irreducible) smooth variety $V/\mathbb{...
10
votes
0answers
638 views

Paths in $\mathrm{Spec} \, \mathbb{Z}$ and Kim's proof of Siegel's theorem for $\mathbb{P}^1 \setminus \{0,1,\infty\}$

This is motivated by a basic number theory question I asked the previous day: Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$? I noted there that the answer to the ...
10
votes
0answers
668 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
0answers
390 views

Is there an algorithm which determines if a curve has good reduction outside a given set of primes

Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$. Is there an algorithm which tells you if $X$ has good reduction outside $S$?...
10
votes
0answers
566 views

An analogue of Deligne-Lusztig theory for positive depth representations?

Deligne-Lusztig theory is an important tool in understanding the depth zero representations of $p$-adic groups. Is there an analogue of Deligne-Lusztig theory that helps in understanding positive ...
9
votes
0answers
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 ...
9
votes
0answers
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 ...
9
votes
0answers
224 views

Examples for a conjecture of Beilinson

Beilinson has conjectured that for a regular, complete, geometrically irreducible curve $C$ of genus $g$ over a number field $k$, $rank(K_2(C))=g[k:\mathbb{Q}]$. As far as I know it is not known in ...
9
votes
0answers
387 views

On Topological Hochschild Homology

Nowdays, I hear talking about Topological Hochschild Homology more and more often, and I was wondering if someone could point out references to explain why it's important and interesting, and what ...
9
votes
0answers
247 views

Do isomorphisms spread out under suitable assumptions?

I'm still trying to wrap my head around the "pathological" algebraic space $\mathbb{A}^1/\mathbb{Z}$; see Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$ and Why is this not an algebraic ...
9
votes
0answers
215 views

Clarification on relationship between Grothendieck-Messing and Honda systems

It's well-known, due to work of Fontaine (see e.g. this), that if $k$ is a perfect field of characteristic $p$, then one can classify $p$-divisible groups over $W(k)$ by Honda systems. Namely, these ...
9
votes
0answers
996 views

Is it worth the efforts to read books/papers written in Weil's algebraic geometry language

There is much important work written in Weil's language of algebraic geometry rather than schemes (besides Weil himself, I can think of Shimura, Neron immediately). My question is: is it worth the ...
9
votes
0answers
374 views

Does bounded-degree base extension yield Zariski-dense Mordell-Weil group?

If $d$ and $n$ are positive integers, does there exist a constant $B=B(d,n)$ with the following property? For any $n$-dimensional abelian variety $A$ over a degree-$d$ number field $K$, there is an ...